european journal of operational researchprotocol television (iptv) and in mobile digital video...

European Journal of Operational Research xxx (2014) xxx–xxx

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Discrete Optimization

Hop constrained Steiner trees with multiple root nodes

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.11.029

⇑ Corresponding author. Present address: Department of Statistics andOperations Research, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria. Tel.: +43 1 4277 38662; fax: +43 1 4277 38699.

E-mail addresses: [email protected] (L. Gouveia), [email protected](M. Leitner), [email protected] (I. Ljubic).

1 Supported by the National Funding from FCT – Fundação para a Ciência eTecnologia, under the project: PEst-OE/MAT/UI0152.

2 Supported by the Austrian Science Fund (FWF) under Grant I892-N23.3 Supported by the APART Fellowship of the Austrian Academy of Sciences.

Please cite this article in press as: Gouveia, L., et al. Hop constrained Steiner trees with multiple root nodes. European Journal of Operational Researchhttp://dx.doi.org/10.1016/j.ejor.2013.11.029

Luis Gouveia a,1, Markus Leitner b,⇑,2, Ivana Ljubic c,3

a DEIO/CIO Faculdade de Ciênçias, Universidade de Lisboa, Bloco C2, Campo Grande, 1749-016 Lisboa, Portugalb Institute of Computer Graphics and Algorithms, Vienna University of Technology, Favoritenstraße 9-11, 1040 Vienna, Austriac Department of Statistics and Operations Research, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

a r t i c l e i n f o

Article history:Received 9 April 2013Accepted 23 November 2013Available online xxxx

Keywords:Integer programmingOR in telecommunicationsSteiner treeHop-constraints

a b s t r a c t

We consider a network design problem that generalizes the hop and diameter constrained Steiner treeproblem as follows: Given an edge-weighted undirected graph with two disjoint subsets representingroots and terminals, find a minimum-weight subtree that spans all the roots and terminals so that thenumber of hops between each relevant node and an arbitrary root does not exceed a given hop limit H.The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes.This article proposes integer linear programming models utilizing one layered graph for each root node.Different possibilities to relate solutions on each of the layered graphs as well as additional strengtheninginequalities are then discussed. Furthermore, theoretical comparisons between these models and to pre-viously proposed flow- and path-based formulations are given. To solve the problem to optimality, weimplement branch-and-cut algorithms for the layered graph formulations. Our computational studyshows their clear advantages over previously existing approaches.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Quality-of-service aspects are among the major issues whendesigning modern telecommunication networks and in particularbounding the maximum overall delay of each relevant communi-cation path is important. It is widely accepted that in many appli-cations the delay along some connection mainly depends on thenumber of intermediate routers, i.e., hops, and that restrictingthe maximum length of each established path by some predefinedthreshold limits the probability of failures (see, e.g., Dahl, Gouveia,& Requejo, 2006; Salama, Reeves, & Viniotis, 1996). Furthermore,whenever redundancy is not of major importance it is usually de-sired that the final network has tree structure in order to ensureunique communication paths and to reduce the maintenance ef-fort, cf. Salama (1996) and Salama et al. (1996).

The literature contains many works dedicated to two problemsthat fit into this framework, namely the ‘‘centralized’’ hop-constrained minimum spanning/Steiner tree problem (HMSTP/HMStTP), see, e.g., Dahl et al. (2006), Gouveia (1995), Gouveiaand Requejo (2001), Gouveia, Paias, and Sharma (2011), Gouveia,

Simonetti, and Uchoa (2011), and Voß (1999) and the referencestherein, and the ‘‘decentralized’’ diameter-constrained minimumspanning/Steiner tree problem (DMSTP/DMStTP), see, e.g., Achu-than, Caccetta, Caccetta, and Geelen (1994), Gouveia and Magnanti(2003), Gouveia, Magnanti, and Requejo (2004, 2006), Gouveiaet al. (2011), and Gruber (2009) and the references therein.

To define the HMSTP consider an undirected, edge-weightedgraph G ¼ ðV ; EÞ with node set V, edge set E, a hop limit H 2 N,and one dedicated central node r 2 V . The objective is to identifya minimum cost spanning tree such that the path between theroot r and any node v 2 V does consist of at most H edges. Forthe Steiner variant (HMStTP) we are further given a set of termi-nals T � V and the aim is to identify a minimum cost Steiner treeconnecting all terminals such that the path between the root rand any terminal node t 2 T does consist of at most H edges. Todefine the DMSTP consider, as before, an undirected, edge-weighted graph. The objective is to identify a minimum costspanning tree such that the path between any two nodes doesconsist of at most D edges, for some given diameter limit D 2 N.Changes to the Steiner variant (DMStTP) are analogous to thehop-constrained problems.

However, several other tree problems with hop constraints ap-pear to be of practical interest and one objective of this work is topropose a more general framework to contextualize these prob-lems. In practice we may have multiple (e.g., replicated) centralservers in which case each server communicates with a subset ofterminals, and lengths of the corresponding communication pathsare limited. One of the important sparse mode multicast routing

(2014),

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Fig. 1. (a) An illustrative instance with R ¼ f0;1g, T ¼ f2;3;4g, and potentialSteiner nodes S ¼ f5;6;7g. (b) A feasible solution for T 0 ¼ T [ R and H ¼ 3. (c) Afeasible solution for T 0 ¼ T and H ¼ 3.

2 L. Gouveia et al. / European Journal of Operational Research xxx (2014) xxx–xxx

protocols is based on core-based trees (CBTs) (Ballardie, Francis, &Crowcroft, 1993). In this protocol, a set of ‘‘core routers’’ is given,and they all multicast the information to a set of other relevantnodes (these correspond to receivers, that can be other routers oreven end users). In classical multicast routing, each core routerbuilds its own communication tree (also known as the source treearchitecture), connecting the core router with the group of its rele-vant nodes. In the sparse mode multicast routing, however, it is re-quired that the union of subtrees associated to the core routersbuilds a single tree. In this latter concept, also known as theshared-tree architecture, a common tree is built that connects allcore routers and their relevant nodes, cf. Gossain, de MoraisCordeiro, and Carlos (2002), Salama (1996), and Salama et al.(1996). CBTs offer better scalability when compared to the sourcetree architecture and their main applications are for the InternetProtocol Television (IPTV) and in Mobile Digital Video Broadcast-ing-Handheld (DVB-H), see Minoli (2007). To ensure that thecommunication delays are not too high and also to ensure a certainreliability of the network, additional hop constraints may beimposed along the communication paths, e.g., between eachserver–receiver pair, cf. Dahl et al. (2006).

In this paper, we provide a new generic mathematical model forthe application described above. The problem is called the HopConstrained Minimum Steiner Tree Problem with Multiple Root nodes(HSTPMR) problem. We are given an undirected graph G ¼ ðV ; EÞ,with node set V, edge set E, edge costs ce P 0, for all e 2 E, and ahop limit H 2 N. The node set V contains two disjoint subsets: rootnodes R; jRjP 1, and terminal nodes T # V n R. Furthermore, we aregiven a set T 0 # T [ R of relevant nodes for which hop limits to allroot nodes need to be considered.

A solution to the HSTPMR is a Steiner tree G0 ¼ ðV 0; E0Þ spanningall root and terminal nodes, i.e., R [ T # V 0, such that the hop con-straints are met for all relevant nodes v 2 T 0. More precisely, foreach relevant node t 2 T 0 and each root r 2 R, the unique path be-tween t and r can contain at most H edges. The objective is to finda feasible subtree yielding minimum total edge costs. If T [ R ¼ V ,the solution will be a spanning tree of G.

In this study we consider two particular cases of this newframework which as far as we know have not been studied before(with exception to the introductory work in Gouveia, Leitner, &Ljubic (2012a)): (a) T 0 ¼ T [ R and (b) T 0 ¼ T. In the first case, delaybounds between roots have to be taken into consideration (e.g.,when roots model replica servers) and in the second case delaysbetween roots are not critical (e.g., when services by different pro-viders are offered to terminals). An illustrative instance of theHSTPMR with two roots and three terminals is given in Fig. 1a,while Fig. 1b and c depict solutions to this instance for T 0 ¼ T [ Rand T 0 ¼ T , respectively, assuming that H ¼ 3. Notice that onecould generalize this problem even further by introducing subsetsof roots and hop limits that would depend on each node from T 0.

However, the two cases already present different characteristicsthat strongly affect the corresponding models. For the caseT 0 ¼ T [ R, it is easy to see that the hop-constrained arborescencesassociated to each root span the same set of nodes and the same setof undirected edges. This property is useful to strengthen the mod-els that will be proposed in the next subsection. Unfortunately, thisproperty may not be satisfied in the case T 0 ¼ T since the maximumdistance between any two roots may exceed H. In fact as can be de-duced from Fig. 1c, the subtree obtained from undirecting the arcsof the hop-constrained arborescence associated to root 0 does notcoincide with the subtree obtained from undirecting the arcs of thehop-constrained arborescence associated to root 1. Thus, many ofthe model enhancements valid for the case T 0 ¼ T [ R that we willdiscuss below, will not be valid for T 0 ¼ T . The following results,however, provide an upper bound on the maximum distancebetween any two roots.

