Impact Of Interference On Multi-hop Wireless Network Performance

Kamal Jain Jitendra Padhye Venkata N. Padmanabhan Lili Qiu

Microsoft ResearchOne Microsoft Way, Redmond, WA 98052.

{kamalj, padhye, padmanab, liliq}@microsoft.com

Abstract

In this paper, we address the following question: given aspecific placement of wireless nodes in physical spaceand a specific traffic workload, what is the maximumthroughput that can be supported by the resulting net-work? Unlike previous work that has focused on comput-ing asymptotic performance bounds under assumptionsof homogeneity or randomness in the network topologyand/or workload, we work with any given network andworkload specified as inputs.

A key issue impacting performance is wireless inter-ference between neighboring nodes. We model such in-terference using aconflict graph, and present methodsfor computing upper and lower bounds on the optimalthroughput for the given network and workload. To com-pute these bounds, we assume that packet transmissionsat the individual nodes can be finely controlled and care-fully scheduled by an omniscient and omnipotent centralentity, which is unrealistic. Nevertheless, using ns-2 sim-ulations, we show that the routes derived from our anal-ysis often yield noticeably better throughput than the de-fault shortest path routes even in the presence of uncoor-dinated packet transmissions and MAC contention. Thissuggests that there is opportunity for achieving through-put gains by employing an interference-aware routingprotocol.

1 Introduction

Multi-hop wireless networks have been a subject of muchstudy over the past few decades [1]. Much of the originalwork was motivated by military applications such as bat-tlefield communications. More recently, however, some

interesting commercial applications have emerged, suchas “community wireless networks” [2, 28], and sensornetworks [8].

A fundamental issue in multi-hop wireless networks isthat performance degrades sharply as the number of hopstraversed increases. For example, in a network of nodeswith identical and omnidirectional radio ranges, goingfrom a single hop to 2 hops halves the throughput of aflow because wireless interference dictates that only oneof the 2 hops can be active at a time.

The performance challenges of multi-hop networkshave long been recognized and have led to a lot of re-search on the medium access control (MAC), routing,and transport layers of the networking stack. In recentyears, there has also been a focus on the fundamentalquestion of what the optimal throughput of a multi-hopwireless network is. The seminal paper by Gupta andKumar [14] showed that in a network comprising ofnidentical nodes, each of which is communicating withanother node, the throughput per node isΘ( 1√

n log n) as-

suming random node placement and communication pat-tern andΘ( 1√

n) assuming optimal node placement and

communication pattern. Subsequent work [10, 11, 20]has considered alternative models and settings, such asthe presence of relay nodes and mobile nodes, and local-ity in inter-node communication, and their results are lesspessimistic.

This paper also deals with the problem of computingthe optimal throughput of a wireless network. However,a key distinction of our approach is that we work withany given wireless network configuration and workloadspecified as inputs. In other words, the node locations,ranges etc. as well as the traffic matrix indicating whichsource nodes are communicating with which sink nodes

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are specified as the input. We make no assumptions aboutthe homogeneity of nodes with regard to radio range orother characteristics, or regularity in communication pat-tern. This is in contrast to previous work that has focusedon asymptotic bounds under assumptions such as nodehomogeneity and random communication patterns.

We use aconflict graphto model the effects of wirelessinterference. The conflict graph indicates which groupsof links mutually interfere and hence cannot be activesimultaneously. We formulate a multi-commodity flowproblem [4], augmented with constraints derived fromthe conflict graph, to compute the optimal throughput thatthe wireless network can support between the sources andthe sinks. We show that the problem of finding optimalthroughput is NP-hard, and present methods for comput-ing upper and lower bounds on the optimal throughput.

We show how our methodology can accommodate adiversity of wireless network characteristics such as theavailability of multiple non-overlapping channels, multi-ple radios per node, and directional antennas. We alsoshow how multiple MAC protocol models as well assingle-path and multi-path routing constraints can be ac-commodated.

We view the generality of our methodology and theconflict graph framework as a key contribution of ourwork.

To compute bounds on the optimal throughput, we as-sume that packet transmissions at the individual nodescan be finely controlled and carefully scheduled by anomniscient and omnipotent central entity. While this isclearly an unrealistic assumption, it gives us a best casebound against which to compare practical algorithms forrouting, medium access control, and packet scheduling.Moreover, ns-2 simulations show that the routes derivedfrom our analysis often yield noticeably better through-put than the default shortest path routes, even in the pres-ence of real-world effects such as uncoordinated packettransmissions and MAC contention. In some cases, thethroughput gain is over a factor of 2. The reason for thisimprovement is that in optimizing throughput, we tendto find routes that are less prone to wireless interference.For instance, a longer route along the periphery of thenetwork may be picked instead of a shorter but more in-terference prone route through the middle of the network.

We use our technique to evaluate how the per-nodethroughput in a multi-hop wireless network varies as thenumber of nodes grows. Previous work (e.g., [14]) sug-gests that the per-node throughput falls as the number of

nodes grows. But this result is under the assumption thatnodes always have data to send and are ready to trans-mit as fast as their wireless connection will allow. In arealistic setting, however, sources tend to be bursty, sonodes will on average transmit at a slower rate than thespeed of their wireless link. In such a setting, we findthat the addition of new nodes can actually improve theper-node throughput because the richer connectivity pro-vides increased opportunities for routing around interfer-ence “hotspots” in the network. This more than offsetsthe increase in traffic load caused by the new nodes.

The rest of this paper is organized as follows. In Sec-tion 2, we discuss related work. In Section 3, we presentdetails of our conflict graph model and methods for com-puting bounds on the optimal network throughput. InSection 4, we present results obtained from applying ourmodel to different network and workload configurations.In Section 5, we discuss ways to incorporate node mobil-ity into our model. In Section 6 we discuss some limita-tions of our work. Section 7 concludes the paper.

2 Related work

A number of papers have been published on the problemof estimating the throughput of a multi-hop wireless net-work. Here, we consider the work that is most closelyrelated to ours.

In their seminal paper [14], Gupta and Kumar studiedthe throughput of wireless networks under two models ofinterference: aprotocolmodel that assumes interferenceto be an all-or-nothing phenomenon and aphysicalmodelthat considers the impact of interfering transmissions onthe signal-to-noise ratio. They show that in a networkcomprising ofn identical nodes, each of which is com-municating with another node, the throughput per nodeis Θ( 1√

n log n) assuming random node placement and

Θ( 1√n) assuming optimal node placement and communi-

cation pattern. These results are shown under the proto-col model, but the latter result also holds in the case of thephysical model under reasonable assumptions. Accord-ing to the intuitive explanation in [20], while the overallone-hop throughput of the network grows asO(n), theaverage path length grows asO(

√n), so the throughput

per node isO( 1√n).

Li et al. [20] have extended the work of Gupta andKumar [14] by considering the impact of different trafficpatterns on the scalability of per node throughput. They

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point out that a random traffic pattern represents the worstcase from the viewpoint of per-node throughput. Theyalso show that for traffic patterns with power law distancedistributions, the per-node throughput stays roughly con-stant as the network size grows, provided the distancedistribution decays more rapidly than the square of thedistance. Liet al. also consider the interactions of packetforwarding with the 802.11 MAC and show that the useof 802.11 instead of a global scheduling scheme does notaffect the asymptotic bound on per-node throughput de-rived in [14].

In [11], Grossglauser and Tse introduce mobility intothe model presented in [14], and show that the averagelong-term throughput per source-destination pair can bekept constant even as the number of nodes per unit areaincreases, provided that we allow for delays on the orderof the time-scale of mobility. This is achieved by exploit-ing mobility to keep data transfers local, and transmittingonly when the transmitter and receiver are close to eachother, at a distance ofO( 1√

n), thereby reducing total re-

source usage and interference. While this is encouraging,in many practical situations such as community wirelessnetworks, mobility may be too infrequent or even non-existent to be exploitable.

Gastpar and Vetterli [10] extend the work of Guptaand Kumar [14] in a different direction. Instead of thesimple point-to-point coding assumption made in [14],which treats each transmitter-receiver pair as being inde-pendent of other pairs, they consider anetwork codingmodel where nodes could cooperate in arbitrary ways,for instance, to boost the transmit power. Further, theyassume that there is a single source and single destina-tion picked at random, and that the rest of the nodes actas relays. They show that the throughput of the networkunder these conditions isO(log n), compared toO(1) forthe point-to-point coding model of [14]. While the useof network coding in this context is a promising line ofresearch, we note that the point-to-point coding modelcorresponds to current radio technology such as 802.11.

The recent work of De Coutoet al. [5], based ontwo experiments in a 802.11b-based multi-hop wirelesstestbed shows that minimizing the hop count of an end-to-end path is not sufficient for achieving good perfor-mance. The reason they point out is that link quality canvary widely and long hops may be included in “short”paths, resulting in a high packet error rate. In our work,we also reach the same conclusion regarding the lim-

itations of the hop count metric, but for a somewhatdifferent reason — because wireless interference limitsthroughput, a circuitous but less interference-prone route,say along the periphery of a network, may perform betterthan the shortest hop count route.

