new hyperbolic set covering problems with competing ground-set … · 2013. 1. 20. · hyperbolic...

g

Hyperbolic set covering problems

with competing ground-set elements

Edoardo Amaldi, Sandro Bosio and Federico Malucelli

Dipartimento di Elettronica e Informazione (DEI), Politecnico di Milano, Italy

XI Workshop on Combinatorial Optimization, Aussois, 2007

.

Outline g

Problems definition

The motivating application: Wireless Local Area Network design

Hyperbolic integer programming formulation

Complexity and Approximability results

Linearizations and Lagrangean Relaxation

Ongoing work and concluding remarks

Sandro Bosio, AUSSOIS 2007, “Hyperbolic set covering problems with competing ground-set elements”

Set Covering notation g



I

I: afinite groundset



J


J : acollectionof subsetsJ = {Ij ⊆ I : j ∈ J}



i

Ji



Ji ⊆ J : subcollection of the subsetscovering an elementi ∈ I



S



Ji ⊆ J : subcollection of the subsetscovering an elementi ∈ I

coverS: a subcollection indexed byS ⊆ J such that⋃

j∈S Ij = I


Set Covering problems g

Classical Set Covering Problem (SCP):

Given an instance(I,J ) and a costcj ∈ R for eachj ∈ J ,





find acoverS that minimizes thetotal costc(S) =∑

j∈S

cj






j∈S

cj

Variants: Set Partitioning forbiddenoverlap

Set Multicover requiredoverlap






j∈S

cj

Variants: Set Partitioning forbiddenoverlap

Set Multicover requiredoverlap

Also: Quadratic objective functions

Maximum coverage

...


Coverage share g

Given a covering instance(I,J ), a coverS and an elementi ∈ I

coverage share: r(S, i) =1

1 + |Ni(S)|


Coverage share g



1 + |Ni(S)|

S


Coverage share g



1 + |Ni(S)|

i

Ni

r(S,i)= 1

3


Coverage share g



1 + |Ni(S)|

i

Ni

r(S,i)= 1

7


Coverage share g



1 + |Ni(S)|

i

Ni

r(S,i)= 1

7

Fraction of resource received byi assumingfair allocation

among thecompeting elements(neighbors ofi)


Coverage share problems g

Maximum Total Coverage Share Problem (TCSP):

Given an instance(I,J ), find acoverS thatmaximizes

ft(S) =∑

i∈I

1

1 + |Ni(S)|





ft(S) =∑

i∈I

1

1 + |Ni(S)|

Maximum Minimum Coverage Share Problem (MCSP):


fm(S) = mini∈I

1

1 + |Ni(S)|





ft(S) =∑

i∈I

1

1 + |Ni(S)|

Maximum Minimum Coverage Share Problem (MCSP):


fm(S) = mini∈I

1

1 + |Ni(S)|

Set covering problems withcompetingground-set elements



Instance



Instance SCP opt

|S| = 2

ft(S) = 1.40

fm(S) =1

11



Instance SCP opt TCSP opt

|S| = 2

ft(S) = 1.40

fm(S) =1

11

|S| = 4

ft(S) = 3.64

fm(S) =1

7



Instance SCP opt TCSP opt MCSP opt

|S| = 2

ft(S) = 1.40

fm(S) =1

11

|S| = 4

ft(S) = 3.64

fm(S) =1

7

|S| = 4

ft(S) = 3.42

fm(S) =1

4



Instance SCP opt TCSP opt MCSP opt

|S| = 2

ft(S) = 1.40

fm(S) =1

11

|S| = 4

ft(S) = 3.64

fm(S) =1

7

|S| = 4

ft(S) = 3.42

fm(S) =1

4

Privilege covers whose subsets havesmall cardinalityandlimited overlaps.


Wireless Local Area Network g

IEEE 802.11 WLAN: a set ofAccess Pointseach able of serving a set ofusers




WLANs are becoming pervasive in airports, trains and train stations, private companies,

universities, hotels, ...




Medium Access Control (MAC) Protocol:

A user can access the network if and only ifno other user

is interferingdirectly or indirectly




Medium Access Control (MAC) Protocol:

A user can access the network if and only ifno other user

is interferingdirectly or indirectly

Assuming uniformpeaktraffic andfair access after collision,

coverage shareof elementi ≈ fraction of timeused by useri




Due to protocol issues, increasing sizes of deployed WLANs and limited resources,

and optimization models and methods can be

very usefulto support the planning decisions.