Please cite this article in press as: Gouveia, L., et al. Hop constrained Steiner treehttp://dx.doi.org/10.1016/j.ejor.2013.11.029

Lemma 1. Let G0 ¼ ðV 0; E0Þ be a feasible solution to an instance of theHSTPMR with T 0 ¼ T and let dðu;vÞ denote the distance between twonodes u;v 2 V 0 in G0. Then, the maximum distance between any pair ofroot nodes in G0 does not exceed 2H � ‘ where ‘ is the maximumdistance between any two terminal nodes in G0, i.e.,‘ ¼maxu;v2T dðu;vÞ.

Proof. If there is a single terminal, two roots can be each at dis-tance H from it, which gives the maximum distance of 2H. Assumethat jTjP 2, let t1 and t2 be two terminals at maximum distanceand let P ¼ ðt1 ¼ v0;v1; . . . ;v ‘ ¼ t2Þ (v i 2 V 0 for 0 6 i 6 ‘, andfv i;v iþ1g 2 E0 for 0 6 i 6 ‘� 1) denote the path between t1 and t2

in G0. Furthermore, let r 2 R be an arbitrary root andv j 2 P;0 6 j 6 ‘, be the node from P such that the path between rand v j is edge disjoint to P. Since, the maximum distance betweena terminal and a root node may not exceed H, we have

dðr;v jÞ 6H � ‘þ j if j 6 ‘=2H � j if j P ‘=2

�

Now let s 2 R be another root and vk 2 P;0 6 k 6 ‘ again be thenode from P such that the path between s and vk is edge disjointto P. Without loss of generality we assume that j 6 k. Then, by casedistinction it is easy to see that

dðr; sÞ ¼ dðr; v jÞ þ dðv j; vkÞ þ dðs;vkÞ 6 2H � ‘

holds and that this bound can be tight. h

The next corollary immediately follows from Lemma 1.

Corollary 1. Let diamðTÞ be the minimum diameter of a subtree of Gspanning all nodes from T. Then, for any feasible solution G0 ¼ ðV 0; E0Þto an instance of the HSTPMR on G with T 0 ¼ T the maximum distancebetween any pair of root nodes in G0 does not exceed H0, whereH0 ¼ 2H � diamðTÞ.

Notice that H0 can be calculated in polynomial time: It sufficesto run breadth-first-search starting from each t 2 T until allremaining terminals are reached. The subtree with the smallestdiameter obtained gives us the value of diamðTÞ. As we will showin Section 3.4, this corollary allows us to provide modified models,where many of the enhancements directly valid for the caseT 0 ¼ T [ R apply. The drawback is that these modified models usemany more variables and constraints than the original model with-out the enhancements.

Our Contribution. In this paper, besides introducing the generaland new problem we present three kinds of results: (a) Complex-ity: We analyze special cases in which the HSTPMR can be reducedto previously studied network design problems, identify specialpolynomial cases, show that the problem is NP-hard in general,and that one cannot guarantee to find an approximation ratio bet-ter than Hðlog jV jÞ unless P = NP. (b) Mixed integer programming(MIP) models: We discuss layered graph reformulations, presentstrengthening valid inequalities and show that the obtained mod-els theoretically dominate flow- and path-based models studied inGouveia et al. (2012a). (c) Computational results: Branch-and-cutalgorithms are developed for layered graph models and computa-

s with multiple root nodes. European Journal of Operational Research (2014),


L. Gouveia et al. / European Journal of Operational Research xxx (2014) xxx–xxx 3

tionally compared to each other and to the best performingapproach from Gouveia et al. (2012a). Computations are carriedout on a set of benchmark instances known from the HMSTP –the results show that the branch-and-cut approaches appear tobe reasonable alternatives to solve these more general cases.

Outline of the Paper. In the remainder of this section we studythe computational complexity of the HSTPMR. In Section 2 wediscuss a generic integer linear programming (ILP) formulationof the problem and review a path-based formulation from ourprevious work (Gouveia et al., 2012a) which outperformed theother flow- and path-based models from Gouveia et al. (2012a)both theoretically, i.e., with respect to the quality of its linear pro-gramming (LP) bounds, and computationally. Afterwards, twopossibilities for reformulating the HSTPMR over layered graphstogether with further valid inequalities for strengthening the LPrelaxations of the resulting models are discussed in Section 3. InSection 4 we compare our models with respect to their LP relax-ation values and also show which variants dominate the previ-ously proposed models. Details of the developed branch-and-cutapproaches are given in Section 5, where we also discuss the re-sults of our computational study. Finally, some conclusions aredrawn in Section 6.

1.1. Computational complexity

Next, we analyze the computational complexity of the HSTPMRand its relationship with other problems. Obviously, for singletonsets R and T the problem becomes the Hop Constrained ShortestPath Problem (HSPP) which can be solved in polynomial time forany H since we are given nonnegative edge costs. If either jRj ¼ 1or jT 0j ¼ 1 (but not both), the problem is either the HMStTP ifV – T [ R or the HMSTP if V ¼ T [ R. These problems are knownto be NP-hard if 2 6 H < jV j � 1, cf. (Gouveia, 1995). If T 0 ¼ R orjTj ¼ 1 and T 0 ¼ T [ R, we have the DMStTP or the DMSTP withthe diameter equal to H which are known to be NP hard if4 6 H < jV j � 1, cf. Garey and Johnson (1979).

In the remainder of this paper, we will consider the most gen-eral case, assuming that H P 3; jRjP 2 and jTjP 2, which isshown to be NP-hard by the following Lemma (see Appendix Afor the proof).

Lemma 2. Assuming that jRjP 2 and jTjP 2, the HSTPMR can besolved in polynomial time for H ¼ 2. For H P 3, the problem isNP-hard, and it cannot be guaranteed to find an approximation ratiobetter than Hðlog nÞ unless P = NP.

An overview on all complexity results regarding the HSTPMRand its relationships with related problems is provided in Table 1where ‘‘2 P’’ is used to denote cases when the problem is solvablein polynomial time and ‘‘–’’ denotes that a particular case is infea-sible or that no previously considered problem corresponds to thatcase.

Notation. Let S ¼ V n ðT [ RÞ denote the set of remaining nodesthat we will refer to as potential Steiner nodes. To model a feasiblesolution G0 ¼ ðV 0; E0Þ on G, we will use binary edge variables, xij, thatare set to one if fi; jg 2 E0, and to zero, otherwise, for all fi; jg 2 E. Inaddition, we will use binary node variables associated to potentialSteiner nodes: yi is set to one if i 2 V 0 \ S, and to zero, otherwise, forall nodes v 2 S. Furthermore, let A ¼ fði; jÞ; ðj; iÞjfi; jg 2 Eg denotethe set of bi-directed arcs in G. For a subset W � V , we usedðWÞ ¼ ffi; jg 2 Eji R W; j 2Wg; d�ðWÞ ¼ fði; jÞ 2 Aji R W; j 2Wg,and dþðWÞ ¼ fði; jÞ 2 Aji 2W; j R Wg to denote the undirectedand directed, ingoing and outgoing cutset, respectively. For a setof arcs A0 and some vector of variables z, we also use notationz½A0� ¼

Pði;jÞ2A0zij. Finally, for a binary vector x 2 f0;1gjEj let EðxÞ

denote the subset of edges for which xe ¼ 1.


2. Generic formulation

Next we present a generic model for the HSTPMR which will bespecialized later on by means of paths and layered graphs.

Let F ¼ fx 2 f0;1gjEjj8s 2 R; 8t 2 T 0; 9 s� t path P in EðxÞ s :t:jPj 6 Hg be the set of incidence vectors that contain at least onefeasible path between each s 2 R and each t 2 T 0 n fsg, i.e., a pathof length at most H. A generic MIP model for the HSTPMR is givenby (1)–(7):

minXfi;jg2E

cijxij ð1Þ

s:t: x 2 F ð2Þxij 6 yi i 2 S; fi; jg 2 E ð3ÞXfi;jg2E

xij ¼ jRj þ jTj þXi2S

yi � 1 ð4ÞXfi;jg2E

xij P 2yi i 2 S ð5Þ

xij 2 f0;1g fi; jg 2 E ð6Þyi 2 f0;1g i 2 S ð7Þ

Constraints (2) ensure that a solution must contain a feasiblepath for each commodity pair ðs; tÞ. These constraints can be mod-eled in several ways by using multi-commodity flows, path vari-ables and constraints, or jump inequalities. To discuss theconstraints (3)–(5) which, together with (2) ensure that the solu-tion is a tree, we first observe that due to constraints (2) the solu-tion subgraph induced by the hop-constrained paths for allcommodities will be connected. Hence, to obtain a valid model,we further add constraints (3) and (4), inequalities (3) to guaranteethat a node variable is set to one whenever an incident edge is cho-sen and Eq. (4) to ensure that the number of edges in the solution isone less than the number of nodes. Finally, constraints (5) guaran-tee that the degree of each Steiner node in a feasible solution is atleast two, i.e., Steiner nodes cannot be leaves of a solution. Due tothe hop constraints these constraints also guarantee that thesolution is not disconnected as illustrated by Fig. 2. Thus (1)–(7)is a feasible model for the HSTPMR and Fig. 2 illustrates that con-straints (5) are not redundant in this formulation since omittingthem we may obtain isolated components.