In [23], Nandagopal et. al. use a construct similar toconflict graphs, called flow contention graph to captureinterference in wireless networks. However, as the nameimplies, the construct is defined on flows rather than onlinks. Moreover, the aim of that paper is to study MACfairness issues, rather than to derive optimal throughputbounds.

Yang and Vaidya [29] also use the notion of a “conflictgraph” in the context of their work on priority schedulingin wireless ad hoc networks. However, like [23], theirconflict graph is also defined on flows rather than links.The graph is used only to interpret experimental resultsshowing that the 802.11 MAC causes flows with a highdegree of conflict to suffer disproportionately comparedto flows with a low degree of conflict. There is no attemptto analyze the conflict graph to derive throughput bounds.

In [19], Kodialam and Nandagopal consider the prob-lem of computing optimal throughput for a given wirelessnetwork with a given traffic pattern. They assume a lim-ited model of interference in which the only constraint isthat node may not transmit and receive simultaneously.With this constraint, they model the problem as a graphcoloring problem. They provide a polynomial time algo-rithm that computes routes and schedules such that theresulting throughput is guaranteed to be at least 67% ofthe optimal throughput. The model we consider in thispaper is much more general and flexible. Our model cantake into account interference from neighboring nodes,impact of directional antennas, availability of multiplenon-interfering channels etc. This generality makes theproblem harder, so our algorithm only provides upper andlower bounds on optimal throughput.

We also note that our approach can compute the op-timal throughput if we assume the limited model of in-terference assumed in [19]. See Appendix B for moredetails.

In summary, there is a large body of work on the multi-hop wireless throughput problem, much of it focused onasymptotic bounds under assumptions such as node ho-mogeneity and random communication patterns. In con-trast, our work focuses on computing throughput boundsfor a given wireless network and traffic workload, using aconflict graph to model the constraints imposed by wire-

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less interference. We do not consider how factors suchas mobility [11] or coding [10]. And like [14], we donotcompute the information theoretic capacity of the net-work.

3 Computing Bounds on OptimalThroughput

We now present our framework for incorporating the con-straints imposed by interference in a multi-hop wirelessnetwork and then present methods for computing boundson the optimal throughput that a given network can sup-port for a given traffic workload. We begin with somebackground and terminology.

3.1 Background and Terminology

Consider a wireless network withN nodes arbitrarily lo-cated on a plane. Letni, 1 ≤ i ≤ N denote the nodes,and dij denote the distance between nodesni and nj.Each node,ni, is equipped with a radio with communica-tion rangeRi and a potentially larger interference rangeR′

i. For ease of explanation, we start by considering thecase of a single wireless channel. (We will generalizethe model in Section 3.5.) We consider two models, theProtocol Modeland thePhysical Model, to define theconditions for a successful wireless transmission. Thesemodels are similar to those introduced in [14].Protocol Model: In the protocol model, if there is a sin-gle wireless channel, a transmission is successful if bothof the following conditions are satisfied:

1. dij ≤ Ri

2. Any nodenk, such thatdkj ≤ R′k, is not transmit-

ting

Note that the second requirement implies that a node maynot send and receive at the same time nor transmit tomore than one other node at the same time. Note alsothat this model differs from the popular 802.11 MACin an important way — it requires only the receiver tobe free of interference, instead of requiring that both thesender and the receiver be free of interference. We dis-cuss how to adapt the model for an 802.11-style MAC inSection 3.5.Physical Model: Suppose nodeni wants to transmit tonodenj. We can calculate the signal strength,SSij, of

ni’s transmission as received atnj. The transmission issuccessful ifSNRij ≥ SNRthresh, whereSNRij de-notes the signal-to-noise ratio at the nodenj for transmis-sions received from nodeni. The total noise,Nj, at nj

consists of the ambient noise,Na, plus the interferencedue to other ongoing transmissions in the network. Noteagain that there is no requirement that the noise level atthe sender also be low.

Our goal is to model wireless interference using a gen-eral framework that would enable us to compute the op-timal throughput the wireless network can support fora given traffic workload. We assume that the work-load consists of greedy sources and destinations, i.e. thesources always have data to send and the destinationnodes are always ready to accept data. The communi-cation between the sources and destinations can be ei-ther direct or be routed via intermediate nodes. We as-sume that packet transmissions at the individual nodescan be finely controlled and scheduled by an omniscientand omnipotent central entity.

We say that a network throughputD is feasible if thereexists a schedule of transmissions such that no two in-terfering links are active simultaneously, and the totalthroughput for the given source-destination pairs isD. Inour problem formulation here, we focus on maximizingthe total throughput between source-destination pairs.

In the rest of this section, we consider the followingthree scenarios in detail: (i) multipath routing under theprotocol interference model, (ii) multipath routing un-der the physical interference model, and (iii) single-pathrouting under both models. We end the section by dis-cussing several other generalizations, and summarizingour framework.

3.2 Multipath Routing Under the Protocol In-terference Model

Given a wireless network withN nodes, we first derive aconnectivity graphC as follows. The vertices ofC cor-respond to the wireless nodes (NC ) and the edges cor-respond to the wireless links (LC ) between the nodes.There is a directed linklij from nodeni to nj if dij ≤ Ri

andi 6= j. We use the terms “node” and “link” in refer-ence to the connectivity graph while reserving the terms“vertex” and “edge” for theconflict graphpresented inSection 3.2.1.

Let us first consider communication between a singlesource,ns, and a single destination,nd. In the absence of

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wireless interference (e.g., on a wired network), findingthe maximum achievable throughput between the sourceand the destination, given the flexibility of using multiplepaths, can be formulated as a linear program correspond-ing to a max-flow problem, as shown in Figure 1. Here,fij denotes the amount of flow on linklij , Capij denotethe capacity of linklij , andLC is a set of all links in theconnectivity graph.

The maximization states that we wish to maximize thesum of flow out of the source. The first constraint rep-resents flow conservation, i.e., at every node, except thesource and the destination, the amount of incoming flowis equal to the amount of outgoing flow. The second con-straint states that the incoming flow to the source node is0. The third constraint states that the outgoing flow fromthe destination node is 0. The fourth constraint indicatesthe amount of flow on a link cannot exceed the capacityof the link. The final constraint restricts the amount offlow on each link to be non-negative.

Note that the above formulation does not take wirelessinterference into account. We turn to this issue next.

3.2.1 Conflict Graph

To incorporate wireless interference into our problemformulation, we define aconflict graph, F , whose ver-tices correspond to the links in the connectivity graph,C. There is an edge between the verticeslij andlpq in Fif the links lij andlpq may not be active simultaneously.Based on the protocol interference model described inSection 3.1, we draw such an edge if any of the follow-ing is true:diq ≤ R′

i or dpj ≤ R′p. This encompasses the

case where a conflict arises because linkslij andlpq havea node in common (i.e.,i == p or i == q or j == por j == q). Note, however, that we do not draw an edgefrom a vertex to itself in the conflict graph.

Before we discuss how to use the conflict graph to addinterference constraints in the linear program in Figure 1,we need to state a hardness result and a few definitions.

3.2.2 Hardness Result

We present a hardness result for computing the optimalthroughput under the protocol interference model. Givena graphH with vertex setVH , anindependent setis a setof vertices such that there is no edge between any two ofthe vertices. Theindependence numberof graphH is the

size of the largest independent set inH. Then, we havethe following hardness result.

Theorem 1 Given a network and a set of source and des-tination nodes, it is NP-hard to find the optimal through-put under the protocol interference model. Moreover, itis NP-hard to approximate the optimal throughput.

Proof : It can be shown that the problem of findingthe independence number of a graph, which is a knownhard problem even to approximate, can be reduced to theoptimal throughput problem. Moreover, this reductionis approximation preserving. Hence the above hardnessresult. We describe the reduction in Appendix A. 2

Since it is NP-hard to approximate the optimalthroughput, we now look at heuristics for obtaining lowerand upper bounds on the optimal throughput. For this, weneed to define some more terms. An independent setI ofa graphH can be characterized using anindependencevector, which is a vector of size|VH |. This vector is de-noted byxI . The jth element of this vector is set to 1if and only if the vertexvj is a member of the indepen-dent setI, otherwise it is zero. We can think ofxI as apoint in a|VH |-dimensional space. The polytope definedby convex combination of independence vectors is calledthe independent set polytopeor thestable set polytope.

3.2.3 Lower Bound

The problem of deriving a lower bound is equivalent tothe problem of finding a network throughputD that hasa feasible schedule to achieve it. We make the follow-ing observation. Links belonging to a given independentset in conflict graphF can be scheduled simultaneously.Suppose there are a total ofK maximal independent setsin graphF . A maximal independent set is one that cannotbe grown further. LetI1, I2, . . . IK denote these maximalindependent sets, andλi, 0 ≤ λi ≤ 1 denote the fractionof time allocated to the independent setIi (i.e., the timeduring which the links inIi can be active). If we addthe schedule restrictions imposed by the independent setsto the original linear program (Figure 1), the resultingthroughput always has a feasible schedule, and thereforeconstitutes a lower bound on the maximum achievablethroughput.