Previous and related work g

WLAN design:

Large-scale WLAN design (Hills 01, ...)

Max average signal quality in test points (Rodrigues, Mateus and Loureiro 00/01)

Max coverage level (Kamenetsky and Unbehaun 02)

Max capacity based on constraint satisfaction (Prommak et al. 02)

...

First hyperbolic model and heuristics (Amaldi, Capone, Cesana and Malucelli 04)


Integer programming formulations g

max∑

i∈I

1

1 + |Ni(S)|

( TCSP) s.t.⋃

j∈S

Ij = I complete coverage

S ⊆ J select subcollection



max∑

i∈I

1

1 + |Ni(S)|

( TCSP) s.t.⋃

j∈S

Ij = I complete coverage

S ⊆ J select subcollection

Variables:

xj = 1 if subsetIj is selected (0 otherwise)

yih = 1 if elementsi andh are neighbors (0 otherwise)



max∑

i∈I

1

1 +∑

h∈Ni

yih

( TCSP) s.t.∑

j∈Ji

xj ≥ 1 i ∈ I

yih ≥ xj i ∈ I, h ∈ Ni, j ∈ Ji ∩ Jh

yih ≤∑

j∈Ji∩Jh

xj i ∈ I, h ∈ Ni

xj ∈ {0, 1} j ∈ J

yih ∈ {0, 1} i ∈ I, h ∈ Ni



max∑

i∈I

1

1 +∑

h∈Ni

yih

→ 0-1 hyperbolic sumproblem

( TCSP) s.t.∑

j∈Ji

xj ≥ 1 i ∈ I


yih ≤∑

j∈Ji∩Jh


xj ∈ {0, 1} j ∈ J

yih ∈ {0, 1} i ∈ I, h ∈ Ni



max mini∈I

1

1 +∑

h∈Ni

yih

(MCSP) s.t.∑

j∈Ji

xj ≥ 1 i ∈ I


yih ≤∑

j∈Ji∩Jh


xj ∈ {0, 1} j ∈ J

yih ∈ {0, 1} i ∈ I, h ∈ Ni



max mini∈I

1

1 +∑

h∈Ni

yih

=1

1 + min maxi∈I

∑

h∈Ni

yih

(MCSP) s.t.∑

j∈Ji

xj ≥ 1 i ∈ I


yih ≤∑

j∈Ji∩Jh


xj ∈ {0, 1} j ∈ J

yih ∈ {0, 1} i ∈ I, h ∈ Ni


Connection with Quadratic SCP g

Quadratic Set Covering Problem (QSCP):

Given(I,J ), Q = {qjℓ ∈ R : j, ℓ ∈ J} (wlog symmetric with zero diagonal)

andc = {cj ∈ R : j ∈ J}, find acoverS ⊆ J thatmaximizes

q(S) =1

2

∑

j∈S

∑

ℓ∈S

qjℓ +∑

j∈S

cj






q(S) =1

2

∑

j∈S

∑

ℓ∈S

qjℓ +∑

j∈S

cj

Choice:

cj =∑

i∈Ij

1

|Ij |qjℓ =

∑

i∈Ij∩Iℓ

(

1

|Ij ∪ Iℓ|−

1

|Ij |−

1

|Iℓ|

)

(for j 6= ℓ)






q(S) =1

2

∑

j∈S

∑

ℓ∈S

qjℓ +∑

j∈S

cj

Choice:

cj =∑

i∈Ij

1

|Ij |qjℓ =

∑

i∈Ij∩Iℓ

(

1

|Ij ∪ Iℓ|−

1

|Ij |−

1

|Iℓ|

)

(for j 6= ℓ)

Then we can verify that:

ft(S) = q(S) if at most two subsets overlap

ft(S) ≥ q(S) otherwise (overhestimated penalty)


Previous and related work g

Unconstrained 0-1 Hyperbolic Programming

Single-ratio: NP-hard, poly with positive denominator (Hammer and Rudeanu 68)

Multiple-ratio: NP-hard; tackled by SA, Tabu, and decomposing into independent

polynomial single-ratio problems (Hansen, Poggi de Aragao and Ribeiro 90/91)

Constrained 0-1 Hyperbolic Programming

Single-ratio (Stancu-Minasian 97)

Multiple-ratio: MILP convex reformulations (Tawarmalani, Ahmed and Sahinidis 02)

Quadratic Set Covering Problem

Various application oriented works (Bazaara 75, Boros, Hammer et al. 00, ...)

