Separation for the Max-Cut Problem on General Graphs · Separation for the Max-Cut Problem on...

Separation for the Max-Cut Problemon General Graphs

Thorsten Bonato

Research Group Discrete and Combinatorial OptimizationUniversity of Heidelberg

Joint work with:

Michael Junger (University of Cologne)Gerhard Reinelt (University of Heidelberg)Giovanni Rinaldi (IASI, Rome)

14th Combinatorial Optimization WorkshopAussois, January 6, 2010

Thorsten Bonato Separation for Max-Cut on General Graphs 1 / 20

Outline

1 Max-Cut Problem

2 Separation using Graph Contraction

3 Computational Results


Outline

1 Max-Cut Problem




Max-Cut Problem

Definition

Let G = (V ,E , c) be an undirected weightedgraph.

Any S ⊆ V induces a set δ(S) of edges withexactly one end node in S . The set δ(S) iscalled a cut of G with shores S and V \S .

Finding a cut with maximum aggregate edgeweight is known as max-cut problem.


Max-Cut Problem

Definition




V \ S

S

δ(S)


Max-Cut Problem

Definition




V \ S

S

δ(S)


Related Polytopes

Cut polytope CUT(G)Convex hull of all incidence vectors ofcuts of G .

Semimetric polytope MET(G)Relaxation of the max-cut IP formulationdescribed by two inequality classes:

CUT(K3)

Odd-cycle: x(F )− x(C \F ) ≤ |F | − 1, for each cycle C of G ,for all F ⊆ C , |F | odd.

Trivial: 0 ≤ xe ≤ 1, for all e ∈ E .


Related Polytopes

Cut polytope CUT(G)Convex hull of all incidence vectors ofcuts of G .

Semimetric polytope MET(G)Relaxation of the max-cut IP formulationdescribed by two inequality classes: CUT(K3)

Odd-cycle: x(F )− x(C \F ) ≤ |F | − 1, for each cycle C of G ,for all F ⊆ C , |F | odd.

Trivial: 0 ≤ xe ≤ 1, for all e ∈ E .


Exact Solution Methods

Algorithms

Branch&Cut,

Branch&Bound using SDP relaxations.

Certain separation procedures only workfor dense/complete graphs.

How to handle sparse graphs

Trivial approach:artificial completion using edges with weight 0.

Drawback:increases number of variables and thus the computational difficulty.



Algorithms

Branch&Cut,



How to handle sparse graphs?





Algorithms

Branch&Cut,



How to handle sparse graphs




Outline

1 Max-Cut Problem




Outline of the Separation using Graph Contraction

Input: LP solution z ∈ MET(G )\CUT(G ).

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i



Transform 1-edges into 0-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i



Transform 1-edges into 0-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i



Contract 0-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i



Contract 0-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Introduce artificial LP values for non-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Introduce artificial LP values for non-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Separate extended LP solution.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Separate extended LP solution.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Project out nonzero coefficients related to non-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Project out nonzero coefficients related to non-edges.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Lift inequality.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

ab

dg e cf

hi



Lift inequality.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i



Switch lifted inequality.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i



Switch lifted inequality.

z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i



z

z

z

z′ (a′, α′)

(c, γ)

(c, γ)

(c, γ)

Separate

G

G

G′

Switch

Contract

Extend Project

Lift

Un-switch

a b c

d e f

g h i


Contraction as Heuristic Odd-Cycle Separator

Assume the end nodes of a 0-edge uv share acommon neighbor w .

Contraction of uv merges the edges uw andvw .

If the LP values of the merged edges differ,e. g., zuw > zvw

then z violates the odd-cycleinequality

xuw − xvw − xuv ≤ 0.

u v

w

Contraction allows heuristic odd-cycle separation.








uv

w









uv

w






If the LP values of the merged edges differ,e. g., zuw > zvw then z violates the odd-cycleinequality


+1 −1

−1u v

w

≤ 0






If the LP values of the merged edges differ,e. g., zuw > zvw then z violates the odd-cycleinequality


+1 −1

−1u v

w

≤ 0



Extension

Given a contracted LP solution z ∈ MET(G ),assign artificial LP values to the non-edges.