We formalize our above observation as follows. Givena conflict graphF , we define ausage vector, U , of size|VF |, whereUi denotes the fraction of time that the link

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max∑

lsi∈LC

fsi

Subject To:

∑

lij∈LC

fij =∑

lji∈LC

fji ni ∈ NC \ {ns, nd} < 1 >

∑

lis∈LC

fis = 0 < 2 >

∑

ldi∈LC

fdi = 0 < 3 >

fij ≤ Capij ∀i, j | lij ∈ LC < 4 >

fij ≥ 0 ∀i, j | lij ∈ LC < 5 >

Figure 1: LP formulation to optimize the throughput for a single source-destination pair.

i can be active. A usage vector isschedulableif the cor-responding links can be scheduled, conflict free, for thefraction of the time indicated in the usage vector. If wethink of the usage vector as a point in a|VF |-dimensionalspace, we have the following theorem.

Theorem 2 A usage vector is schedulable if and only ifit lies within the independent set polytope of the conflictgraph.

Proof : Let us first show that a schedulable usage vec-tor lies in the independent set polytope of the conflictgraph. In other words, we want to show that the usagevector is a convex combination of independence vectors.

Consider a schedulable usage vector,U . Consider oneunit of time, and assume that we have scheduled the linksover fractions of this unit time, such that the usage vectorhas been satisfied. Since the vector is schedulable, sucha schedule must exist. This schedule will tell us whichlinks are active at any given instance of time. Also, sincethe usage vector is schedulable, at any instance in thisschedule, the links that are active are not in conflict witheach other. That is, the vertices corresponding to theselinks must form an independent set in the conflict graph.Find each such independent setI and denote its indepen-dence vector byxI . DefineλI as the fraction of the unittime independent setI is active. Since the total time isone unit, the sum ofλI ’s over all the independent sets

equals to one. Thus:

U =∑

I is an independent setλIxI .

Now we show that a usage vector that is a convex com-bination of independence vectors is always schedulable.Consider a usage vectorU that is obtained by a convexcombination of independence vectors:

U =∑

I is an independent setλIxI

It follows that U is schedulable since each independentsetI can be scheduled forλI fraction of the time. 2

Theorem 2 implies that the optimal network through-put problem is a linear program, no matter how manysender-receiver pairs we have. In fact, the problem isthat of maximizing a linear objective function over a fea-sible polytope. This feasible polytope can be describedas the intersection of two polytopes — the flow polytopeand the independent set polytope of the conflict graph.Theflow polytopeis the collection of feasible points de-scribed by the flow constraints (Figure 1), ignoring wire-less conflicts. The flow polytope is a simple structure onwhich a linear objective function can easily be optimized.Independent set polytope, on the other hand, is a difficultstructure and no simple characterization of it is knownbecause there may be exponentially many independentsets.

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Theorem 2 implies that any convex combination of in-dependence vectors is schedulable. In general, however,an arbitrary point inside the independent set polytope willbe a convex combination of an exponentially many in-dependence vectors. To get around this computationalproblem, we only want to pick “easy” points in the in-dependent set polytope. An obvious notion of “easy” isthat the point picked should be a convex combination of asmall number of (i.e., polynomially many) independencevectors. We will be using this notion explicitly in thealgorithm as follows. We derive a lower bound on theoptimal throughput by findingK ′ independence vectorsin the conflict graphF , and adding the following con-straints to the LP formulation shown in Figure 1.

• ∑K ′

i=1 λi ≤ 1 (because only one maximal indepen-dent set can be active at a time)

• fij ≤∑

lij∈IiλiCapij (because the fraction of time

for which a link may be active is constrained by thesum of the activity periods of the independent sets itis a member of).

Note the solution produced by solving this linear pro-gram is always feasible (i.e., schedulable). This is dueto the fact that all links belonging to independent setIi

can be simultaneously active forλi fraction of time, andwe have required that the

∑K ′

i=1 λi ≤ 1. Moreover, The-orem 2 assures us that when we include all independentsets, the solution will be exact, i.e., this will be the maxi-mum value ofD that is feasible. To help tighten the lowerbound more quickly, we should consider using maximalindependence sets. While findingall maximal indepen-dent sets is also NP-hard [9], the lower bound obtainedby considering a subset of the maximal independent setshas the nice property that as we add more constraints, thebound becomes tighter, eventually converging to the op-timal (i.e., the maximum feasible) throughput when weadd all the constraints.

3.2.4 Upper Bound

In this section, we derive an upper bound on the networkthroughput. Consider the conflict graph. Aclique in theconflict graph is a set of vertices that mutually conflictwith each other. Theorem 2 implies that the total usageof the links in a clique is at most 1. This gives us a con-straint on the usage vector. We can find many cliques andwrite corresponding constraints to define a polytope. We

1

2

34

5

1

2

34

5

Figure 2: A pentagon and its complementgraph. The former is an odd hole, and the latteris an odd anti-hole.

Figure 3: An example that shows it is not suf-ficient even if we add all clique, hole, anti-holeconstraints.

can then maximize the throughput over the intersectionof this polytope with flow polytope. This will give us anupper bound on the throughput.

Unfortunately, it is computationally expensive to findall the cliques, and even if we could find them all, thereis still no guarantee that our upper bound will be tight.This can be illustrated by the following example. Sup-pose the conflict graph is the pentagon depicted in Fig-ure 2. As we can see, the only cliques in the graph areformed by the adjacent pairs of nodes. Adding the cliqueconstraints alone to the LP would suggest that a sum oflink utilization equal to 2.5 is achievable. But actually atmost 2 links can be active at a time. This suggests thatwe need to add constraints corresponding toodd holesandodd anti-holes. An odd hole is a cycle formed by anodd number of edges, without a chord in between. Forexample, the pentagon in Figure 2 is an odd hole. Thesum of the link utilizations in an odd hole containingkvertices can be no more than⌊k

2⌋. An odd anti-hole is the

complementary graph of an odd hole. Figure 2 shows anexample of an anti-hole with 5 nodes. The sum of linkutilizations in an odd anti-hole can be no more than 2.

Unfortunately, even if we consider the constraints im-posed by the odd holes and odd anti-holes (in additionto those imposed by the cliques), we are not guaranteedto have a feasible solution. For example, consider theconflict graph, as shown in Figure 3. We can assign a uti-

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lization of 0.4 to all the vertices on the pentagon and 0.2to the center of the pentagon, while satisfying all clique,hole, and anti-hole constraints. But there is no feasibleschedule to achieve this, because this solution does notlie in the stable-set polytope. In fact, the upper boundbased only on clique constraints is tight only for a spe-cial class of conflict graphs called perfect graphs.Per-fect graphsare the graphs without any odd holes or oddanti-holes. Thus, in our present formulation, the upperbounds may not always be tight. We discuss this furtherin Appendix C.

3.3 Multipath Routing Under the Physical In-terference Model

As before, we begin by creating a connectivity graphC, whose vertices correspond to the nodes in the net-work. Based on the physical interference model, thereexists a link,lij, from ni to nj if and only if SSij/Na ≥SNRthresh (i.e., the SNR exceeds the threshold at leastin the presence of just the ambient noise).

Using the connectivity graph, we can write an LP for-mulation to optimize network throughput for a wired net-work. As discussed before, the solution to the linear pro-gram, as shown in Figure 1, provides an upper boundon network throughput. However, this bound is not veryuseful since it does not take interference effects into ac-count.

To take interference effects into account, we constructa conflict graphF . Unlike in the protocol model, con-flicts in the physical model are not binary. Rather, the in-terference gradually increases as more neighboring nodestransmit, and becomes intolerable when the noise levelreaches a threshold. This gradual increase in interferencesuggests that we should have a weighted conflict graph,where the weight of a directed edge from verticeslpq toverticeslij (denoted bywpq

ij ) indicates what fraction ofthe maximum permissible noise at nodenj (for link lijto still be operational) is contributed by activity on linklpq (i.e., nodenp’s transmission to nodenq). Specifically,we have

wpqij =

SSpj

SSij

SNRthresh− Na

whereSSpj andSSij denote the signal strength at nodenj of transmissions from nodesp andi, respectively, and

SSij

SNRthresh−Na is the maximum permissible interference

noise at nodenj that would still allow successful recep-tion of nodeni’s transmissions. The edges of the conflict

graph are directed, and in generalwpqij may not be equal

to wijpq.

3.3.1 Lower Bound

In the protocol model, we derive a lower bound on thenetwork throughput by finding independent sets in theconflict graphF , and adding the constraints associatedwith the independent sets to the LP for the wired network.Analogous to independent sets, we introduce the notionof schedulable setsin the physical model. A schedulablesetHx is defined as a set of vertices such that for everyvertex lij ∈ Hx,

∑lpq∈Hx

wpqij ≤ 1. It follows that all

links in a schedulable set can be active simultaneously.Suppose we schedule the links belonging toHx for timeλx, 0 ≤ λx ≤ 1. We now take the original LP for thewired network (in Figure 1), and include the followingconstraints:

• ∑K ′

x=1 λx ≤ 1, whereK ′ is the number of schedula-ble sets found

• fij ≤∑

lij∈HxλxCapij

To tighten the bound, we should consider using max-imal schedulable setsin graphF (i.e., a schedulable setsuch that adding additional vertices to the set will violatethe schedulable property). We have the following theo-rem, which is similar to the Theorem 2 in the protocolmodel.

