Separation for the Max-Cut Problem on General Graphs Thorsten - - PowerPoint PPT Presentation

Separation for the Max-Cut Problem

• n General Graphs

Thorsten Bonato

Research Group Discrete and Combinatorial Optimization University of Heidelberg

Joint work with: Michael J¨ unger (University of Cologne) Gerhard Reinelt (University of Heidelberg) Giovanni Rinaldi (IASI, Rome) 14th Combinatorial Optimization Workshop Aussois, 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

Thorsten Bonato Separation for Max-Cut on General Graphs 2 / 20

Outline

1

Max-Cut Problem

2

Separation using Graph Contraction

3

Computational Results

Thorsten Bonato Separation for Max-Cut on General Graphs 3 / 20

Max-Cut Problem

Definition Let G = (V , E, c) be an undirected weighted graph.

Thorsten Bonato Separation for Max-Cut on General Graphs 4 / 20

Max-Cut Problem

Definition Let G = (V , E, c) be an undirected weighted graph. Any S ⊆ V induces a set δ(S) of edges with exactly one end node in S. The set δ(S) is called a cut of G with shores S and V \S. V \ S S δ(S)

Thorsten Bonato Separation for Max-Cut on General Graphs 4 / 20

Max-Cut Problem

Definition Let G = (V , E, c) be an undirected weighted graph. Any S ⊆ V induces a set δ(S) of edges with exactly one end node in S. The set δ(S) is called a cut of G with shores S and V \S. Finding a cut with maximum aggregate edge weight is known as max-cut problem. V \ S S δ(S)

Thorsten Bonato Separation for Max-Cut on General Graphs 4 / 20

Related Polytopes

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

CUT(K3)

Thorsten Bonato Separation for Max-Cut on General Graphs 5 / 20

Related Polytopes

Cut polytope CUT(G) Convex hull of all incidence vectors of cuts of G. Semimetric polytope MET(G) Relaxation of the max-cut IP formulation described 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.

Thorsten Bonato Separation for Max-Cut on General Graphs 5 / 20

Exact Solution Methods

Algorithms Branch&Cut, Branch&Bound using SDP relaxations. Certain separation procedures only work for dense/complete graphs.

Thorsten Bonato Separation for Max-Cut on General Graphs 6 / 20

Exact Solution Methods

Algorithms Branch&Cut, Branch&Bound using SDP relaxations. Certain separation procedures only work for dense/complete graphs. How to handle sparse graphs?

Thorsten Bonato Separation for Max-Cut on General Graphs 6 / 20

Exact Solution Methods

Algorithms Branch&Cut, Branch&Bound using SDP relaxations. Certain separation procedures only work for 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.

Thorsten Bonato Separation for Max-Cut on General Graphs 6 / 20

Outline

1

Max-Cut Problem

2

Separation using Graph Contraction

3

Computational Results

Thorsten Bonato Separation for Max-Cut on General Graphs 7 / 20

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction

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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction 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

Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Contraction as Heuristic Odd-Cycle Separator

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

u v w

Thorsten Bonato Separation for Max-Cut on General Graphs 9 / 20

Contraction as Heuristic Odd-Cycle Separator

Assume the end nodes of a 0-edge uv share a common neighbor w. Contraction of uv merges the edges uw and vw.

uv w

Thorsten Bonato Separation for Max-Cut on General Graphs 9 / 20

Contraction as Heuristic Odd-Cycle Separator

Assume the end nodes of a 0-edge uv share a common neighbor w. Contraction of uv merges the edges uw and vw. If the LP values of the merged edges differ,

• e. g., zuw > zvw

uv w

Thorsten Bonato Separation for Max-Cut on General Graphs 9 / 20

Contraction as Heuristic Odd-Cycle Separator

Assume the end nodes of a 0-edge uv share a common neighbor w. Contraction of uv merges the edges uw and vw. If the LP values of the merged edges differ,

• e. g., zuw > zvw then z violates the odd-cycle

inequality xuw − xvw − xuv ≤ 0.

+1 −1 −1 u v w ≤ 0

Thorsten Bonato Separation for Max-Cut on General Graphs 9 / 20

Contraction as Heuristic Odd-Cycle Separator

Assume the end nodes of a 0-edge uv share a common neighbor w. Contraction of uv merges the edges uw and vw. If the LP values of the merged edges differ,

• e. g., zuw > zvw then z violates the odd-cycle

inequality xuw − xvw − xuv ≤ 0.

+1 −1 −1 u v w ≤ 0

Contraction allows heuristic odd-cycle separation.

Thorsten Bonato Separation for Max-Cut on General Graphs 9 / 20

Extension

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

′). ab dg e cf hi

Thorsten Bonato Separation for Max-Cut on General Graphs 10 / 20

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

ab dg e cf hi

Thorsten Bonato Separation for Max-Cut on General Graphs 10 / 20

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 of a former non-edge

ab dg e cf hi

Thorsten Bonato Separation for Max-Cut on General Graphs 10 / 20

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 of a former non-edge and a connecting path.

ab dg e cf hi

Thorsten Bonato Separation for Max-Cut on General Graphs 10 / 20

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 of a 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 }.