2.1. Disaggregated path formulation

In this section, we briefly recall model UPathDI from Gouveia et al.(2012a) which turned out to be the best model, both from a theoreticalas well as from a computational perspective among all models pre-sented in Gouveia et al. (2012a). Next to already introduced edgeand node decision variables, UPathDI used disaggregated arc variablesas

ij;8s 2 R;8ði; jÞ 2 A, to indicate whether or not arc ði; jÞ is used wheninterpreting the solution as an outgoing arborescence rooted at s.Furthermore, the set of all hop constrained paths Wst # 2A; jpj6 H;8p 2 Wst , from each root s 2 R to each relevant terminalt 2 T 0 n fsg is considered and an exponential number of path variables0 6 kst

p 6 1 one for each commodity pair ðs; tÞ; s 2 R; t 2 T 0 n fsg, andeach feasible path p 2 Wst is introduced. Then, a valid path model isobtained by replacing (2) by (8)–(12) in model (1)–(7).

asij þ as

ji ¼ xij s 2 R; fi; jg 2 E ð8ÞXp2Wst

kstp ¼ 1 s 2 R; t 2 T 0 n fsg ð9Þ

Xp2Wst :ði;jÞ2p

kstp 6 as

ij s 2 R; t 2 T 0 n fsg; ði; jÞ 2 A ð10Þ

kstp P 0 s 2 R; t 2 T 0 n fsg; p 2 Wst ð11Þ

asij 2 f0;1g s 2 R; ði; jÞ 2 A ð12Þ



Fig. 2. A solution feasible for (1)–(7) without constraints (5) if T 0 ¼ T [ R and H ¼ 3 that is infeasible for the HSTPMR.

Fig. 3. Layered graphs corresponding to the instance given in Fig. 1(a) for H ¼ 3 andT 0 ¼ T [ R. Edges that map back to the solution in Fig. 1(b) are drawn in bold.


Finally, for UPathDI we add the strengthening constraints (13) and(14) ensuring that the indegree of each node i R R is identical forall arborescences and that Steiner nodes cannot be leaves in them.In turn, we remove (4) and (5) since these constraints were shownto be redundant in the resulting model (Gouveia et al., 2012a).

as½d�ðiÞ� ¼yi; i 2 S

0; i ¼ s

1; else

8><>: s 2 R; i 2 V ð13Þ

as½dþðiÞ�P yi s 2 R; i 2 S ð14Þ

Fig. 4. (a) An illustrative instance with R ¼ f0;1g and terminals T ¼ f2;3g that isinfeasible for H ¼ 3 and T 0 ¼ T [ R. (b) Solution feasible for LG without (4) and (5).(c) Solution feasible for LG without (5). (d) and (e) Arborescences in G0

L and G1L

corresponding to (b) and (c).

3. Layered graph formulations

Reformulating hop-constrained network design problems usinglayered graphs recently became a popular technique for obtainingtheoretically strong ILP models yielding tight LP bounds. Branch-and-cut algorithms used to solve these models are frequentlyamong the leading approaches for the underlying problems, cf.Gouveia et al. (2011) and Ljubic and Gollowitzer (2013). In thissection we show two different layered graph approaches that canbe used to model the HSTPMR.

3.1. Layered graphs with H layers

We now introduce one layered graph GsL ¼ Vs

L;AsL

� �for each root

node s 2 R. For every s 2 R;VsL is defined by its root node s0, to-

gether with nodes ih;1 6 h 6 H � 1, for all original nodesi 2 V n fsg and nodes tH for all other relevant terminalst 2 T 0 n fsg. For each pair of nodes ih; jhþ1 2 Vs

L we add an arcðih; jhþ1Þ to As

L if ði; jÞ 2 A. Formally, for eachs 2 R;Vs

L ¼ fs0g [ fih : i 2 V n fsg;1 6 h 6 H � 1g [ ftH : t 2 T 0 n fsggand As

L ¼ ðih; jhþ1Þ : ih 2 VsL; jhþ1 2 Vs

L; ði; jÞ 2 A� �

; see Fig. 3 for anexample.

In addition to the previously introduced node and edge designvariables, we use two new sets of binary variables to model theproblem in the layered graph framework. Variables Xsh

ij , are associ-ated to arcs ðih; jhþ1Þ 2 As

L and are set to one if the corresponding arcis part of the rooted Steiner arborescence in Gs

L, for each s 2 R.Variables Ysh

i are associated to nodes ih 2 VsL; i 2 V ;0 6 h 6 H, and

are set to one if the corresponding node is part of the rooted Steinerarborescence in Gs

L, for each s 2 R. The resulting MIP model towhich we will refer to as LG is given by (15)–(21) together with(3)–(7).


minXfi;jg2E

cijxij ð15Þ

s:t: Xs½d�ðihÞ� ¼ Yshi s 2 R; ih 2 Vs

L; i – s ð16Þ

XH

h¼1

Yshi

¼ 1 i 2 T 0 n fsg6 1 i 2 R n ðT 0 [ fsgÞ6 yi i 2 S

8><>: s 2 R; i 2 V ð17Þ

Xðih�1 ;jhÞ2As

L ; i–k

Xs;h�1ij P Xsh

jk s 2 R; ðjh; khþ1Þ 2 AsL; j – s ð18Þ

XH�1

h¼0

Xshij þ Xsh

ji

� �6 xij s 2 R; fi; jg 2 E ð19Þ

Xshij 2 f0;1g s 2 R; ðih; jhþ1Þ 2 As

L ð20Þ

Yshi 2 f0;1g s 2 R; ih 2 Vs

L ð21Þð3Þ—ð7Þ

Indegree constraints (16) link arc to node variables on each lay-ered graph, while constraints (17) ensure that each original node isused at most once on each layered graph and link node variables onthe layered graph to original node variables for potential Steinernodes. Since layered graphs are acyclic, inequalities (18) ensureconnectivity on each layered graph, i.e., they guarantee that anarc ðjh; khþ1Þ emanating from node jh may only be used if at leastone ingoing arc ðih�1; jhÞ with i – k is selected. Constraints (19) linkarc variables on each layered graph to undirected edge variables onthe original graph. Fig. 4 shows that in the context of this model,that is after adding all the information provided by the layeredgraph variables, constraints (4) and (5) are still necessary to guar-antee that the final solution will be a tree.

3.2. Model enhancements (general)

Acyclicity of each layered graph, allows to eliminate subtours inmodel LG using a polynomial number of constraints (18). Hence,LG is a compact model which can be solved by LP-based branch-and-bound. It is well known, however, that one can strengthenlayered graph based models by adding directed cutset constraints(22) where Vs

i ¼ ih 2 VsLj1 6 h 6 H

� �.

Xs½d�ðWÞ�P 1; s 2 R; t 2 T 0 n fsg; W # VsL n fs0g; Vs

t # W ð22Þ

The resulting model will be denoted by LGC. It contains anexponential number of constraints and can be solved by branch-and-cut (B&C).




3.3. Model enhancements for T 0 ¼ T [ R

As pointed out in the introduction, the arborescences for eachroot share the same set of edges. Thus, we can replace inequalitiesby equations in (17) and (19), i.e., consider Eqs. (23) and (24)instead.

XH

h¼1

Yshi ¼

yi i 2 S

1 i 2 ðR [ TÞ n fsg0 i ¼ s

8><>: s 2 R ð23Þ

XH�1

h¼0

Xshij þ Xsh

ji

� �¼ xij s 2 R; fi; jg 2 E ð24Þ

We use LGCI to refer to model LGC where inequalities (17) and(19) are replaced by Eqs. (23) and (24). To make sure that the inde-gree of each Steiner node does not exceed its outdegree, we canfurther add (25) to obtain model LGCIO.

Xs½dþðihÞ�P Yshi s 2 R; ih 2 Vs

L; i 2 S ð25Þ

Another set of valid inequalities is derived from the fact that thedistance between two roots must not depend on the arborescenceconsidered, i.e., on the chosen root. Root-depth constraints (26)which are further added to obtain model LGCIOR simply state thatif root q 2 R is on level h w.r.t. Gs

L, then root s 2 R must be on thesame level w.r.t. Gq

L.