Theorem 3 A usage vector is schedulable if and only if itlies in the schedulable set polytope of the conflict graph.

Proof : The proof is similar to that of Theorem 2.2

3.3.2 Upper Bound

To derive an upper bound, we consider maximal sets ofvertices inF such that for any pair of verticeslpq andlij,wpq

ij ≥ 1. These correspond to the cliques in the protocolinterference model. Therefore for each such set, we adda constraint that the sum of their utilization has to be nomore than 1.

These constraints may result in a loose bound sincethere may not be very many cliques. To tighten the upperbound, we further augment the linear program with thefollowing additional constraints. After we find a max-imal schedulable set, say verticesv1, v2, ..., vt, adding

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any additional vertex, denoted asva, to the set will makethe set unschedulable. Therefore we have the followingconstraint:U1 + U2 + ...Ut + Ua ≤ t, where as beforeUi denotes the fraction of time for which physical linkli(corresponding to vertexvi in the conflict graph) is ac-tive. By adding as many such constraints as possible, wecan tighten the upper bound. Still, the bound is not guar-anteed to converge to the optimal even if we include allsuch sets.

3.4 Single-path Routing

So far we have considered multipath routing. As manyexisting routing algorithms [17, 27, 26, 25] are confinedto single-path routing, it is useful to derive a through-put bound for single-path routing so that we can comparehow much the current protocols deviate from the theo-retical achievable throughput under the same routing re-striction. The way we enforce the single-path restrictionfor the flow from a source to a destination is by addingthe following additional constraints to the LP problem forthe wired network (shown in Figure 1):

• For each linklij, fij ≤ Capij · zij , wherezi,j ∈{0, 1}

• At each nodeni,∑

zij ≤ 1

Herezij is a 0–1 variable that indicates whether or notlink lij is used for transmissions, andfij is the amountof flow on the link. The basic intuition for these con-straints is that in a single-path routing, at any node in thenetwork, there is at most one out-going edge that has anon-zero flow. Sincezij can have only one of two val-ues, either 0 or 1, the two conditions ensure that at nodeni at most onezij will have a value of 1.

In theory, solving integer linear program is a NP-hard [9], but in practice, software such as lpsolve [3]and CPLEX [6] can solve mixed-integer programs.

3.5 Other Generalization

The basic conflict graph model is quite flexible, and canbe generalized in many ways.

• Multiple source-destination pairs: We can ex-tend our formulations in the previous sections froma single source-destination pair to multiple source-destination pairs using a multi-commodity flow for-mulation [4] augmented with constraints derived

from the conflict graph. We assign a connectionidentifier to each source-destination pair. Instead ofthe flow variablesfij, we introduce the variablefijk

to denote the amount of flow for connectionk onlink lij. Referring to Figure 1, the flow conservationconstraints at each node apply on a per-connectionbasis (constraint<1>); the total incoming flow intoa source node is zero only for the connection(s) orig-inating at that node (constraint<2>); likewise, thetotal outgoing flow from a sink node is zero onlyfor the connection(s) terminating at that node (con-straint<3>); and the capacity constraints apply tothe sum of the flows over all connections traversinga link (constraint<4>).

• Multiple wireless channels:It may be the case thatinstead of just one channel, each node can tune toone ofM channels,M ≥ 1. This can be easily mod-eled by introducingM links between nodesi andj,instead of just 1. In general, links corresponding todifferent channels do not conflict with each other,reflecting the fact that the channels do not mutuallyinterfere. However, the links emanating from thesame node do conflict, reflecting the constraint thatthe single radio at each node can transmit only onone channel at a time.

• Multiple radios per node: Each wireless node maybe equipped with more than one radio. If each nodehasM radios, this can be modeled by introducingM links between each pairs of nodes. If we assumethat each of these radios is tuned to a separate chan-nel, and that a node can communicate on multipleradios simultaneously, then the conflict graph willshow no conflict among theM links between a pairof nodes.

• Directional antennas: We can combine the use ofdirectional antennas with the basic protocol modelof communication. Instead of specifying a rangefor each node, we simply specify a list of nodes (orpoints in space) where transmissions or interferencefrom this node can be perceived. The connectivitygraph and the conflict graph are modified to take thisinto account.

• Multirate radios: Many wireless technologies sup-port multirate radios, which can switch between aset of discrete data rates depending on the quality of

9

the RF channel. For instance, 802.11b supports 4rates: 1, 2, 5.5, and 11 Mbps. We can model this inour framework by creating multiple “virtual” linkscorresponding to a physical link in the connectiv-ity graph, one for each rate. The conflict graph isaugmented to reflect the fact that only one of thevirtual links corresponding to a physical link can beactive at a time. The weights assigned to the edgesof the conflict graph (under the physical interferencemodel) would reflect the specific noise tolerance ofthe virtual link corresponding to each rate.

• Multiple transmit power levels: We have thus farassumed that transmitters used a fixed power level.However, we can extend our framework to incor-porate a discrete set of transmitter power levels. Wecreate multiple “virtual” links corresponding to eachphysical link in the connectivity graph, one for eachpower level. Depending on the environment and theproximity between the transmitter and the receiver,the different power levels may also correspond todifferent modulation schemes and/or different datarates. Using the physical model, we create edgesin the conflict graph whose weights are a functionof the power level of the links in the connectivitygraph. This would, for instance, allow modelingof the case where a transmitter communicating witha nearby receiver switches to a lower power, whileperhaps still maintaining a high data rate, given theproximity of the nodes, to minimize interference onother nodes.

• Other models of interference: In the simple ex-ample, we considered an optimistic model of inter-ference that did not require the sender to be freeof interference. But a more realistic model, whichmore closely reflects the situation in802.11, wouldrequire both the sender and the receiver to be free ofinterference. This reflect the fact that802.11 mayperform virtual carrier sensing using an RTS–CTSexchange, and that for successful communication,the sender must be able to hear the link layer ac-knowledgment transmitted by the receiver. There-fore, we draw an edge in the conflict graph be-tween verticeslij and lpq if dab ≤ R′

a for ab =iq, qi, ip, pi, jp, pj, jq, or qj.

• Non-greedy sources or destinations:We can eas-ily accommodate the case where the rate at which

nodes generate data or are willing to accept data isbounded. We do so by creating avirtual source orsink node and connecting it to the real source or sinkvia avirtual link of speed equal to the source or sinkrate. The virtual link is special in that it is assumednot to interfere with any other link in the network.The virtual link is just a convenient construct to helpus model the bound on the source or sink rate.

• Other objective functions: Our framework is notlimited to maximizing the total network through-put. We can accommodate any objective that canbe expressed as a linear function. For example, wecan assign a linear revenue function to each source-destination pair, and then maximize the revenue in-stead of maximizing the total network throughput.We can also maximize the minimum throughputacross all source-destination pairs, to provide a de-gree of fairness.

Many of these generalizations can be combined with eachother to model complex networking scenarios. We willsee examples of such combinations in Section 4.

3.6 Summary

In this section, we presented the concept of a conflictgraph, and discussed how it could be used to derive upperand lower bounds on the optimal throughput that a wire-less network can support, for a given set of sources anddestinations. We show that the conflict graph model canbe generalized to handle a wide range of scenarios. Wehave shown that the lower bound derived from our frame-work is always schedulable, and will be optimal once allthe independent set constraints are incorporated. If theupper and lower bounds are equal, then these correspondto the optimal solution. We have not dealt with the ques-tion of node mobility so far, but we will present someideas in Section 5.

4 Results

This section presents several results based on our model.The section is organized as follows. In Section 4.1, wepresent illustrative results that demonstrate the flexibilityof our model. In Section 4.2, we use our model to provideinsights into the tradeoff between the richer connectivityprovided by the increase in the size of a wireless mesh

10

network and the increase in cumulative traffic load dueto the new mesh participants. In Section 4.3, we illus-trate how optimal routing can bring benefits even in ab-sence of optimal scheduling (i.e., in the presence of MACcontention and other inefficiencies). In Section 4.4, wediscuss the issue of convergence of the upper and lowerbounds to the optimal throughput. Finally, in Section 4.5,we present a discussion of the computational costs of ourmodel.

4.1 Illustrative Results

In this section, we present several illustrative results todemonstrate the capabilities of our model. We begin bydefining a metric for computational effort. In Section 3,we have described the procedure for finding upper andlower bounds on throughput. Let us consider the proto-col model of interference, and focus on the lower bound.We have shown that as we include more distinct indepen-dent sets, the lower bound becomes progressively tighter.In other words, the moreeffort we spend looking for in-dependent sets in our conflict graph, the better the boundwill be. Since we can not always hope to find optimalsolutions, any upper or lower bounds discovered by ourmodel need to be presented along with the amount of ef-fort required to find those bounds. Thus we require ametric to measure thiseffort. We use the following sim-ple algorithm to find distinct independent sets:

1. Start with an empty independent setIS.

2. Consider a random ordering of vertices in the con-flict graph.

3. Consider the vertices of the graph in that order. Al-ways add the first vertex toIS.

4. Add a new vertex if and only if it does not have anedge to any of the vertices added toIS so far. Oncewe consider all the vertices,ISwill be of size at leastone.