Thorsten Bonato Separation for Max-Cut on General Graphs 10 / 20

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 of a 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 a lower (resp. upper) inequality of uv.

Thorsten Bonato Separation for Max-Cut on General Graphs 10 / 20

Projection

Consider a valid inequality a′Tx′ ≤ α′ violated by the extended LP solution z′. Non-edges may have nonzero coefficients!

(· · · a′

uv · · ·

a′

st · · · , α′) Thorsten Bonato Separation for Max-Cut on General Graphs 11 / 20

Projection

Consider a valid inequality a′Tx′ ≤ 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. an

upper inequality if a′

uv < 0.

(· · · a′

uv · · ·

a′

st · · · , α′)

(· · · a′

uv · · ·

· · · · · · , β

′ 1)

− (· · · · · · · · · a′

st · · · , β ′ 2)

− + +

Thorsten Bonato Separation for Max-Cut on General Graphs 11 / 20

Projection

Consider a valid inequality a′Tx′ ≤ 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. an

upper inequality if a′

uv < 0.

(· · · a′

uv · · ·

a′

st · · · , α′)

(· · · a′

uv · · ·

· · · · · · , β

′ 1)

− (· · · · · · · · · a′

st · · · , β ′ 2)

− (· · · · · · · · · , γ) + + =

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

Thorsten Bonato Separation for Max-Cut on General Graphs 11 / 20

Projection

Consider a valid inequality a′Tx′ ≤ 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. an

upper inequality if a′

uv < 0.

(· · · a′

uv · · ·

a′

st · · · , α′)

(· · · a′

uv · · ·

· · · · · · , β

′ 1)

− (· · · · · · · · · a′

st · · · , β ′ 2)

− (· · · · · · · · · , γ) + + =

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

Problem

If the added inequalities are not tight at z′ then the projection reduces the initial violation a′Tz′ − α′.

Thorsten Bonato Separation for Max-Cut on General Graphs 11 / 20

Adaptive Extension

Artificial LP values z′

uv adapt to the sign of the corresponding

coefficient in a given inequality a′Tx′ ≤ α′, i. e., z′

uv =

• Luv

if a′

uv > 0,

Uuv

• therwise.

Thorsten Bonato Separation for Max-Cut on General Graphs 12 / 20

Adaptive Extension

Artificial LP values z′

uv adapt to the sign of the corresponding

coefficient in a given inequality a′Tx′ ≤ α′, i. e., z′

uv =

• Luv

if a′

uv > 0,

Uuv

• therwise.

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

Thorsten Bonato Separation for Max-Cut on General Graphs 12 / 20

Adaptive Extension

Artificial LP values z′

uv adapt to the sign of the corresponding

coefficient in a given inequality a′Tx′ ≤ α′, i. e., z′

uv =

• Luv

if a′

uv > 0,

Uuv

• therwise.

Advantage: Violation remains unchanged during projection. Drawback: Separation procedures may need to be modified. Trivial modification case For a given class of inequalities, all nonzero coefficients have identical sign.

Thorsten Bonato Separation for Max-Cut on General Graphs 12 / 20

Adaptive Extension

Artificial LP values z′

uv adapt to the sign of the corresponding

coefficient in a given inequality a′Tx′ ≤ α′, i. e., z′

uv =

• Luv

if a′

uv > 0,

Uuv

• therwise.

Advantage: Violation remains unchanged during projection. Drawback: Separation procedures may need to be modified. Trivial modification case For a given class of inequalities, all nonzero 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

Thorsten Bonato Separation for Max-Cut on General Graphs 12 / 20

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.

Thorsten Bonato Separation for Max-Cut on General Graphs 13 / 20

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 aTyj ≤ 0, for all j = 1, . . . , t a ∈ Rm

Thorsten Bonato Separation for Max-Cut on General Graphs 13 / 20

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 aTyj ≤ 0, for all j = 1, . . . , t a ∈ Rm For max-cut we set Q = CUT

• G(W )
• for a subset W ⊆ V .

Thorsten Bonato Separation for Max-Cut on General Graphs 13 / 20

Adaptive Extension: Target Cuts (2/2)

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

Thorsten Bonato Separation for Max-Cut on General Graphs 14 / 20

Adaptive Extension: Target Cuts (2/2)

Modified input W.l.o.g. let the last ℓ vector entries correspond to the non-edges. z′ := (z1, . . . , zm−ℓ, L1, . . . , Lℓ, U1, . . . , Uℓ), x′

i := (xi1, . . . , xi,m−ℓ,

xi,m−ℓ+1, . . . , xim, xi,m−ℓ+1, . . . , xim), q′ := (q1, . . . , qm−ℓ, qm−ℓ+1, . . . , qm, qm−ℓ+1, . . . , qm), Q′ := conv {x′

1, . . . , x′ s} + cone {−em−ℓ+k, em+k | k = 1, . . . , ℓ}.