Generic: not2p(|I|)-approximable for any polynomialp() (Escoffier and

Convex: approximable withinO(ln2(|I|)) but not withinρ ln2(|I|) Hammer 05)


Complexity and Approximability (TCSP) g



Generic instances



Generic instances

Strongly NP-hard (Amaldi et al. 04)

Not approximable withinρ (√

|I|)1−ε or ρ (√

|J |)1−ε for a givenρ > 0

and anyε > 0, unless NP= ZPPReduction from Max Independent Set



Generic instances



|I|)1−ε or ρ (√



Euclidean 2D Instances



Generic instances



|I|)1−ε or ρ (√




Strongly NP-hard (does not admit a FPTAS unless P= NP)Adapting and extending a reduction for Disc-Cover (Fowler et al. 81)

Under a reasonable restriction, admits a PTASUsing the shifting lemma



Generic instances



|I|)1−ε or ρ (√






Euclidean 1D Instances (or instances with C1C covering matrix)



Generic instances



|I|)1−ε or ρ (√






Euclidean 1D Instances (or instances with C1C covering matrix)

Polynomial-time solvableLongest path on an appropriate directed acyclic digraph


Complexity and Approximability (MCSP) g



Generic instances



Generic instances


Polynomial-time solvable if|Ij | = 2

Reduction to perfect b-matching



Generic instances







Generic instances





Strongly NP-hard (does not admit a FPTAS unless P= NP)Adapting the reduction for TCSP

Not approximable within3/2 unless P= NPConsequence of the above reduction

Under a reasonable restriction, approximable within a factor 3

Tiling with hexagons


Linearization g

For each ratio1

1 +∑

h 6=i

yih

is introduced a variableri ≥ 0 and the quadratic constraint

ri =1

1 +∑

h 6=i

yih

≡ ri +∑

h 6=i

riyih = 1


Linearization g

For each ratio1

1 +∑

h 6=i

yih


ri =1

1 +∑

h 6=i

yih

≡ ri +∑

h 6=i

riyih = 1

ri · yih is standardly linearized with a variablezih ≥ 0 and the constraints

zih ≤ uiyih

zih ≥ liyih

zih ≥ ri + ui(yih − 1)

zih ≤ ri + li(yih − 1)


Linearization g

For each ratio1

1 +∑

h 6=i

yih


ri =1

1 +∑

h 6=i

yih

≡ ri +∑

h 6=i

riyih = 1

ri · yih is standardly linearized with a variablezih ≥ 0 and the constraints

zih ≤ uiyih

zih ≥ liyih

zih ≥ ri + ui(yih − 1)

zih ≤ ri + li(yih − 1)

NB: ri is continuous and bounded, andy is binary


Tightening linearization of bilinear terms g

Z = {(r, y, z) : z = r · y, r ∈ [l, u], y ∈ {0, 1}}

conv(Z) =

{

z ≥ ly, z ≥ r + u(y − 1),

z ≤ uy, z ≤ r + l(y − 1)

}

1

0

yz

r

l

u



Z = {(r, y, z) : z = r · y, r ∈ [l, u], y ∈ {0, 1}}

conv(Z) =

{

z ≥ ly, z ≥ r + u(y − 1),

z ≤ uy, z ≤ r + l(y − 1)

}

1

0

yz

r

l

u

Z = ∪ {(r, y, z) : z = 0, r ∈ [l , u ], y = 0}

Z ′ =∪ {(r, y, z) : z = r, r ∈ [l , u ], y = 1}



Z = {(r, y, z) : z = r · y, r ∈ [l, u], y ∈ {0, 1}}

conv(Z) =

{

z ≥ ly, z ≥ r + u(y − 1),

z ≤ uy, z ≤ r + l(y − 1)

}

1

0

yz

r

l

u

Z ′ = ∪ {(r, y, z) : z = 0, r ∈ [l0, u0], y = 0}

Z ′ =∪ {(r, y, z) : z = r, r ∈ [l1, u1], y = 1}



Z = {(r, y, z) : z = r · y, r ∈ [l, u], y ∈ {0, 1}}

conv(Z) =

{

z ≥ ly, z ≥ r + u(y − 1),

z ≤ uy, z ≤ r + l(y − 1)

}

1

0

yz

r

l

u

Z ′ = ∪ {(r, y, z) : z = 0, r ∈ [l0, u0], y = 0}

Z ′ =∪ {(r, y, z) : z = r, r ∈ [l1, u1], y = 1}

conv(Z ′) =

{

z ≥ l1y, z ≥ r + u0(y − 1),

z ≤ u1y, z ≤ r + l0(y − 1)

}

1

0

yz

r

l0

l1

u0

u1


Lagrangean relaxation g

By applying Lagragean relaxation to an appropriate reformulation

the problem is decomposed intosmallerandeasiersubproblems.