5. We check to see if we have previously discoveredthis independent set, and if not, we add constraintsbased on this independent set to our linear program.Otherwise we discard the set.

We consider this entire sequence as one unit ofeffort.Note that one unit of effort does not always result in ad-dition of a constraint or variable to the linear program.

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Figure 4: 3x3 Grid

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Bidirectional MAC

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Figure 5: Throughput with a bidirectional MAC

Moreover, there is a complex relationship between thenumber of variables and constraints in a linear program,and the amount of time required to solve it. Thus, themetric is only a rough guide for the amount of actual time(or CPU cycles) spent while finding the bound. In Sec-tion 4.5, we will provide further discussion about the re-lationship between the effort metric and actual time spentin computation. The effort metric is defined in a similarmanner by considering cliques in case of searching forthe upper bound, and by considering schedulable sets incase of the physical model.

4.1.1 A Simple Topology

We consider the topology shown in Figure 4. The net-work consists of 9 nodes, placed in a 3x3 grid. We makeno claims that this topology is representative of typicalwireless networks. We have deliberately chosen a small,

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link 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 230 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 01 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 02 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 03 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 04 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 06 1 0 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 17 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 08 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 19 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 0 110 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 111 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 112 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 113 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 114 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 115 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 116 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 117 1 1 1 0 1 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 118 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 119 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 120 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 121 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 122 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 1 1 1 0 123 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0

Table 1: Conflict Graph in matrix form

simple topology, to facilitate detailed discussion of theresults.

We start with several simplifying assumptions. Wewill relax these assumptions as we proceed through thesection. We assume that the range of each node is oneunit, i.e., just enough to reach its lateral neighbors, butnot the diagonal ones. We also assume that the inter-ference range is equal to the communication range. Weassume an 802.11-like protocol model of interference de-scribed in Section 3.5. This model requires both thesender and the receiver to be free of interference forsuccessful communication. We term this abidirectionalMAC. The resulting conflict graph for this scenario isshown in the matrix form in Table 1. A 0 indicates thatthe links are not in conflict with each other, while 1 indi-cates otherwise. For example, when node 0 is transmit-ting to node 3, node 1 can hear these transmissions, andhence can not transmit to node 2. Thus, links 1 (0 → 3)and 3 (1 → 2) are in conflict.

We allow multipath routing. We assume that all wire-less links have an identical capacity (i.e., speed) of 1 unitand that all nodes have infinite buffers. We designatenode 0 to be the sender, and node 8 to be the receiver.The sender always has data to send, and the receiver isalways willing to consume the data.

In this scenario, it is easy to see that the optimalthroughput is 0.5. A convenient way to visualize theoptimal transmission schedule is to imagine that timeis divided into slots of equal size, and in each slot wecan transmit one packet between neighboring nodes, sub-ject to constraints imposed by the conflict graph. Then,the following transmission schedule will achieve optimal

throughput: (i)0 → 1 (ii) 1 → 2 (iii) 0 → 3 and2 → 5(iv) 3 → 6 and 5 → 8 (v) 0 → 1 and 6 → 7 (vi) . . .We can continue in this manner indefinitely. It is easy tosee that in alternate timeslots, node 0 gets to transmit toeither node 1 or 3. Hence the optimal throughput is 0.5.

In Figure 5, we show the upper and lower boundon throughput calculated by our model, as we devoteincreasing amount of effort. As shown, the upper boundquickly converges to the stable value of 0.667, which issomewhat higher than the optimal value. This is a clearindication of the fact that clique constraints alone are notsufficient to guarantee optimality, even in such a smallgraph, as noted in Section 3.2.4. The lower bound, onthe other hand, steadily converges to the optimal valueof 0.5. We have verified that our program has discoveredall independent sets and cliques with 100 units of efforts.

4.1.2 Community Networking Scenario

Our model can also incorporate single path routing, mul-tiple source-destination pairs, multiple channels as wellas multiple radios. We demonstrate this flexibility witha community mesh networking scenario, in which mul-tiple users share an Internet connection, using a multi-hop wireless network. We consider a map of a real sub-urban neighborhood shown in Figure 6. There are 252houses in an area of 1 square kilometer. We select 35 ofthese houses at random, and assume that these housesare equipped with hardware that enables them to par-ticipate in a wireless mesh network. We assume thatcommunication range of the wireless technology is 200

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Houses

Figure 6: Neighborhood Map

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House with wireless connectionWireless links

Internet access pointSenders

Figure 7: Mesh formation in the neighborhood

Scenario Optimal ThroughputI 0.5

II 0.5III 1IV 1

Table 2: Throughput for neighborhood mesh in variousscenarios

meters, while the interference range is 400 meters. InFigure 7, we show the resulting network. We select anode that is roughly at the center of the area and des-ignate it as the Internet access point. We assume thatthere are four senders, located as shown in the Figure. Allthe senders communicate with the Internet access point,and the metric of interest is the cumulative throughput ofthese senders. We assume that all wireless links are ofunit capacity.

We begin with a baseline case, for which we assumea bidirectional MAC and single path routing. Our linearprogram is set to optimize the sum of the throughputs ofthe four flows, with no consideration of fairness. In thiscase, with about 5000 units of effort, upper and lowerbounds converge, and our model indicates that the maxi-mum possible cumulative throughput is 0.5.

We may now ask what we can do to improve the cu-mulative throughput. We consider four possibilities: (I)Employ multi-path routing. (II) Double the range of eachradio. We also double the interference range. (III) Leavethe radio range unchanged, but use two non-overlappingchannels instead of one. A node may communicate ononly one of the two channels at any given time, but mayswitch between channels as often as necessary. (IV) Usetwo radios instead of one at each node. The radios areassumed to be tuned to two fixed, non-overlapping chan-nels, so a node may communicate on the two channelssimultaneously. The throughput bounds in each of thefour scenarios are shown in Table 2. In each case, theupper and the lower bounds converge to the same value,which indicates that the solution is optimal.

The results indicate that neither multipath routing nordoubling the range of the radio increases cumulativethroughput in the scenario we considered. On the otherhand, by using two channels instead of one, the networkmay achieve the maximum possible throughput of 1. Themaximum possible throughput is 1 because the Internetaccess point has only one radio. On the other hand, evenif we use two radios, the throughput remains at one. It is

13

not hard to see why. The situation is equivalent to hav-ing two separate copies of the baseline network, and thenadding up their throughputs. These scenarios illustratethat the model we have developed can be used as a toolfor analysis and capacity planning of wireless multi-hopnetworks.

4.2 Tradeoff Between Connectivity andThroughput

In Section 3, we discussed how our model can accommo-date nodes which do not send data in a greedy fashion,i.e. they have a lower send rate. In [15, 20], the authorshave shown that the per node throughput in the networkdecreases as the number of nodes in the network goes up.These results, however, were derived under the assump-tion that each node sends data as fast as it can. In otherwords, the desired sending rate of the node is assumed tobe 1. However, if each node has a lower desired send-ing rate, the richer connectivity provided by additionalnodes might help increase per node throughput, by al-lowing better routes to be discovered. We now explorethis hypothesis using our model.

We consider a 7x7 grid, whose nodes are 200 metersapart horizontally, and vertically. We assume that thecommunication range is 250 meters, and the interferencerange is 500 meters. We set the link capacity to 1. Weassume a bidirectional MAC, similar to the one used toplot Figure 5. We use single-path routing.

We pick N nodes from the 49 available nodes, at ran-dom, and without replacement. Half of these nodes aredesignated as senders, and the other half are designatedas receivers. The senders and the receivers formN/2flows in the network. Each sender is paired with onlyone receiver. We first calculate the fraction of flows forwhich the source and the destination lie in the same con-nected component of the topology. We call this fractiontheconnectivity ratio. The connectivity ratio for variousvalues ofN is shown in Figure 8. The results show thatafter 24 nodes (i.e. 12 flows) are selected, the connectiv-ity ratio becomes 1.

We then assign a sending rate ofD to each sender.Then, using our model, we calculate the optimal through-put using single-path routing. We divide the cumulativethroughput by the number of flows (i.e.N/2) to ob-tain average per-flow throughput, and normalize it furtherby dividing it by D. The resulting normalized per-flowthroughput for various values ofN andD is plotted in

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Figure 9: Normalized per-flow throughput

Figure 9.