Yshq ¼ Yqh

s s 2 R; q 2 R n fsg;1 6 h 6 H ð26Þ

3.4. The case T 0 ¼ T: layered graphs with H0 ¼ 2H � diamðTÞ layers

As pointed out above, the strengthening inequalities (23) and(24) are not valid in this case since we cannot simply ensure thatthe arborescences for each root share the same set of edges.Corollary 1 permits us to introduce a different layered graph modelbGs

L containing H0 layers for the case T 0 ¼ T in which all arborescencesuse the same set of original nodes and edges. In each such graph bGs

L

the maximum layer HðiÞ of some original node i 2 V is defined as

HðiÞ ¼

0 if i ¼ s

H0 if i 2 R n fsgH0 � 1 if i 2 S

H if i 2 T

8>>><>>>:

Formally, for each s 2 R; bGsL ¼ bV s

L;bAs

L

� �is defined by bV s

L ¼ fs0g[fih : i 2 V n fsg;1 6 h 6 HðiÞg and bAs

L ¼ ðih; jhþ1Þ : ih 2 bV sL; jhþ1 2

nbV s

L; ði; jÞ 2 Ag.Based on this observations, model (27)–(33) to which we will

refer to as LGE uses the same set of variables as the previous model.Note, however, that for this case we consider H0 layers which isalmost as twice as the number of layers in the original graph. Onthe other hand, the model defined in this extended layered graph

Table 1Complexity of the HSTPMR.


permits us to use the strengthening inequalities that have beenused for the T 0 ¼ T [ R case.

minXfi;jg2E

cijxij ð27Þ

s:t: Xs½d�ðihÞ� ¼ Yshi s 2 R; ih 2 bV s

L; i – s ð28Þ

XHðiÞh¼1

Yshi ¼

1 i 2 ðT [ RÞ n fsgyi i 2 S

�s 2 R; i 2 V ð29Þ

Xðih�1 ;jhÞ2bAs

L ; i–k

Xs;h�1ij P Xsh

jk s 2 R; ðjh; khþ1Þ 2 bAsL; j – s ð30Þ

XHðiÞh¼0

Xshij þ

XHðjÞh¼0

Xshji ¼ xij s 2 R; fi; jg 2 E ð31Þ

Xshij P 0 s 2 R; ðih; jhþ1Þ 2 bAs

L ð32Þ

Yshi P 0 s 2 R; ih 2 bV s

L ð33Þð5Þ—ð7Þ

Note that, we do not include constraints (3) and (4) in modelLGE since they are redundant as we will prove in Section 4. As pre-viously discussed, by considering directed cutset constraints (34)we can obtain a stronger model LGC

E which contains an exponentialnumber of constraints. To avoid that Steiner nodes may be leavesin any of the arborescences, we further add inequalities (35) yield-ing model LGCO

E . Finally, by the same arguments as before root-depth constraints (36) are valid and we will use LGCOR

E to refer tothe resulting model.

Xs½d�ðWÞ�P 1 s 2 R; t 2 ðT [ RÞ n fsg; W # bV sL n fs0g;

fth : 1 6 h 6 HðtÞg# Wð34Þ

Xs½dþðihÞ�P Yshi s 2 R; ih 2 bV s

L; i 2 S ð35Þ

Yshq ¼ Yqh

s s 2 R; q 2 R n fsg;1 6 h 6 H0 ð36Þ

4. Polyhedral comparison

In this section we compare the different formulations withrespect to the value of their LP relaxation. In Section 4.1 weaddress the case T 0 ¼ T [ R and in Section 4.2 we addressthat case T 0 ¼ T . We also show that some set of constraintsbecomes redundant after the addition of some sets of validinequalities.

By PM we will denote the convex hull of all feasible LP solutionsof a MIP formulation M and by proja1 ;...;an ðPMÞ, the orthogonal pro-jection of the convex hull of LP solutions of M onto the spacedefined by variables a1, . . ., an. Furthermore, by vLPð:Þ we denotethe value of the LP relaxation of some model. When comparingtwo formulations F1 and F2, we say F1 is stronger than F2 ifvLPðF2Þ 6 vLPðF1Þ and strictly stronger if there additionally existinstances for which strict inequality holds. Furthermore, if fortwo formulations, none of them is stronger than the other, wesay that they are incomparable. In many cases strict dominancefollows due to the computational results that will be discussed inSection 5.1. Usually, however, we will additionally providesolutions that are feasible for one model and not feasible for theother.

For better readability, Table 2 provides a summary of all modelvariants and their definitions.



Table 2Overview on the considered models.

Model T 0 Definition

UPathDI 2 fT; T [ Rg (1), (3), (6) and (7), (8)–(14)

LG 2 fT; T [ Rg (15)–(21), (3)–(7)

LGC 2 fT; T [ Rg LG, (22)

LGCI T [ R LGC, (23) and (24)

LGCIO T [ R LGCI, (25)

LGCIOR T [ R LGCIO, (26)LGE T (27)–(33), (5)–(7)

LGCE

T LGE, (34)

LGCOE

T LGCE , (35)

LGCORE

T LGCOE , (36)


4.1. Polyhedral comparison for T 0 ¼ T [ R

The following theorem which is proved by a series of subse-quent lemmas summarizes the obtained relations between theconsidered models when T 0 ¼ T [ R.

Theorem 4.1. For T 0 ¼ T [ R, the following relations hold:

Thereby, an arrow indicates that the formulation at the target isstrictly stronger than the one at the source while a dashed edgeindicates that the corresponding formulations are incomparable.

Lemma 3. Formulation LGC is strictly stronger than formulation LG.We skip the proof of this lemma, since it is well known that the

result holds for the case of a single root and the result easilyextends to multiple roots.

Lemma 4. Formulation LGCI is strictly stronger than formulation LGC.Furthermore, constraints (3) and (4) are redundant in LGCI.

Proof. Since LGCI contains all constraints of model LGC it is suffi-cient to consider the example given in Fig. 5 which shows that con-straints (23) and (19) improve the LP bound. Computational resultsgiven in Section 5.1 further show this relation.

To prove the second result, consider an arbitrary root s 2 R andedge fi; jg 2 E incident to some potential Steiner node i 2 S. To seethat constraints (3) are redundant we use Eq. (24), (18), (16) and(23) together with the fact that the minimum and maximum layerof nodes ih corresponding to potential Steiner nodes i is 1 andH � 1, respectively:

xij ¼ð24ÞXH�1

h¼0

Xshji þ Xsh

ij

� �¼XH�2

h¼0

Xshji þ

XH�1

h¼1

Xshij 6ð18Þ

6

XH�2

h¼0

Xshji þ

XH�2

h¼0

Xðkh; ihþ1Þ 2 As

L;

k–j

Xshki ¼

XH�1

h¼1

Xs½d�ðihÞ� ¼ð16ÞXH�1

h¼1

Yshi ¼ð23Þ

yi

To show that Eq. (4) is implied:

Xfi;jg2E

xij ¼ð24Þ X

fi;jg2E

XH�1

h¼0

Xshij þ Xsh

ji

� �¼Xi2V

XH

h¼1

Xs½d�ðihÞ� ¼ð16Þ

¼ð16ÞXi2V

XH

h¼1

Yshi ¼ð23ÞjRj þ jTj þ

Xi2S

yi � 1 �

Lemma 5. Formulation LGCIO is strictly stronger than formulationLGCI. Furthermore, inequalities (5) are redundant in LGCIO.

Proof. Since LGCIO contains all constraints of LGCI, it only remainsto show that LGCIO is strictly stronger. This relation can be seenfrom the computational results discussed in Section 5.1. In addi-tion, Fig. 6 illustrates an example that is feasible for the LGCI model,but it violates inequalities (25).


Finally, for each potential Steiner nodes i 2 S, inequalities (5)are redundant since:Xfi;jg2dðiÞ

xij ¼ð24Þ X

fi;jg2dðiÞ

XH�1

h¼0

Xshij þ Xsh

ji

� �¼XH

h¼1

Xs½d�ðihÞ� þXH�1

h¼0

Xs½dþðihÞ�

Pð16Þ and ð25ÞXH

h¼1

Yshi þ

XH�1

h¼0

Yshi ¼ð24Þ

2yi �

Lemma 6. Formulation LGCIOR is strictly stronger than formulationLGCIO.

Proof. Since LGCIOR contains all constraints of LGCIO, it only remainsto show that root-depth constraints (26) can be violated in an opti-mal LP solutions of model LGCIO. Consider the solution shown inFig. 7 feasible for LGCIO (H ¼ 5; T 0 ¼ T [ R). Clearly, inequalities(26) are violated since eY 05

1 – eY 150 and also eY 04

1 – eY 140 .

It remains to show that, for solution values ð�x; �yÞ corresponding toFig. 7a we cannot find a different set of feasible vectors ðeXs; eYsÞ;s 2 f0;1g, that satisfy constraints (26). This is established by thefollowing two observations which will be proved in the following:

1. For any solution vector ðeX0; eY0Þ feasible w.r.t. ð�x; �yÞ; eY 051 P 1=3

holds.2. For any solution vector ðeX1; eY1Þ feasible w.r.t. ð�x; �yÞ; eY 15

0 ¼ 0 holds.

To see that eY 051 ¼ 1=3 must hold, note that each path in Fig. 7a

between nodes 0 and 3 consists of at least three edges. Since H ¼ 5and we need to establish a feasible connection from 0 to 9, thisimplies that eY 03

3 ¼ 1 and thus through the path 0–5–7–3 and 0–1–6–3 we send 2=3 and 1=3 units of flow, respectively. Since indegreeof 6 and 7 needs to be one, 1=3 units of flow are sent throughð6;7Þ; ð3;6Þ; ð2;6Þ. Consequently, 1=3 units of flow has to be sentalong 0–5–7–3–6–1 to reach node 1 which means that Y05

1 P 1=3.Note that eY 15

0 ¼ 0 means that node 0 cannot be at the last layer.To see that this holds observe that terminal 4 can only be reachedthrough node 0. Therefore, when 1 is taken as the root, themaximal layer for node 0 is four. h

Lemma 7. Formulation UPathDI is strictly stronger than formulationLG.