Goal: extended LP solution z ′ ∈ MET(G′).

New cycles in the extended graph

consist ofa former non-edge and a connecting path.

ab

dg e cf

hi

Feasible artificial LP values of non-edge uv

Range: [ max{0, Luv}, min{Uuv , 1} ] ⊆ [0, 1] with

Luv := max { z(F ) − z(P \ F ) − |F |+ 1 | P (u, v)-path, F ⊆ P, |F | odd },Uuv := min {−z(F ) + z(P \ F ) + |F | | P (u, v)-path, F ⊆ P, |F | even }.

Odd-cycle inequality derived from arg max (resp. arg min) is called alower (resp. upper) inequality of uv .


Extension



New cycles in the extended graph

consist ofa former non-edge and a connecting path.

ab

dg e cf

hi






Extension



New cycles in the extended graph consist ofa former non-edge

and a connecting path.

ab

dg e cf

hi






Extension



New cycles in the extended graph consist ofa former non-edge and a connecting path.

ab

dg e cf

hi






Extension




ab

dg e cf

hi






Extension




ab

dg e cf

hi






Projection

Consider a valid inequality a′T x ′ ≤ α′violated by the extended LP solution z ′.

Non-edges may have nonzero coefficients!

Project out coefficient of non-edge uv

Add a lower inequality if a′uv > 0 resp. anupper inequality if a′uv < 0.

(· · · a′

uv· · · a

′

st· · · , α

′)

In the projected inequality, all non-edge coefficients are 0 and can betruncated.

Problem

If the added inequalities are not tight at z ′ then the projectionreduces the initial violation a′T z ′ − α′.


Projection





(· · · a′

uv· · · a

′

st· · · , α′)

(· · · a′

uv· · · · · · · · · , β

′

1)−

(· · · · · · · · · a′

st· · · , β

′

2)−

+

+


Problem



Projection





(· · · a′

uv· · · a

′

st· · · , α′)

(· · · a′

uv · · · · · · · · · , β′

1)−

(· · · · · · · · · a′

st · · · , β′

2)−

(· · · 0 · · · 0 · · · , γ)

+

+

=


Problem



Projection





(· · · a′

uv· · · a

′

st· · · , α′)

(· · · a′

uv · · · · · · · · · , β′

1)−

(· · · · · · · · · a′

st · · · , β′

2)−

(· · · 0 · · · 0 · · · , γ)

+

+

=


Problem



Adaptive Extension

Artificial LP values z ′uv adapt to the sign of the correspondingcoefficient in a given inequality a′T x ′ ≤ α′, i. e.,

z ′uv =

{Luv if a′uv > 0,

Uuv otherwise.

Advantage: Violation remains unchanged during projection.Drawback: Separation procedures may need to be modified.

Trivial modification case

For a given class of inequalities, allnonzero coefficients have identical sign.

E. g., bicycle-p-wheel inequalities: x(B) ≤ 2p(set z ′uv = Luv for all non-edges uv).

1

2

3

4

p


Adaptive Extension


z ′uv =

{Luv if a′uv > 0,

Uuv otherwise.





1

2

3

4

p


Adaptive Extension


z ′uv =

{Luv if a′uv > 0,

Uuv otherwise.





1

2

3

4

p


Adaptive Extension


z ′uv =

{Luv if a′uv > 0,

Uuv otherwise.





1

2

3

4

p


Adaptive Extension: Target Cuts (1/2)

Input for separation framework [Buchheim, Liers, and Oswald]

Associated polyhedron Q = conv {x1, . . . , xs}+ cone {y1, . . . , yt},Interior point q ∈ Q,

Point z /∈ Q to be separated.