Note that when the sending rate is 0.01, the normalizedper-flow throughput continues to rise even after the con-nectivity has reached 1. This means that the richer con-nectivity provided by additional nodes allows for newerroutes, and allows extra traffic to be sent through thenetwork. However, if each node sends at rate 1, thenode might have little capacity left to forward traffic fromother nodes. Thus, the average per-flow throughput peaksearly (i.e the network is saturated), and then declinesslowly, as new nodes join the network, but fail to trans-mit most of their desired traffic. For sending rate of 0.1,the results are between these two cases. Note that thenon-monotonic nature of the graphs is due to fluctuationin random runs. As part of our future work, we plan to

14

State of art

(ns simulation)

State of art under optimal scheduling

(solve LP)

Shortest-path

Alternative routing scheme

(ns simulation)

Optimal throughput

(solve LP)

Optimal

802.11Optimal Scheduling

Routing

Figure 10: Four scenarios.

verify the generality of this result using a wide variety oftopologies.

We stress that these results have been derived by as-suming optimal routing, as well as optimal scheduling ofpackets. In the next section, we further discuss the impactof these two assumptions.

4.3 Benefits of Optimal Routing in Absence ofOptimal Scheduling

As shown in the previous sections, the optimal through-put is achieved by selecting optimal routes and schedul-ing the links on the routes appropriately. A natural ques-tion to ask is how much performance improvement isdue to the optimal route selection, and how much is dueto the optimal scheduling. Motivated by this question,we empirically examine four scenarios shown in Fig-ure 10. They correspond to (i) optimal routing with op-timal scheduling, (ii) shortest-path routing with optimalscheduling, (iii) “optimal” routing under 802.11 MAC1,(iv) shortest-path routing under 802.11 MAC. We firstbriefly describe the approach we use to derive through-put for each case, and then present the results.

Given a network topology, we apply the algorithm de-scribed in Section 3 to compute the optimal throughputunder single-path routing. This corresponds toscenario(i).

To derive the performance of optimal routing under802.11, we runns-2[24] simulations. To ensure that thepackets follow the optimal routes, we specify the opti-mal routes obtained in Scenario (i) as the static routes

1This means routes derived in (i) used with 802.11 MAC. Itmay also be possible to derive optimal routes for contention-basedscheduling, but that is not our intent here.

in ns-2. The throughput numbers from these simulationscorrespond toscenario (iii).

We then repeat our simulation using AODV [27], astandard shortest path routing protocol. The resultingthroughput corresponds to the performance of thesce-nario (iv). To minimize the impact of AODV routingoverhead, all nodes are static and simulations are run for50 seconds, long enough to make the initial route setupoverhead negligible.

Based on the AODV simulation results, we obtain aset of links that are used in the shortest paths betweensources and destinations. We then modify the LP for-mulation in Section 3 to compute bounds on the optimalthroughput by excluding all but those links that lie onone or more of the shortest paths. We do so by settingthe capacity of the excluded links to zero. We solve theresulting LP, and obtain the throughput forscenario (ii).

Our aim is to compare throughput in scenario (i) tothroughput in scenario (ii). Similarly, we compare sce-narios (iii) and (iv) against each other. Note that wedonot compare the throughput obtained by solving the LPmodel with the throughput obtained from ns-2 simula-tions.

We consider these four scenarios in a 7x7 grid (49nodes). The horizontal and vertical separation betweenadjacent nodes is 200 meters. We assume the communi-cation range to be 250 meters, and the interference rangeto be 500 meters. All other parameters are at their defaultsettings inns-2. For each simulation run, we randomlypick a few pairs of nodes as sources and destinations; thesource sends packets to the corresponding destination ata constant bit rate equal to the wireless link capacity.

Table 3 shows the throughput ratios between optimalrouting and shortest path routing, under optimal schedul-ing. These numbers are derived from our LP formulation.In all cases, optimal routing yields comparable or betterthroughput than the shortest path routing when optimalscheduling is used. The benefit of optimal routing varieswith the number of flows, as well as with the locations ofcommunicating nodes. For instance, when the two flowsare far apart and do not interfere with each other, the op-timal path achieves the same throughput as the shortestpath (e.g., numFlow=2 and run=1, 5); when the two flowsinterfere with each other, the optimal path takes a detour,which results in reduced interference and hence higherthroughput (e.g., the case of numFlow=2 and run= 2, 3,4).

Table 4 shows the throughput ratios between “optimal”

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numFlow run 1 run 2 run 3 run4 run 52 1.00 1.25 1.60 1.38 1.004 1.41 1.00 1.44 1.43 1.148 2.10 1.00 1.05 1.11 1.11

Table 3: Throughput ratios between optimal routing andshortest path routing, both under optimal scheduling in a7x7 grid.

numFlow run 1 run 2 run 3 run4 run 52 1.08 2.43 1.53 1.80 1.194 1.07 1.54 0.79 1.02 1.558 3.55 1.22 0.50 1.14 0.40

Table 4: Throughput ratios between “optimal” path rout-ing and shortest path routing, both under 802.11 MAC ina 7x7 grid.

routing and shortest path routing, under under the 802.11MAC. These numbers are based on ns-2 simulations. Op-timal path outperforms the shortest path even under the802.11 MAC when number of flows in the network issmall. On the other hand, the optimal path routing doesnot always outperform the shortest path routing under802.11 MAC when the number of flows is higher. Thisoccurs because as network load increases, it is harder tofind paths that do not interfere with other flows in theabsence of optimal scheduling.

The above results are encouraging, and suggest thatthere is a potential to improve throughput by makingroute selection interference-aware. In ongoing work, weare continuing to investigate the benefits of interference-aware routing under a wider range of scenarios.

4.4 Convergence of Upper and Lower Bounds

In most of the previous results in this section, the upperand the lower bounds converged, assuring us of the opti-mality of the solution. When they did not converge, e.g.,Figure 5, we were able to assure ourselves of optimalityof the lower bound by manual verification. In general,however, the bounds may not converge, as there is noguarantee that even after adding all the clique constraintsthe upper bound will beschedulable. This leads to thequestion: how do we decide when to stop looking foreven tighter bounds? Given that the conflict graph may

Grid Size Lower Bound Upper Bound Time3x3 0.25 0.25 25x5 0.5 0.5 27x7 0.495 0.5 259x9 0.474 0.5 35

11x11 0.479 0.5 40

Table 5: Lower and upper bounds after 150,000 units ofeffort. Time in minutes.

Effort Lower Bound Upper Bound Time10000 0.443 0.5 250000 0.48 0.5 5

100000 0.49 0.5 13150000 0.495 0.5 25200000 0.5 0.5 41

Table 6: Lower and upper bounds after varying effort fora 7x7 grid. Time in minutes.

have an arbitrarily complex structure, we cannot wait un-til the upper and lower bounds are within a small per-centage of each other since this may never happen. Evenafter all the cliques are found, the upper bound may stillbe well above the optimal feasible solution. Thus, there isno easy way to decide when to stop the calculations. Thedata we present next does indicate, however, that conver-gence is quite good in many scenarios.

4.5 Computational Costs

In Section 4.1, we mentioned that theeffort metric pro-vides only a rough indication of the computational costsof finding the bounds. We now provide more data in thisregard. Note that much of the data provided is for theMATLAB [21] solver to which we had ready access; asnoted below,the CPLEX [6] solver reduced the compu-tation time by a factor of 7, albeit on a somewhat fasterCPU. Unfortunately, we only had limited access to theCPLEX resource and were able to use it for only a few ofour experiments. So it is important to note that there isthe potential for significant improvements over the com-putational costs (for MATLAB) reported here.

In Table 5, we consider the relationship between thesize of the network and the amount of time required tocompute upper and lower bounds. The table shows thebounds computed after 150,000 units of efforts for sev-eral grid sizes, and the time required to compute them. In

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Flows Lower Bound Upper Bound Time2 0.578 0.583 343 0.707 0.75 314 0.758 0.833 295 0.799 0.875 316 0.849 0.925 347 0.861 1.00 36

Table 7: 7x7 grid, multiple flows, 150,000 units of effort.Time in minutes.

Flows Lower Bound Upper Bound Time6 0.849 0.925 57 0.861 1.00 5

Table 8: 7x7 grid, multiple flows, 150,000 units of effort,with CPLEX. Time in minutes.

each case, there is a single flow in the network, with itssource and destination nodes at diagonally opposite cor-ners of the grid. The rest of the parameters are similar tothose used to plot Figure 5. Note that the upper and lowerbounds are not equal in all cases (but they are all close),which indicates that we might not have found the optimalsolution in all cases. The computations were done usingMATLAB 6.1 [21], on a machine with 1.7Ghz Pentiumprocessor, and 1.7GB of RAM.

In Table 6, we consider the relationship between theamount ofeffort, and the closeness of upper and lowerbounds, as well as the time required to compute thosebounds. The results are based on the 7x7 grid, with restof the parameters similar to those used for Table 5. Aswe discussed in Section 4.1, with more effort, we arelikely to add more variables as well as more restrictiveconstraints in the linear program. So the bounds becometighter.

In Table 7, we consider the relationship between thenumber of flows in the network, and the amount of timerequired to compute bounds for a given amount of ef-fort. The results are based on a 7x7 grid, with multipleflows. For each flow, the source is in the bottom row ofthe grid, and it communicates with a destination locatedin the same column, but in the top row. All other param-eters are the same as Table 5.