Proof. Let ð�x; �y; �a; �kÞbe an optimal solution to the LP relaxation of for-mulation UPathDI. We first show how to derive a solutionð�x; �y;X;YÞ 2 PLG with the same objective value. The main difficultyin this derivation is that the linking constraints (19) of model LGsum over all copies of one edge e 2 E while the linking constraints(8) of model UPathDI consider each terminal individually. Thus, foreach s 2 R and each arc ði; jÞ 2 A, we use values of the variables as

ij toobtain the values of the variables Xsh

ij ; ðih; jhþ1Þ 2 AsL;0 6 h 6 H � 1 as

follows:



Fig. 5. A feasible solution ð�x; �y;X;YÞ to the LP relaxation of LGC to an instance withR ¼ f0;1g; T ¼ f2;3g, T 0 ¼ T [ R, and H ¼ 3. (a) Subgraph induced by variable values�xe; e 2 E, and �yi; i 2 S. (b) and (c) Subgraphs induced by variable valuesXsh

ij ; s 2 f0;1g; ðih; jhþ1Þ 2 AsL, respectively. Solid (dashed) edges and arcs indicate a

corresponding variable value of 1 (0:5); �y5 ¼ �y7 ¼ 1; �y4 ¼ �y8 ¼ 0:75, and �y6 ¼ 0:5.Observe that arc ð7;8Þ is used in the arborescences rooted at 1 but neither arc ð7;8Þnor arc ð8;7Þ can be part of a feasible arborescence rooted at 0 since potentialSteiner nodes do not exist at layer H, i.e., iH R Vs

L;8i 2 S;8s 2 R. Similarly, arc ð5;8Þis used in the arborescence rooted at 0 but neither ð5;8Þ nor ð8;5Þ can be part of afeasible arborescence rooted at 1. Further note, that in order to satisfy constraints(3)–(5) the unique feasible assignment of node variable values is �y5 ¼ �y7 ¼ 1,�y4 ¼ �y8 ¼ 0:75, and �y6 ¼ 0:5 and it is not possible to derive feasible arborescenceson the layered graphs such that

PH�1h¼1 Ysh

i ¼ yi , 8s 2 R, 8i 2 S.


Xshij ¼

�asij if i¼ s

max 0;min �asij;

Xðkh�1 ;ihÞ2As

L :k–j

Xs;h�1ki

8<:

9=;�

Xh�1

h0¼0

Xsh0

ij

8<:

9=; otherwise

8>>><>>>:

ð37Þ

For each root s and for each arc ði; jÞ, the values Xshij are defined

recursively w.r.t. the layers starting from the root. Available capac-ities �as

ij are distributed among the layers while respecting the

connectivity constraints (18) and ensuring thatPH�1

h¼0 Xshij 6

�asij. We

note that since �as½d�ðiÞ� 6 1;8i 2 V , cf. (13), we can use Eq. (16) toset variable values Ysh

i ;8s 2 R;8ih 2 VsL. To see that inequalities

(17) hold, we first observe that for each node i 2 V , we have

XH

h¼1

Yshi ¼ð16ÞXH

h¼1

Xs½d�ðihÞ� ¼Xðj;iÞ2A

XH

h¼1

Xshji 6ð37Þ X

ðj;iÞ2A

�asji ¼ð13Þ

�yi i 2 S

0 i ¼ s

1 else

8><>:

It remains to prove that for each root s 2 R and each terminal

t 2 T 0 n fsg;PH

h¼1Ysht ¼ 1 does hold. First observe that due to (9),

(10), and (13) for each arc ðu; tÞ we haveP

p2Wst :ðu;tÞ2p�kst

p ¼ �asut.

Furthermore, for each arc ði; jÞ; i – s, contained in the used set ofpaths from s to t, i.e., in the set fp 2 Wst j�kp > 0g, we haveP

p2Wst jði;jÞ2p�kp 6

Pp2Wst jðk;iÞ2p;k–j

�kp since flow balance holds for eachpath. Thus, due to (37) we will distribute the total available capacity�as

it for each arc ði; tÞwith t 2 T 0 n fsg on the arcs of the layered arbores-

cence with root s, i.e.,PH

h¼1Xs½d�ðthÞ� ¼PH

h¼1Ysht ¼ �as½d�ðtÞ� ¼ 1.

To see that the inequality can be strict consider the LP-solutionð�x; �y;X;YÞ given in Fig. 5 feasible for LG. It is, however, not possibleto derive assignments of variable values a0 and a1 that satisfy allconstraints of model UPathDI. h

Lemma 8. Formulations UPathDI and LGC are incomparable.

Proof. We first consider the solution given in Fig. 5 which is a feasibleLP-solution of LGC but infeasible for the LP relaxation of UPathDI.Hence, it suffices to additionally consider an LP-solution feasible forUPathDI which is infeasible for LGC. As already observed by Gouveiaet al. (2011) for the single root case a path formulation allows to usethe full capacity of arcs at different positions in paths to different ter-minals, while in a layered graph formulation total capacity must beequal to the sum of capacities on different positions independently


of the considered terminal. Their example can be generalized to themultiple root case in a straightforward way. h

Lemma 9. Formulation LGCIO is strictly stronger than formulationUPathDI.

Proof. We show that given an LP solution ð�x; �y;X;YÞ of LGCIO wecan construct a solution ð�x; �y; �a; �kÞ 2 PUPathDI using

�asij :¼

XH�1

h¼0

Xshij 8s 2 R; 8ði; jÞ 2 A ð38Þ

Hereby, to simplify the notation we assume that Xshij ¼ 0 if

ðih; jhþ1Þ R AsL. From Lemma 4 we conclude that inequalities (3)

are satisfied since they are implied by model LGCIO. Constraints(8) follow due to (38) and (24). Furthermore, from the directedcutset constraints (22), using the max-flow min-cut theo-rem together with the path decomposition of the flow we canconstruct the necessary set of paths on the layered graph foreach root s 2 R and each relevant terminal t 2 T 0 n fsg. Sincehop constraints are implicitly satisfied in the structure of thelayered graph, constraints (9) and (10) are satisfied. Using(38), (16) and (23) we show that Eq. (13) are satisfied asfollows:

Xðj;iÞ2A

�asji ¼ð38Þ X

ðj;iÞ2A

XH�1

h¼0

Xshji ¼

XH

h¼1

Xs½d�ðihÞ� ¼ð16ÞXH

h¼1

Yshi ¼ð23Þ

�yi i 2 S

0 i ¼ s

1 else

8><>:

Finally, using the fact that potential Steiner nodes i 2 S do not existin any layered graph at layer H, inequalities (14) hold for each roots 2 R since

Xði;jÞ2A

�asij ¼ð38Þ X

ði;jÞ2A

XH�1

h¼0

Xshij ¼

XH�1

h¼0

Xs½dþðihÞ�Pð25ÞXH�1

h¼0

Yshi ¼ð23Þ

�yi:

To see that the inclusion can be strict, we refer again to the pre-viously mentioned straightforward generalization of the exampleprovided by Gouveia et al. (2011) for the single root case. h

4.2. Polyhedral comparison for T 0 ¼ T

In this subsection we prove similar results for the case T 0 ¼ T.Again, the following theorem is proved by a series of subsequentlemmas.

Theorem 4.2. For T 0 ¼ T, the following relations hold:

Thereby, an arrow indicates that the formulation at the target isstrictly stronger than the one at the source while a dashed edge indi-cates that the corresponding formulations are incomparable.

In what follows, we will prove only those results stated in thelatter theorem that are non-trivial and cannot be derived in a sim-ilar way as for the case T 0 ¼ T [ R.



Fig. 8. A feasible solution ð�x; �y;X;YÞ to the LP relaxation of LG or LGC of an instancewith R ¼ f0;1g; T ¼ f2;3;4;5g; T 0 ¼ T, and H ¼ 3. (a) Subgraph induced by variablevalues �xe; e 2 E. (b) and (c) Subgraphs induced by variable values


Lemma 10. Formulation LGC is strictly stronger than formulation LG.Furthermore, formulation LGC

E is strictly stronger than formulationLGE.

We skip the proof of this result since it is well known for thecase of one root and it is easy to find examples showing that thedirected cutset constraints can be violated in optimal LP solutionsof models LG and LGE, respectively.

Lemma 11. Formulation LGE is strictly stronger than formulation LGand formulation LGC

E is strictly stronger than formulation LGC.Furthermore, constraints (3) and (4) are redundant in LGE.

Xshij ; s 2 f0;1g; ðih; jhþ1Þ 2 As

L, respectively. Solid edges indicate a correspondingvariable value of 1 while dashed edges and arcs indicate a variable valueof 1/2.
Proof. To see that LGE is stronger than LG, we observe that it
essentially differs from model LG by equations Eq. (29) which arelifted variants of inequalities (17) and Eq. (31) which are strongerversions of inequalities (19). To see that the relation is strict, weconsider the LP-solution of LG corresponding to Fig. 8 and note thatwe cannot find a layered arborescence with root 1 feasible for LGE,i.e., such that the indegree of node 0 is one, without increasing thevariable value x01 or x03.

The same arguments can be used when considering the

formulations with directed cutset constraints, i.e., LGCE and

LGCE. Redundancy of constraints (3) and (4) in LGE can be shown

using an analogous deduction as in Lemma 4 for the caseT 0 ¼ T [ R. h

Fig. 7. A feasible solution ð�x; �y;X;YÞ to the LP relaxation of LGCI of an instance withR ¼ f0;1g, T ¼ f3;4;5;6;7;8;9g;H ¼ 5, and T 0 ¼ T [ R. (a) Subgraph induced byvariable values �xe; e 2 E; �y3 ¼ 1=3. (b) and (c) Subgraphs induced by variable valuesXsh

ij ; s 2 f0;1g; ðih; jhþ1Þ 2 AsL, respectively. Solid edges and arcs indicate a variable

value of 1, dashed edges and arcs of 2/3 and dotted edges and arcs of 1/3.