Expanded formulation obtained by adding for eachi ∈ I a vector

χi = {χij : j ∈ Ji} of binary variables, one for each covering subset.

Incidence vector of alocal covering solutionfor i.








i

Ji








i



Expanded formulation:

maxX

i∈I

1

1 +P

h∈Ni

yih

s.t.X

j∈Ji

xj ≥ 1 i ∈ I (1)

yih ≥ xj i ∈ I, h ∈ Ni, j ∈ Ji ∩ Jh (2)

X

j∈Ji

χij ≥ 1 i ∈ I (3)

yih ≥ χij i ∈ I, h ∈ Ni, j ∈ Ji ∩ Jh (4)

yih ≤X

j∈Ji∩Jh

χij i ∈ I, h ∈ Ni (5)

xj = χij i ∈ I, j ∈ Ji (6)

yih = yhi i ∈ I, h ∈ Ni : h > i (7)

xj , χij , yih ∈ {0, 1}

Several possibilities, depending on which constraints aredeleted/dualized




maxX

i∈I

1

1 +P

h∈Ni

yih

s.t.X

j∈Ji

xj ≥ 1 i ∈ I (1)


X

j∈Ji

χij ≥ 1 i ∈ I (3)


yih ≤X

j∈Ji∩Jh

χij i ∈ I, h ∈ Ni (5)

xj = χij i ∈ I, j ∈ Ji (6)

yih = yhi i ∈ I, h ∈ Ni : h > i (7)

xj , χij , yih ∈ {0, 1}

Without (2), (6) and (7): one SCP and|I| independent hyperbolic subproblems




maxX

i∈I

1

1 +P

h∈Ni

yih

s.t.X

j∈Ji

xj ≥ 1 i ∈ I (1)


X

j∈Ji

χij ≥ 1 i ∈ I (3)


yih ≤X

j∈Ji∩Jh

χij i ∈ I, h ∈ Ni (5)

xj = χij i ∈ I, j ∈ Ji (6)

yih = yhi i ∈ I, h ∈ Ni : h > i (7)

xj , χij , yih ∈ {0, 1}

LAGa: remove (2) and dualize (6) and (7)→ NP-hard hyperbolic subproblems




maxX

i∈I

1

1 +P

h∈Ni

yih

s.t.X

j∈Ji

xj ≥ 1 i ∈ I (1)


X

j∈Ji

χij ≥ 1 i ∈ I (3)


yih ≤X

j∈Ji∩Jh

χij i ∈ I, h ∈ Ni (5)

xj = χij i ∈ I, j ∈ Ji (6)

yih = yhi i ∈ I, h ∈ Ni : h > i (7)

xj , χij , yih ∈ {0, 1}

LAGb: remove (5), (6) and dualize (2), (7)→ polynomial hyperbolic subproblems


Lagrangean subproblem for LAGb g

Problem for a given elementi:

max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t.X

j∈Ji

χj ≥ 1

yh ≥ χj h ∈ Ni, j ∈ Ji ∩ Jh

χj ∈ {0, 1} j ∈ Ji

yh ∈ {0, 1} h ∈ Ni



Fix one variableχℓ to 1 (try all). This covers allh ∈ Iℓ.

max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t.X

j∈Ji

χj ≥ 1


χj ∈ {0, 1} j ∈ Ji

yh ∈ {0, 1} h ∈ Ni




max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. χℓ = 1


χj ∈ {0, 1} j ∈ Ji

yh ∈ {0, 1} h ∈ Ni




max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. χℓ = 1

yh = 1 h ∈ Iℓ

yh ≥ χj h ∈ Ni \ Iℓ, j ∈ Ji ∩ Jh

χj ∈ {0, 1} j ∈ Ji

yh ∈ {0, 1} h ∈ Ni



Fix all otherχj to 0 (no o.f. contribution).