The software used to solve the linear program is alsoa significant factor in the amount of time required to findthe optimal solution. In Table 8, we show the amountof time taken by CPLEX [6] to solve the 7x7 grid case,

with 6 and 7 flows on a 2.7GHz Pentium machine, with3.7GB of RAM. While we can not compare these en-tries directly with the corresponding entries in Table 8,as the machines used to run MATLAB and CPLEX aredifferent, the speedup is still quite significant: a reduc-tion by a factor of 7, from 34-36 minutes down to 5 min-utes. Moreover, MATLAB cannot solve the Mixed In-teger Programs resulting from the formulation of single-path routing. We could only solve these using CPLEX.Unfortunately, we only had limited access to the CPLEXsoftware, so we are unable to report the full set of num-bers for CPLEX.

Since these numbers are based on a single run, and arebased only on grid graphs, which have a regular connec-tivity pattern, we cannot draw general conclusions fromthem. However, some trends are useful to note. We ob-serve that for grid networks, the amount of time requiredto solve the problem increases with the number of nodes.We also see that for a given effort level, the time requiredto compute the bounds does not depend significantly onthe number of flows in the network. However, the dif-ference between the upper and lower bounds for a givenamount of effort tends to increase with increase in thenumber of flows.

In case of irregular graphs, such as the neighborhoodgraph shown in Figure 7, we have observed that theamount of time required to solve depends significantlyon connectivity and interference patterns.

Finally, we note that we have not included any resultsinvolving physical model of communication in this sec-tion. We have also not included results that demonstratethe use of links of different capacities. While we havesolved such networks (physical models of interference,links of different capacities etc.), we could not do a de-tailed study due to resource constraints. Therefore, wehave chosen to focus on the protocol model of interfer-ence in this section.

5 Dealing with Node Mobility

So far in this paper, we have assumed that all the nodesin the wireless network are static. we now discuss somepossible ways to incorporate node mobility in our model.One way is to repeat the whole process all over: re-construct connectivity graph and conflict graph based onthe new topology, and solve the resulted LP problem.A more efficient way is to take advantage of the efforts

17

spent for the old topology, and incrementally computenew bounds. We propose the following incremental ap-proach for the protocol model.

When a node moves from one place to another, it re-sults in adding and/or removing links in a connectivitygraph. For ease of discussion, suppose a linkAB is re-moved, and a linkAC is added as a result of nodeA’smovement. The following discussions can be easily gen-eralized to the case when more than one link is added orremoved.

The above change in the connectivity graph leads tothe following changes in the conflict graph: (i) addinglink AC in the connectivity graph results in adding itscorresponding vertex to the original conflict graph; in ad-dition, we need to identify the links that conflict with thenew linkAC , and add edges to the original conflict graphto reflect these conflicts; and (ii) removing linkAB in theconnectivity graph results in deleting its correspondingvertex and all its incident edges from the original conflictgraph.

Based on the above changes in the conflict graph, wecan incrementally update the independent set constraintsto derive a new lower bound as follows. To account forthe addition of linkAC in connectivity graph, we add in-dependent set constraints that containAC to the originalLP. To account for the deletion of linkAB , we remove in-dependent set constraints containingAB from the orig-inal LP. If the topology change does not cause changesto the objective function (i.e., links between sources andtheir neighbors remain the same or links between sinksand their neighbors remain the same), then LP solvers,such as lpsolve inc [16], can take advantage of incre-mental changes in the linear constraints, and more effi-ciently derive solution to the new LP than solving it start-ing from scratch.

Similarly, to derive a new upper bound, we incremen-tally update the clique constraints as follows. To accountfor the addition of linkAC , we add clique constraints in-volving AC to the original LP; to account for the deletionof link AB, we remove clique constraints involvingAB

from the original LP. As before, as long as the topologychanges do not affect the objective function, LP solverscan more efficiently derive a solution to the new LP basedon the incremental changes in the linear constraints.

Incrementally computing the lower and upper boundsis hard under the physical model, because in extreme anode’s movement can affect noise level experienced byall nodes, thereby having a global impact on the conflict

graph. We plan to further investigate this problem as partof our future work.

6 Discussion of Limitations

Our results have demonstrated the flexibility of ourmodel and methodology for computing throughputbounds. However, our work does have some limitations,as we discuss below.

Time-varying channels pose a problem for our model.Time-varying channel characteristics could result eitherfrom the interference caused by other nodes or fromphysical effects, e.g. mobility-induced fading. Ourmodel does account for fluctuations in the noise level at anode due to the interfering transmissions of other nodes.However, it does not accommodate fluctuations causedby phenomena such as fading. As with mobility, it maybe feasible to recompute from scratch if the fluctuationshappen slowly.

The computational cost numbers presented in Section4.5 suggest that our methodology is feasible for mod-est sized networks of the order of a few hundred nodes,which may be typical of a neighborhood wireless net-work. However, the methodology in its current form islikely to be too expensive for large-scale networks con-taining thousands or millions of nodes, e.g. sensor net-works. Since energy consumption rather than through-put is often the metric of interest in such large-scale net-works, this limitation may be moot.

7 Conclusion and future work

In this paper we have presented a model and methodol-ogy for computing bounds on the optimal throughput thatcan be supported by a multi-hop wireless network. A keydistinction compared to previous work is that we workwith any given wireless network configuration and work-load specified as inputs. No assumptions are made on thehomogeneity of nodes with regard to radio range or othercharacteristics, or regularity in communication pattern.We use aconflict graphto model wireless interferenceunder various conditions (multiple radios, multiple chan-nels, etc.). We view the generality of our methodologyand the conflict graph framework as a key contributionof our work.

Although the bounds that we compute on the optimalthroughput assume the ability to finely control and care-

18

fully schedule packet transmissions, the optimal routesyielded by our analysis often outperform shortest pathroutes even under “real-world” conditions such as unco-ordinated scheduling and MAC contention. In ns-2 sim-ulations, we have observed a throughput improvement ofover a factor of 2 in some cases. The reason for this sig-nificant improvement is that the optimal routes often tendto be less interference-prone than the default shortest pathroutes.

We have also considered the impact of new nodes onthe per-node throughput in multi-hop wireless networks.Contrary to previous results, we have found that the ad-dition of new nodes can be beneficial for all nodes, un-der the (realistic) assumption that each node is active foronly a small fraction of the time. The richer connectiv-ity enabled by new nodes presents increased opportuni-ties for routing around interference “hotspots” in the net-work. This more than offsets the increase in traffic loadcaused by the new nodes.

In ongoing work, we are continuing to investigate thebenefits of interference-aware routing under a wide rangeof scenarios. Our next step after that would be to designa practical interference-aware routing protocol, whichaddresses interesting challenges such as constructingthe conflict graph and computing optimal routes in adistributed manner.

Acknowledgments

We thank Jayavel Shanmugasundaram and LaszloLovasz for helpful discussions. We also thank Jon How-ell for his helpful comments on earlier drafts of this pa-per.

References

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[4] V. ChavtaL. Linear Programming. W. H. Freeman andCompnay, 1983.

[5] D. S. J. D. Couto, D. Aguayo, B. A. Chambers, andR. Morris. Performance of multihop wireless networks:Shortest path is not enough. In1st Workshop on Hot Top-ics in Networks, Oct. 2002.

[6] Ilog cplex suite, 2003.http://www.ilog.com/products/cplex/.

[7] J. Edmonds. Maximum matching and a polyhedron with0,1-vertices.Journal of Research of the National Bureauof Standards, B69:125–130, 1965.

[8] D. Estrin, R. Govindan, J. Heidemann, and S. Kumar.Next century challenges: Scalable coordination in sensornetworks. InACM MOBICOM, Aug. 1999.

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[10] M. Gastpar and M. Vetterli. On the capacity of wirelessnetworks: the relay case. InIEEE INFOCOM, Jun. 2002.

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[12] M. Grotschel, L. Lovasz, and A. Schrijver. The ellipsoidmethod and its consequences in combinatorial optimiza-tion. Combinatorica, 1:169–197, 1981.

[13] M. Grotschel, L. Lovasz, and A. Schrijver.Geomet-ric Methods in Combinatorial Optimization. AcademicPress, 1984.

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[16] R. Haemmerle. Lp solve - incremental version.http://contraintes.inria.fr/haemmerl/lpsolve inc/.

[17] D. B. Johnson and D. A. Maltz. Dynamic source rout-ing in ad-hoc wireless networks. In T. Imielinski andH. Korth, editors,Mobile Computing. Kluwer AcademicPublishers, 1996.

[18] L. G. Khachiyan. A polynomial algorithm in linear pro-gramming. Soviet Mathematics Doklady, 20:191–194,1979.

[19] M. Kodialam and T. Nandagopal. Charaterizing achiev-able rates in multi-hop wireless newtorks: The joint rout-ing and scheduling problem. InACM MOBICOM, Sep.2003.

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[20] J. Li, C. Blake, D. S. J. D. Couto, H. I. Lee, and R. Mor-ris. Capacity of ad hoc wireless networks. InACM MO-BICOM, Jul. 2001.