Fig. 6. A feasible solution ð�x; �y;X;YÞ to the LP relaxation of LGCI to an instance withR ¼ f0;1g, T ¼ f2;3;9g; T 0 ¼ T [ R, and H ¼ 5. (a) Subgraph induced by variablevalues �xe; e 2 E, and �yi; i 2 S. (b) and (c) Subgraphs induced by variable valuesXsh

ij ; s 2 f0;1g; ðih; jhþ1Þ 2 bAsL, respectively. Dashed (solid) edges and arcs indicate a

corresponding variable value of 1=2 (1); �y4 ¼ �y6 ¼ �y10 ¼ 0:5, and �yi ¼ 1, i 2 f5;7;8g.This solution clearly violates inequalities (25) since Steiner nodes 7 is a leave in thearborescence on G0

L . Further note, that for the given values of �x and �y each feasibleset of arborescences must contain Steiner nodes as leaves to satisfy constraints (23)and (19), i.e., the solution cannot satisfy inequalities (25).


Lemma 12. Formulations LGE and LGC are incomparable.

Proof. We first observe that if T 0 ¼ T , the solution given in Fig. 8is a valid LP solution for LGC. As argued before, however, wecannot find a feasible arborescence with root 1 such that theindegree of 0 (and all terminals) is one. Since this argument doesnot depend on the maximum allowed path length (and thus onthe number of layers on a layered graph) this solution is infeasi-ble for LGE. On the other hand, it is well known that the directedcutset constraints (22) can be violated in optimal LP-solutionsof LGE. h

Lemma 13. Formulation LGCOE is strictly stronger than formulation

LGCE . Formulation LGCOR

E is strictly stronger than formulation LGCOE .

Furthermore, constraints (5) are redundant in LGCOE .

Proof. LGCORE contains all constraints of LGCO

E which in turn con-tains all constraints of LGC

E. Strict inequality can be seen by modi-fying the previously discussed exemplary solutions given in Fig. 6and Fig. 7 to the case T 0 ¼ T. Redundancy of inequalities (5) inLGCO

E can be shown in an analogous way as for the case T 0 ¼ T [ Rin Lemma 5. h

Lemma 14. Formulation UPathDI is strictly stronger than formulationLG. Furthermore, formulation LGCO

E is strictly stronger than formula-tion UPathDI.

Proof. One can prove that UPathDI is stronger than LG and thatLGCO

E is stronger than UPathDI using analogous arguments thanfor the case T 0 ¼ T [ R, cf., Lemmas 7 and 9. To see that the firstrelation can be strict consider the previously discussed examplegiven in Fig. 8 which provides a feasible LP solution of LG whichis infeasible for UPathDI. For the second relation, again considerthe previously mentioned straightforward generalization of theexample from Gouveia et al. (2011) to the case with more thanone root node. h

Lemma 15. Formulations LGC and UPathDI are incomparable.

Proof. As mentioned before for the case T 0 ¼ T [ R an exemplaryLP-solution feasible for UPathDI but infeasible for LGC can be con-structed as a straightforward generalization from the single rootcase (Gouveia et al., 2011). On the other hand, as discussed inLemma 12 the solution given in Fig. 8 is feasible for LGC but wecannot find a feasible orientation with root 1 such that the inde-gree of 0 is one, i.e., the solution is infeasible for the LP-relaxationof UPathDI. h




5. Computational study

In this section, we detail all components of the implementedbranch-and-cut algorithms (B&C) for the different variants of LGand LGE and of the column generation approach (CG) used to solvethe LP relaxation of UPathDI. All approaches are implemented inC++ using IBM CPLEX 12.4 and all experiments have beenperformed on a single core of an Intel Xeon processor with2.53 gigahertz using at most 3 gigabytes RAM.

For the separation of directed cutset constraints (22) or (34),respectively, we run the maximum flow algorithm of Cherkasskyand Goldberg (1994). In all separation variants, we use nestedand backcuts, cf. (Ljubic et al., 2006), and insert at most 100 vio-lated cuts in each iteration. If a particular model considers outde-gree constraints on potential Steiner nodes, cf. (25) and (35), orroot-depth inequalities, cf. (26) or (36), they are separated dynam-ically, rather than statically inserted in the beginning, since preli-minary tests showed that typically only very few of them will beviolated.

In the column generation approach of UPathDI, a hop con-strained shortest path problem between each root and each rele-vant node on a graph with nonnegative arc costs needs to besolved in order to solve the pricing subproblem. As originally pro-posed by Gouveia, Paias, and Sharma (2008) for a spanning treeproblem with distance constraints we potentially add multiplepath variables for each root terminal pair by considering the short-est paths to all nodes adjacent to a currently considered relevantnode for all hop values 0 6 h 6 H � 1, for more details see Gouveiaet al. (2012a).

Benchmark Instances. Evaluation and comparison of the ap-proaches and models is conducted on benchmark instances fromGouveia et al. (2011) that are typically used for testing HMSTPand DMSTP approaches. We choose the first instance from eachof the groups of random (R) and Euclidean instances (C) with 31,41, and 61 nodes. For the sake of simplicity we will use 30, 40,and 60 to refer to them. All graphs are complete and we use thefirst jTj nodes as terminals and the last jRj nodes as roots. In ourexperiments, we choose jRj 2 f2;4;6;8g; jTj 2 f5;10;15;20g, andtest all possible combinations for hop limits H ¼ 3; . . . ;6 andT 0 2 fT; T [ Rg.

Table 3Results for solving LP relaxations in case T 0 ¼ T [ R grouped by instance set, jRj, and H. Numof instances solved by all approaches and where IP optimum is known (#all), numbers of insin %. P and L are used as abbreviations for UPathDI and LG, respectively; time limit: 7200

# #solved CPU-time (seconds) #all #int

P L LC LCI LCIO LCIOR P L LC LCI LCIO LCIOR P L L

SetC30 64 33 64 61 60 60 59 826 18 59 49 49 55 33 25 4 2C40 64 23 64 51 52 52 48 2324 80 374 296 298 333 23 18 1 2C60 64 14 54 38 37 35 36 4704 465 1559 1579 1613 1608 13 8 0R30 64 38 64 64 64 64 64 1014 9 15 12 13 16 38 20 17 1R40 64 25 64 59 59 58 57 2277 54 166 178 197 234 24 12 3 1R60 64 15 59 50 46 45 43 4438 249 473 621 652 744 15 3 2

jRj2 96 86 96 96 96 96 96 183 4 9 8 8 8 86 55 13 54 96 46 96 93 93 92 92 2719 46 115 108 121 136 44 22 8 26 96 13 93 77 73 72 68 5835 193 601 592 605 705 13 8 58 96 3 84 57 56 54 51 7066 556 1949 2002 2034 2352 3 1 1

H3 96 51 96 94 94 94 92 1134 18 51 55 58 68 51 15 0 14 96 38 96 86 83 81 78 2145 52 168 184 198 228 37 23 7 25 96 31 92 75 74 73 71 2717 105 330 300 317 367 31 23 7 26 96 28 85 68 67 66 66 3097 182 421 316 325 327 27 25 13 2


5.1. Computational results

Tables 3 and 4 detail our results regarding the LP relaxations ofall proposed models for T 0 ¼ T [ R and T 0 ¼ T , respectively. Resultsare grouped by instance sets, numbers of root nodes, and the hoplimit. The tables provide information on: the total number of in-stances in each group (#), the number of instances for which theLP relaxation of a particular model could be solved within 7200CPU-seconds (#solved), geometric means of the correspondingCPU-times, the numbers of instances for which the LP relaxationis integral (#int), average and maximum LP gaps in percent calcu-lated by ðOPT� vLPð:ÞÞ=OPT. Notice, that #all denotes the numberof instances for which the LP-relaxation could be solved by allmodels and for which the optimal IP solution is known. The valuesfor #int, average and maximum gaps are calculated only amongthose instances.

We first observe that solving the LP relaxation of UPathDI needssignificantly more CPU-time than solving the LP-relaxation of anyof the layered graph models both for T 0 ¼ T [ R and for T 0 ¼ T.Furthermore, model UPathDI is not only theoretically dominated bythe stronger layered graph variants but also exhibits significantlylarger LP gaps in our test cases. Thus, it is clearly not competitiveto the layered graph approaches. Since in the integer case the latterare additionally expected to benefit much more from built-in pre-processing of state-of-the-art ILP solvers we will not consider modelUPathDI in the remainder of this study. Comparing the different lay-ered graph models, the largest improvement on the reported LP gapsis obtained with the inclusion of the directed cutset constraints. Onthe other hand, their inclusion also significantly increases the run-time. From Table 3 we further observe that when T 0 ¼ T [ R, that isin the case where the arborescences of each root share the sameset of nodes and edges, the addition of Eqs. (23) and (24), often yieldsa further significant reduction on the reported LP gaps. This reduc-tion is usually obtained with no cost or only with little cost in termsof CPU-time. Adding outdegree constraints (25) for potential Steinernodes on each layered graph and root-depth constraints (26) furtherreduces the obtained gaps in many cases. This improvement, how-ever, is typically rather small. On the other hand, since we onlydynamically separate these constraints the additional CPU-time isalmost negligible.

bers of solved instances (#solved), geometric means of CPU-times in seconds, numberstances where LP relaxation is integral (#int), average and maximum LP relaxation gapsCPU-seconds. Best values are marked bold.