Obtain facet defining inequality aT (x − q) ≤ 1 by solving the LP

max aT (z − q)

s.t. aT (xi − q) ≤ 1, for all i = 1, . . . , s

aT yj ≤ 0, for all j = 1, . . . , t

a ∈ Rm

For max-cut we set Q = CUT(G (W )

)for a subset W ⊆ V .







max aT (z − q)


aT yj ≤ 0, for all j = 1, . . . , t

a ∈ Rm









max aT (z − q)


aT yj ≤ 0, for all j = 1, . . . , t

a ∈ Rm





Modified input

W.l.o.g. let the last ` vector entries correspond to the non-edges.

z ′ := (z1, . . . , zm−`, L1, . . . , L`, U1, . . . , U`),x ′i := (xi1, . . . , xi ,m−`, xi ,m−`+1, . . . , xim, xi ,m−`+1, . . . , xim),q′ := (q1, . . . , qm−`, qm−`+1, . . . , qm, qm−`+1, . . . , qm),

Q ′ := conv {x ′1, . . . , x ′s}+ cone {−em−`+k , em+k | k = 1, . . . , `}.

Resulting target cut separation LP

max a′T (z ′ − q′)

s.t. a′T (x ′i − q′) ≤ 1, for all i = 1, . . . , s

−a′m−`+k , a′m+k ≤ 0, for all k = 1, . . . , `

a′ ∈ Rm+`

In an optimum solution a′∗ at most one of a′∗m−`+k and a′∗m+k can benonzero for each k = 1, . . . , `.



Modified input





max a′T (z ′ − q′)


−a′m−`+k , a′m+k ≤ 0, for all k = 1, . . . , `

a′ ∈ Rm+`




Modified input





max a′T (z ′ − q′)


−a′m−`+k , a′m+k ≤ 0, for all k = 1, . . . , `

a′ ∈ Rm+`




Modified input





max a′T (z ′ − q′)


−a′m−`+k , a′m+k ≤ 0, for all k = 1, . . . , `

a′ ∈ Rm+`



Outline

1 Max-Cut Problem




Computational Experiments

Used max-cut solver based on B&C framework ABACUS.

Problem classes1 Unconstrained quadratic 0/1-optimization problems.2 Spin glass problems on toroidal grid graphs with:

Uniformly distributed ±1-weights.Gaussian distributed integral weights.

Separation schemes

Standard:odd-cycles (spanning-tree heuristic, 3-/4-cycles, exact separation).

Contraction:standard scheme + contraction as heuristic OC-separator.

Extension:contraction scheme + separation of bicycle-p-wheels, hypermetricinequalities and target cuts on the extended LP solution.






Separation schemes









Separation schemes









Separation schemes









Separation schemes





Unconstrained Quadratic 0/1-Optimization Problems

0

0.5

1

1.5

2

2.5

3b250-1

b250-3

b250-5

b250-7

b250-9

Runnin

g tim

e [h]

Instance

Running time of Beasley instances (250 nodes, density 0.1)

Standard

Contraction

Extension

[Intel Xeon 2.8 GHz, 8GB shared RAM.]


Spin Glass Problems with Uniformly Distributed ±1-Weights

1s

1m

1h

10h

302

402

502

602

702

802

Avera

ge r

unnin

g tim

e (

log. scalin

g)

Number of grid nodes

Average running time of 10 random instances per grid size

Standard

Contraction

Extension

[Intel Xeon 2.8 GHz, 8GB shared RAM. Running time capped to 10h per instance.]


Spin Glass Problems with Gaussian Distributed Integral Weights

1s

1m

1h

10h

402

602

802

1002

1202

1402

1602

1802

Avera

ge r

unnin

g tim

e (

log. scalin

g)

Number of grid nodes

Average running time of 10 random instances per grid size

Standard

Contraction

Extension

[Intel Xeon 2.8 GHz, 8GB shared RAM. Running time capped to 10h per instance.]


Conclusion and Future Work

Separation using graph contraction

Enables the use of separation techniques for dense/completegraphs on sparse graphs.

Accelerates the exact solution of the max-cut problem for theexamined classes of spin glass problems.

Acceleration is mainly due to the use of contraction as heuristicodd-cycle separator.

Future work

Develop special branching rules.

Determine good parameter settings.

Further computational experiments.

Thank you for your attention!







Future work











Future work






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