max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. χℓ = 1

yh = 1 h ∈ Iℓ

yh ≥ χj h ∈ Ni \ Iℓ, j ∈ Ji ∩ Jh

χj ∈ {0, 1} j ∈ Ji

yh ∈ {0, 1} h ∈ Ni



Fix all otherχj to 0 (no o.f. contribution).

max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. yh = 1 h ∈ Iℓ

yh ∈ {0, 1} h ∈ Ni



Remains an unconstrained problem with hyperbolic+linear o.f..

max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. yh = 1 h ∈ Iℓ

yh ∈ {0, 1} h ∈ Ni




max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. yh = 1 h ∈ Iℓ

yh ∈ {0, 1} h ∈ Ni

Since hyperbolic depends only onhow manyand not onwhich:




max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. yh = 1 h ∈ Iℓ

yh ∈ {0, 1} h ∈ Ni


1) sortch coefficients in nonincreasing order. f




max1

1 +P

h∈Ni

yh

+

X

h∈Ni

chyh

s.t. yh = 1 h ∈ Iℓ

yh ∈ {0, 1} h ∈ Ni


1) sortch coefficients in nonincreasing order. f

2) fix the firstk variables to 1, the remaining to0 (try all k).


Comparison - our department g

Our Department



Our Department

TCSP best solution

Standard Linearization Improved Linearization LAGb

|J| |I| den gap time gap time gap time

(%) (sec) (%) (sec) (%) (sec)

81 84 13.79.28 − 5.08 − 0.87 237.2

− : time limit exceeded



Our Department

TCSP best solution

SCP optimal solution



Our Department

TCSP best solution

SCP optimal solution

Tests with a WLAN simulator (ns-2):

2.58 Mb/s for SCP solution,15.8 Mb/s for TCSP solution


Comparison - synthetic instances g

Standard Linearization Improved Linearization LAGb

|J| |I| den gap stdev time stdev gap stdev time stdev gap stdev time stdev

GEOMETRIC INSTANCES(LOW DENSITY)50 50 6.6 ∗ 0.4 0.3 ∗ 0.1 0.1 ∗ 0.5 0.350 100 6.4 ∗ 7.2 4.1 ∗ 4.1 4.1 0.11 0.15 5.3 2.8

100 100 5.3 0.79 1.51 1826.61639.3 ∗ 177.1 203.1 0.28 0.28 24.8 13.6100 200 5.1 9.42 2.57 − 3.07 1.88 3363.0530.0 0.37 0.22 409.6 129.550 300 6.3 ∗ 561.6 557.5 ∗ 395.3 353.9 ∗ 621.8 137.9

GEOMETRIC INSTANCES(HIGH DENSITY)50 50 10.5 ∗ 15.5 20.3 ∗ 6.1 5.7 0.17 0.32 2.0 1.750 100 10.3 ∗ 798.6 452.7 ∗ 313.6 188.9 0.10 0.14 52.5 29.8

100 100 10.8 27.76 3.42 − 12.26 3.43 − 1.89 0.61 215.0 24.8100 200 10.6 33.17 3.50 − 18.96 1.08 − 2.26 1.14 1111.2 44.550 300 11.1 27.12 7.28 − 26.43 6.69 − 0.98 0.69 1494.4140.7

STANDARD SCPINSTANCES (CLASS SCP4*)1000 200 2.0 11.53 1.69 − 6.13 1.65 − 0.12 0.09 3304.6403.6

STANDARD SCPINSTANCES (CLASS SCPE*)500 50 20.0 71.22 4.69 − 35.75 16.62 − 6.14 0.78 1765.9148.1

∗ : the primal-dual gap is zero (proven optimality)− : time limit exceeded for all instances of the class


Concluding remarks g

This presentation:



This presentation:

New interesting class: set covering problems withcompeting ground-set elements



This presentation:


Complexity and approximability for generic and geometric versions



This presentation:



Improved linearization and efficient Lagrangean relaxation



This presentation:




Ongoing work:



This presentation:




Ongoing work:

Linearization byDantzig-Wolfedecomposition



This presentation:




Ongoing work:


Refined hyperbolic models, accounting for relevant features of WLANs



This presentation:




Ongoing work:



Direct interference and node association



This presentation:




Ongoing work:



Direct interference and node association

Multiple frequenciesandadaptive rate


new hyperbolic set covering problems with competing ground-set … · 2013. 1. 20. · hyperbolic...

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