Avg. gap (%) Max. gap (%)

C LCI LCIO LCIOR P L LC LCI LCIO LCIOR P L LC LCI LCIO LCIOR

7 27 27 27 0.6 5.8 0.4 0.4 0.4 0.3 6.3 14.9 4.2 4.2 4.2 4.20 20 20 20 0.4 8.9 0.2 0.2 0.2 0.2 4.8 15.5 2.3 2.3 2.3 2.38 8 8 8 0.9 10.6 0.7 0.7 0.7 0.7 3.6 17.9 3.3 3.3 3.3 3.39 23 24 24 2.2 4.1 2.5 1.7 1.6 1.5 10.4 14.0 12.8 8.7 8.7 8.63 14 14 15 3.6 8.6 4.3 2.2 2.2 2.1 19.9 29.3 22.5 11.8 11.8 11.53 5 5 6 8.3 13.1 10.7 4.3 4.0 3.6 25.2 30.8 28.5 14.3 13.9 13.8

8 61 62 63 1.5 7.5 1.8 1.0 1.0 0.9 25.2 30.0 25.7 14.3 13.9 13.84 26 26 27 3.7 7.9 4.3 2.2 2.1 2.0 20.7 30.8 28.5 8.7 8.7 8.67 9 9 9 2.5 5.8 2.7 1.4 1.4 1.2 19.9 29.3 22.5 11.8 11.8 11.51 1 1 1 4.0 8.4 3.2 2.0 2.0 1.9 6.3 13.1 6.4 3.1 3.1 2.9

6 21 21 23 3.9 9.2 4.7 2.1 2.1 1.9 20.7 30.8 28.5 11.8 11.8 11.55 25 26 26 3.1 8.5 3.4 2.2 2.0 1.9 25.2 30.0 25.7 14.3 13.9 13.83 25 25 25 0.6 5.8 0.7 0.6 0.6 0.5 8.6 17.0 10.6 8.6 8.5 8.16 26 26 26 0.1 4.7 0.0 0.0 0.0 0.0 0.7 17.9 0.3 0.3 0.3 0.3



Table 4Results for solving LP relaxations in case T 0 ¼ T grouped by instance set, jRj, and H. Numbers of solved instances (#solved), geometric means of CPU-times in seconds, numbers of instances solved by all approaches and where integeroptimum is known (#all), numbers of instances where LP relaxation is integral (#int), average and maximum LP-gaps in %. P and L are used as abbreviations for UPathDI and LG, respectively; time limit: 7200 CPU-seconds. Best values aremarked bold.

# #solved CPU-time (seconds) #all #int Avg. gap (%) Max. gap (%)

P L LC LE LCE LCO

E LCORE

P L LC LE LCE LCO

E LCORE

P L LC LE LCE LCO

E LCORE

P L LC LE LCE LCO

E LCORE

P L LC LE LCE LCO

E LCORE

SetC30 64 40 64 57 64 57 57 56 600 16 116 21 93 114 121 40 29 0 27 4 32 32 33 0.6 11.5 0.7 5.8 0.4 0.4 0.3 5.0 21.1 6.8 14.9 4.2 4.2 4.2C40 64 28 64 46 62 44 45 40 2031 61 487 132 552 620 703 26 19 0 20 2 21 21 23 0.6 13.9 0.6 8.3 0.3 0.3 0.2 5.5 26.1 6.9 15.5 3.2 3.2 2.5C60 64 16 60 31 42 28 29 26 4413 364 2082 853 2657 2672 2747 13 8 0 7 0 8 8 8 0.9 15.2 0.9 9.8 0.7 0.7 0.7 3.6 25.4 3.3 17.9 3.3 3.3 3.3R30 64 48 64 64 64 64 64 63 754 10 19 14 26 31 36 48 22 16 21 19 22 26 27 2.8 5.7 3.2 4.6 2.6 2.4 2.2 11.1 25.8 13.4 18.6 10.1 9.6 9.4R40 64 31 64 57 62 51 48 48 1904 42 176 109 350 425 487 26 12 2 11 3 12 12 12 3.8 10.8 4.8 8.0 3.3 3.0 2.9 16.8 35.2 21.1 28.9 14.8 14.0 13.8R60 64 16 61 45 49 39 35 33 3992 191 532 502 994 1432 1540 15 3 2 3 2 3 4 4 9.2 14.1 11.0 12.1 8.3 5.7 5.4 25.7 35.2 29.9 30.0 24.8 16.9 16.8

jRj2 96 86 96 96 96 96 96 96 188 4 11 4 13 15 16 86 55 11 57 13 58 61 61 1.6 8.5 1.8 7.3 1.4 1.0 1.0 25.2 30.0 25.7 30.0 24.8 14.1 13.94 96 52 96 91 93 87 85 83 2027 39 160 85 242 306 359 49 22 5 22 9 23 24 24 4.2 13.2 4.8 7.5 3.6 3.2 3.0 25.7 35.2 29.9 28.9 20.8 16.9 16.86 96 28 96 68 84 59 60 56 4399 145 855 409 1227 1435 1592 22 10 4 8 6 10 11 15 2.5 11.7 3.1 5.6 2.1 2.0 1.8 11.3 26.1 14.9 18.2 11.1 11.0 10.78 96 13 89 45 70 41 37 31 5870 394 2446 1070 3088 3421 3765 11 6 0 2 2 7 7 7 2.3 13.7 3.5 6.3 2.0 2.0 1.8 8.5 25.8 10.4 18.6 8.3 8.2 7.9

H3 96 63 96 92 96 92 90 84 840 14 59 32 123 151 173 57 16 0 18 0 19 20 22 4.2 12.0 4.9 8.8 3.3 2.9 2.7 25.7 35.2 29.9 28.9 20.8 16.9 16.84 96 43 96 78 90 71 71 68 1810 44 231 106 362 467 524 42 24 5 23 6 26 27 29 3.5 12.5 4.1 8.6 3.3 2.6 2.5 25.2 30.0 25.7 30.0 24.8 14.1 13.95 96 38 96 68 82 62 58 56 2420 87 431 177 534 624 674 37 25 5 22 8 25 28 28 0.9 8.9 1.2 5.2 0.9 0.8 0.7 9.6 24.1 13.2 17.0 9.6 9.2 8.96 96 35 89 62 75 58 59 58 2670 166 601 249 485 523 555 32 28 10 26 16 28 28 28 0.2 7.7 0.2 4.0 0.2 0.2 0.1 2.7 26.1 2.2 17.9 1.8 1.8 1.8

Table 5Results for T 0 ¼ T [ R grouped by instance set, jRj, and H. Numbers of instances (#solved) solved to proven optimality, geometric means of CPU-times in seconds, average optimality gaps in %, numbers of instances (#L) for which the LPrelaxation could be solved by all layered graph variants, and average optimality gaps in % on them; time limit: 7200 CPU-seconds. Best values are marked bold.

# #solved CPU-time (seconds) Avg. gap (%) #L Avg. gap (%) (LP solved)

LG LGC LGCI LGCIO LGCIOR LG LGC LGCI LGCIO LGCIOR LG LGC LGCI LGCIO LGCIOR LG LGC LGCI LGCIO LGCIOR

SetC30 64 48 58 58 59 57 273 79 62 70 72 8.9 5.7 7.9 7.8 10.3 59 6.9 0.8 0.0 1.7 2.7C40 64 32 42 48 44 47 1371 417 286 369 355 20.3 21.5 17.5 19.8 20.8 47 8.7 1.4 0.3 3.1 2.6C60 64 14 38 38 37 35 4550 1626 1495 1830 1826 42.9 39.1 36.6 38.6 41.7 35 14.7 0.0 0.0 0.0 0.0R30 64 58 53 56 58 60 60 86 38 45 44 7.0 13.6 8.3 8.3 6.3 64 7.0 13.6 8.3 8.3 6.2R40 64 33 27 33 35 35 698 823 569 574 550 35.2 46.9 39.1 39.0 39.3 56 28.7 39.9 30.7 30.5 30.6R60 64 25 23 35 32 31 1597 2007 943 1063 1087 49.2 56.8 39.3 44.5 46.6 43 29.6 37.4 15.1 22.0 22.8

jRj2 96 86 93 96 96 96 41 18 11 13 14 1.1 2.1 0.0 0.0 0.0 96 1.1 2.1 0.0 0.0 0.04 96 64 75 84 84 83 702 390 202 254 232 17.3 15.9 9.1 10.0 11.1 92 15.1 15.5 6.2 7.2 7.26 96 37 46 54 52 52 2063 1465 1026 1146 1169 41.7 41.9 33.4 39.0 41.1 68 27.3 23.1 15.1 20.9 21.18 96 23 27 34 33 34 3945 3654 2907 3145 3095 48.9 62.4 56.6 56.3 57.7 48 27.2 33.9 27.4 28.2 26.5

H3 96 69 67 77 78 79 300 282 100 118 113 15.4 20.9 10.5 11.3 11.6 92 15.2 19.5 10.6 11.2 11.44 96 43 49 60 60 58 1052 581 431 472 475 32.3 38.0 29.8 31.1 34.0 77 25.4 27.0 16.8 19.1 20.25 96 48 58 65 63 66 944 546 412 497 500 28.8 34.5 28.6 32.5 30.9 70 11.5 12.3 7.9 10.4 8.16 96 50 67 66 64 62 786 430 370 433 428 32.5 29.1 30.1 30.4 33.5 65 7.6 1.4 1.4 3.1 3.1

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Please cite this article in press as: Gouveia, L., et al. Hop constrained Steinehttp://dx.doi.org/10.1016/j.ejor.2013.11.029

r tree

For T 0 ¼ T , cf. Table 4, we conclude that despite the reportedtighter LP bounds, the additional CPU-time needed to solve LGE isnot negligible compared to LG.