[21] Matlab version 6.1. http://www.matlab.com/.

[22] M.Chudnovsky, N. Robertson, P.D.Seymour, andR.Thomas. The Strong Perfect Graph Theorem. Sub-mitted for publication., February 2003.

[23] T. Nandagopal, T. Kim, X. Gao, and V. Bharghavan.Achieving mac layer fairness in wireless packet net-works. InACM MOBICOM, Aug. 2000.

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A Proof of Theorem 1

Suppose we are given a graphG and we want to com-pute the cardinality of its maximum independent set. Wenow construct a wireless network such that the optimalthroughput it can support under the protocol interferencemodel is the same as the cardinality of the maximum in-dependent set ofG. Create two wireless nodes, a sources and a receiverr. For every vertex inG add a wirelesslink of unit capacity betweens and r. For every edgebetween two nodes inG, assume a conflict between thecorresponding wireless links in the network. (Such a net-work may arise, for instance, if nodess andr are eachequipped with multiple radios set either to the same (i.e.,interfering) channel or to separate (i.e., non-interfering)channels. It is not hard to see that the optimal throughputis achieved if and only if a maximum independent set inG is scheduled. Thus finding the optimal throughput ofthe wireless network is equivalent to finding the cardinal-ity of the maximum independent set of graphG, which isknown to be a hard problem.

A

E B

D C

V

W

X

Y

Z

Figure 11: A 6x6 grid connectivity graph. ABCDE andVWXYZ are examples of odd holes in the correspondingconflict graph, assuming an 802.11-style MAC, commu-nication range equal to the lateral spacing between neigh-bors, and interference range equal to twice the commu-nication range. These odd holes also happen to be oddanti-holes.

The above proof may come across as contrived sincethe wireless network we constructed is unlikely to arisein practice. This raises an interesting question of whetherrealistic wireless networks could give rise to complexconflict graphs? Our answer is both yes and no. Ouranswer is “yes” because the maximum independent setproblem is hard due to the existence of odd holes and oddanti-holes in the given graph2. As shown in Figure 11,very realistic and simple grid graphs could have conflictgraphs with many odd holes and odd anti-holes. On theother hand, our answer is “no” because realistic conflictgraphs may have some special property or structure thatcould make the problem of finding the maximum inde-pendent set easy. We have been unable to identify anysuch property, but our failure does not mean that no suchproperty exists (though the complex conflict graphs aris-ing from the simple grid graphs, as in Figure 11, diminishour optimism). In view of this, we believe that the heuris-tic approach presented in Section 3 is reasonable. In thefollowing subsection, we discuss certain special cases inwhich optimal solution may be found in polynomial time.

2If a graph does not have any odd holes or anti-holes then thegraph is termedperfect[22], and for perfect graphs there are polyno-mial time algorithms to solve the maximum independent set problem[12].

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B Polynomial Time Algorithm in Spe-cial Cases

Even in special cases where polynomial time algorithmsmay exist, they may be too expensive to be of practical in-terest. One such special case arises in the context of gridgraphs when the conflict radius is zero. By zero conflictradius we mean that two links conflict if and only if theyshare an endpoint. In this simple and somewhat unrealis-tic setting, the conflict graph is nothing but theline graphof the underlying grid network. The line graph,L(G), ofa graph,G, is a graph on the edges ofG, i.e., the verticesof L(G) correspond to the edges ofG. There is an edgebetween two vertices ofL(G) if the corresponding edgesin G have a vertex in common.

Note that we have assume that our network in thiscase is a grid. A grid is a bipartite graph, and bipartitegraphs are perfect. The line graph of a perfect graphis perfect too. Hence the conflict graph of a grid graphwith a zero conflict radius is a perfect graph. A perfectgraph has the property that its set of clique constraintsdefine its independent set polytope. So if we write a lin-ear program with all the clique constraints together withthe flow constraints then we can find the optimal networkthroughput. The problem, however, is that the numberof cliques could still be exponentially many. (Althoughthis does not happen with grid graphs, it could very wellhappen with other perfect graphs.) A solution is to usethe ellipsoid algorithm [18] to optimize linear functionsover a polytope. This algorithm does not require all theconstraints in an explicit form to optimize a linear func-tion over a polytope, hence we do not have to enumer-ate the exponentially many clique constraints. The el-lipsoid algorithm only needs a subroutine that given apotential solution indicates whether the constraints aresatisfied or not, and if not identifies at least one con-straint which is not satisfied. Such a subroutine is calledseparation oracle. The separation oracle for our prob-lem would be one that finds a violated clique constraintgiven a usage vector. This can be accomplished using theGrotschel semidefinite programming algorithm for find-ing the heaviest clique [13]. However, both the ellipsoidalgorithm and the semidefinite algorithm have a runningtime of O(n3), so in combination their running time isO(n6). Thus this polynomial time algorithm is not verypractical.

As discussed in Section 2, Kodialam and

Nandagopal [19] present an approximation algo-rithm for a similar case. They also assume zero conflictradius but the underlying conflict graph can be arbitrarygraphs instead of a grid graph. We note that our algo-rithm still finds the optimal solution for the problemwithin polynomial time. Since the conflict radius is zero,conflict graph is just a line graph of the connectivitygraph. Independent set polytope of the conflict graphis just the matching polytope of the connectivity graph.Edmonds [7] gave a linear program describing thematching polytope of an arbitrary graph. Hence, for thisproblem, we can describe the independent set polytopeby a linear program. This implies that our algorithmcan compute the optimal solution for the case when theconflict radius is zero.

C Theoretical limits on the upper-bound

One of the questions, which our upper-bounding heuris-tic raises, is how good is the upperbound if we include allthe clique constraints, which can itself take exponentialcomputation time. We have given examples (odd holes)earlier which show that unlike the lowerbound, the up-perbound may not always be tight. The next questionis, is the upperbound even likely to be within any con-stant factor of the optimal. Unfortunately, the answercan be no. Consider the two nodes example discussedin Appendix A. Let the conflict graph beG. We havealready shown that the maximum throughput froms to ris the cardinality of the maximum independent set inG.From probabilistic graph theory we know that there aretriangle-free graphs onn nodes which have independentsize ofO(

√n). Suppose the conflict graphG is one of

those graphs. SinceG is triangle-free, only clique con-straints are edges. So using each wireless link for halfthe time satisfies all the cliques, the upperbound can’t bebetter thatn/2, whereas the optimal isO(

√n). So the

upperbound is not within any constant factor of optimumthroughput.

The next question, then, is whether the quality of theupperbound improves when we add all the odd-hole andthe odd-anti-hole constraints. We do not explore thisquestion, but do believe that the upperbound will not bewithin any constant factor even then. The reason for ourbelief is that the stable set polytope can include compli-cated structures such as odd wheels, that will need to be

21

considered even after all odd-hole and odd anti-hole con-straints have been discovered. Instead, we now present atechnique that may improve the convergence rate of ouralgorithm, but does not guarantee tightness.

The technique is a separation oracle that given a con-flict graphG and a candidate solutionλ finds a violatedodd hole constraint, if any. Such an oracle could be usedto improve the convergence rate of the algorithm pre-sented in Section 3. Note that this separation oracle is ap-plicable to general graphs; for the perfect conflict graphconsidered in Section B above, there are no odd holesanyway.

Consider an odd hole,H, of the given conflict graphG. Any vectorλ inside the independent set polytope ofG must satisfy the following:

∑i∈H λi ≤ (|H| − 1)/2.

A violated odd hole is one for which this constraint is notsatisfied. Before attempting to find a violated odd hole,we may assume that the givenλ satisfies all the edge con-straints, i.e.,λi + λj ≤ 1 for every edge inG, becauseif it does not then we can include the violated edge con-straint to shrink the upperbounding polytope. After mak-ing this assumption we define a weight function on theedges. For every edgeij of the graphG, we define itsweight to be1− λi − λj, which is guaranteed to be non-negative. With this weight function we find the lightest(i.e., least-weight) odd cycle in the graph. The lightestodd cycle can be found using a bipartite graph constructas explained in the next paragraph. LetC be the light-est odd cycle.

∑ij∈C(1 − λi − λj) < 1 is equivalent to

∑i∈C λi > |C|−1

2. So, if the weight of the lightest odd

cycle is less than 1 then the cycle is a violated odd hole.If the weight of the lightest odd cycle is 1 or more thenthere is no violated odd hole.

Now we come to the question of efficiently finding thelightest odd cycle. LetG be the graph in which we needto find the lightest odd cycle. We construct a bipartitegraph,B, as follows. For every vertexv in G we puttwo verticesvl andvr in B (the subscriptsl andr canconceptually be thought of as representing the left andright “halves” of the the bipartite graphB). For everyedgeuv in G we put two edgesulvr andurvl in B. Nowan odd cycle inG becomes an odd length path inB e.g.,uvwu becomesulvrwlur. So for every vertexu in G wefind the shortest path fromul to ur in B. The shortestsuch path inB yields the lightest odd cycle inG.

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