Similar to our previous observations w.r.t. LGCI in caseT 0 ¼ T [ R, the bounds of model LGC

E are clearly better than thoseof LGE or LGC. Outdegree and root-depth constraints produce a fur-ther, although usually rather small, improvement on the reportedLP bounds. Their influence seems to be more significant for the lar-ger random instances.

Overall, we conclude that the LP gaps usually increase with the in-stance size, increasing number of root nodes or decreasing hop limitand that the gaps resulting from random instances (R) are usuallysignificantly larger than those resulting from the Euclidean in-stances (C). Furthermore, all proposed model enhancements arenot only of theoretical importance but clearly tighten the LP boundsin many cases. In general the additional CPU-time needed to com-pute the tighter gaps is reasonable and, as we will see below, usingthe enhancements often contributes for solving the integer models.

Tables 5 and 6 summarize the computational results for solvingthe layered graph models in the integer case for T 0 ¼ T [ R andT 0 ¼ T , respectively. Here, we report the numbers of instances ineach set (#), the numbers of instances solved to proven optimalitywithin the time limit of 7200 CPU-seconds (#solved), geometricmeans of CPU-times in seconds, average optimality gaps in per-cent, numbers of instances where the LP relaxation could be solvedby each layered graph model (#L), and average optimality gaps inpercent on them. Hereby, optimality gaps are calculated asðUB� LBÞ=UB where UB and LB denote the obtained upper andlower bounds, respectively.

From Table 5, i.e., in case T 0 ¼ T [ R, we first observe that modelLGCI clearly outperforms the weaker variants LG and LGC with re-spect to all analyzed criteria. Whether LGCI or the two even stron-ger models LGCIO and LGCIOR perform best heavily depends on theconsidered instance and its parameters. On the one hand, LGCI of-ten yields the lowest CPU-times and optimality gaps after twohours. On the other hand, LGCIO and LGCIOR successfully solve someinstances to proven optimality that could not be solved by the the-oretically weaker models. For T 0 ¼ T the benefit of the strongermodels including the various enhancements is intensified. LGCOR

E

solved more instances to proven optimality than any of the othermodels. LGC

E and LGCOE also perform almost as good. With respect

to needed CPU-times, these three models are usually quite closeto each other and frequently exhibit a better overall performancethan the weaker variants. In particular, we conclude that consider-ing the extended layered graph with 2H � diamðTÞ layers whichenables most of the strengthening techniques clearly pays off inpractice. When comparing the average gaps grouped by the num-bers of root nodes, we observe that only for jRj ¼ 2 we were ableto solve all instances to optimality. The problem becomes more dif-ficult to solve already for jRj ¼ 4, where the average gaps are 11.1%and 21.3% for T 0 ¼ T [ R and T 0 ¼ T , respectively. Note that for theinstances of manageable size (i.e., where the LP relaxations couldbe solved by all layered graph models) the average gaps are signif-icantly smaller. When jRj ¼ 6 or jRj ¼ 8 the remaining gaps remain,however, quite large (e.g., around 26% for jRj ¼ 8). Overall, we con-clude that the models including the proposed enhancementsoften outperform their weaker variants and in particular allow tosolve more instances to proven optimality within the given timelimit.

6. Conclusions

In this article, we studied a generalization of the hop- and diam-eter constrained Steiner tree problems which arises by introducingmultiple central, i.e., root, nodes. After introducing the general casewe draw our attention to two particular cases which are motivated



Fig. 9. Transformation of a set cover instance with elements X ¼ fx1; . . . ; xng andsubsets Y ¼ fy1; . . . ymg to an instance of HSTPMR with R ¼ f0;1g; T ¼ fx1; . . . ; xng,and H ¼ 3. Edge costs are set to c0yi

¼ nþ 1;1 6 i 6 m, cyi xj¼ 1, if set yi;1 6 i 6 m,

contains element xj;1 6 j 6 n, and c01 ¼ 1.


from practical applications. For them we identified special polyno-mially solvable cases and proved that the problem is NP-hard ingeneral. Furthermore, we discussed MIP models for the two casesbased on layered graph reformulations together with strengthen-ing valid inequalities, established a hierarchy with respect to theirLP relaxation values, and also compared them theoretically to apreviously proposed path model. A computational study carriedout on a set of benchmark instances known from the literatureshows that the branch-and-cut approaches based on the layeredgraph reformulations clearly outperform the previously leadingpath model. Our results clearly indicate that all proposed modelenhancements reduce the LP gaps in practice. Furthermore, in spiteof the additional time needed to solve the LP relaxations, the stron-ger ILP models often lead to a better overall performance.

The results of our computational study also indicate two direc-tions for potential future research: (a) Since even for the strongestamong the proposed models the bounds of the linear program-ming relaxation are sometimes quite large, one may try toidentify further strengthening valid inequalities or even differentmodeling approaches. (b) We observed that for several instances,no reasonably good primal solutions could be obtained leading tolarge optimality gaps. Hence, obtaining high-quality heuristicsolutions is another interesting topic for future research. It is,however, an open question whether we can always find a feasiblesolution to the HSTPMR in polynomial time. We conclude this pa-per by pointing out that other variants of the more general prob-lem introduced at the beginning of the paper may be worthstudying. Consider the variant with T 0 ¼ R and V ¼ T [ R. This cor-responds to the problem of finding a minimum cost spanning treethat includes a diameter constrained Steiner tree with terminalset R. This variant is closely related to the two-level problemdescribed and studied in Gouveia, Leitner, and Ljubic (2012b). Itis also worth pointing out the particular case with jRj ¼ 2 wherewe obtain the problem of finding a minimum cost spanning treesuch that the length of the path between the two given root nodesdoes not exceed H. To the best of our knowledge it is not knownand it does not appear to be obvious whether this problem can besolved in polynomial time.

Appendix A

Proof of Lemma 2. H ¼ 2: For jRjP 3, each optimal solution mustbe a star centered at a node v 2 V with all roots and terminalsdifferent from v being leaves. Thus, we enumerate all such starsand each one yielding lowest cost is an optimal solution. In casejRj ¼ 2 there are additional feasible solutions that can be obtainedby assigning each terminal t 2 T to the closest of the two roots andconnecting the roots by an edge. Thus, we can still obtain anoptimal solution by enumeration. This argument holds for bothcases, T 0 ¼ T and T 0 ¼ T [ R.

H P 3: To show this non-approximability result, we use anerror-preserving reduction (see, e.g., Crescenzi, Kann, Silvestri, &Trevisan (1999)) from the SET COVER problem. Given a set coverinstance with the set of elements X ¼ fx1; . . . ; xng and a collectionof subsets Y ¼ fy1; . . . ; ymg, we transform it into a HSTPMRinstance with two roots 0;1 and H ¼ 3 as follows: Construct agraph G with the set of nodes V ¼ fx1; . . . ; xn; y1; . . . ; ym;0;1g. Weinsert an edge of cost one between xi and yj whenever the set yi

contains the element xi. We connect all y nodes to the root 0 andset the cost of those edges to nþ 1. Finally, we connect 0 and 1with an edge of cost one (see Fig. 9). It is not difficult to see thatthere is a one-to-one correspondence between the set of feasiblesolutions of the set cover and the set of feasible solutions of theHSTPMR on G with H ¼ 3. This transformation can be done inpolynomial time. To show that this transformation also preserves


the approximation ratio, observe that if the cost of the set coversolution is k, so is the cost of the corresponding HSTPMR solutionequal to FðkÞ ¼ ðnþ 1Þðkþ 1Þ. Let S be a feasible HSTPMR solutionwith the objective value equal to FðksÞ where ks is the value of thecorresponding set cover solution, and let k be the value of theoptimal set cover solution. Then we have:

FðksÞ � OPTOPT

¼ ðnþ 1Þðks þ 1Þ � ðnþ 1Þðkþ 1Þðnþ 1Þðkþ 1Þ ¼ ks � k

kþ 1P b

ks � kk

which is true for, e.g., b ¼ 1=2. Therefore, any approximation algo-rithm for the HSTPMR that runs in polynomial time cannot have abetter approximation ratio than Hðlog nÞ since ks�k

k P Hðlog nÞ holdsfor the set cover problem unless P = NP (Lund & Yannakakis, 1994).

For jRjP 3, we attach all further roots to 0 and set the edgecosts to one. The rest of the proof works similarly. h

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