Thorsten Bonato Separation for Max-Cut on General Graphs 14 / 20

Adaptive Extension: Target Cuts (2/2)

Modified input W.l.o.g. let the last ℓ vector entries correspond to the non-edges. z′ := (z1, . . . , zm−ℓ, L1, . . . , Lℓ, U1, . . . , Uℓ), x′

i := (xi1, . . . , xi,m−ℓ,

xi,m−ℓ+1, . . . , xim, xi,m−ℓ+1, . . . , xim), q′ := (q1, . . . , qm−ℓ, qm−ℓ+1, . . . , qm, qm−ℓ+1, . . . , qm), Q′ := conv {x′

1, . . . , x′ s} + cone {−em−ℓ+k, em+k | k = 1, . . . , ℓ}.

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

a′ ∈ Rm+ℓ

Thorsten Bonato Separation for Max-Cut on General Graphs 14 / 20

Adaptive Extension: Target Cuts (2/2)

Modified input W.l.o.g. let the last ℓ vector entries correspond to the non-edges. z′ := (z1, . . . , zm−ℓ, L1, . . . , Lℓ, U1, . . . , Uℓ), x′

i := (xi1, . . . , xi,m−ℓ,

xi,m−ℓ+1, . . . , xim, xi,m−ℓ+1, . . . , xim), q′ := (q1, . . . , qm−ℓ, qm−ℓ+1, . . . , qm, qm−ℓ+1, . . . , qm), Q′ := conv {x′

1, . . . , x′ s} + cone {−em−ℓ+k, em+k | k = 1, . . . , ℓ}.

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

a′ ∈ Rm+ℓ In an optimum solution a′∗ at most one of a′∗

m−ℓ+k and a′∗ m+k can be

nonzero for each k = 1, . . . , ℓ.

Thorsten Bonato Separation for Max-Cut on General Graphs 14 / 20

Outline

1

Max-Cut Problem

2

Separation using Graph Contraction

3

Computational Results

Thorsten Bonato Separation for Max-Cut on General Graphs 15 / 20

Computational Experiments

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

Thorsten Bonato Separation for Max-Cut on General Graphs 16 / 20

Computational Experiments

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

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

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

Thorsten Bonato Separation for Max-Cut on General Graphs 16 / 20

Computational Experiments

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

1 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:

• dd-cycles (spanning-tree heuristic, 3-/4-cycles, exact separation).

Thorsten Bonato Separation for Max-Cut on General Graphs 16 / 20

Computational Experiments

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

1 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:

• dd-cycles (spanning-tree heuristic, 3-/4-cycles, exact separation).

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

Thorsten Bonato Separation for Max-Cut on General Graphs 16 / 20

Computational Experiments

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

1 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:

• dd-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, hypermetric inequalities and target cuts on the extended LP solution.

Thorsten Bonato Separation for Max-Cut on General Graphs 16 / 20

Unconstrained Quadratic 0/1-Optimization Problems

0.5 1 1.5 2 2.5 3 b250-1 b250-3 b250-5 b250-7 b250-9 Running time [h] Instance Running time of Beasley instances (250 nodes, density 0.1) Standard Contraction Extension [Intel Xeon 2.8 GHz, 8GB shared RAM.] Thorsten Bonato Separation for Max-Cut on General Graphs 17 / 20

Spin Glass Problems with Uniformly Distributed ±1-Weights

1s 1m 1h 10h 302 402 502 602 702 802 Average running time (log. scaling) 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.] Thorsten Bonato Separation for Max-Cut on General Graphs 18 / 20

Spin Glass Problems with Gaussian Distributed Integral Weights

1s 1m 1h 10h 402 602 802 1002 1202 1402 1602 1802 Average running time (log. scaling) 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.] Thorsten Bonato Separation for Max-Cut on General Graphs 19 / 20

Conclusion and Future Work

Separation using graph contraction Enables the use of separation techniques for dense/complete graphs on sparse graphs. Accelerates the exact solution of the max-cut problem for the examined classes of spin glass problems. Acceleration is mainly due to the use of contraction as heuristic

• dd-cycle separator.

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

Conclusion and Future Work

Separation using graph contraction Enables the use of separation techniques for dense/complete graphs on sparse graphs. Accelerates the exact solution of the max-cut problem for the examined classes of spin glass problems. Acceleration is mainly due to the use of contraction as heuristic

• dd-cycle separator.

Future work Develop special branching rules. Determine good parameter settings. Further computational experiments.

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

Conclusion and Future Work

Separation using graph contraction Enables the use of separation techniques for dense/complete graphs on sparse graphs. Accelerates the exact solution of the max-cut problem for the examined classes of spin glass problems. Acceleration is mainly due to the use of contraction as heuristic

• dd-cycle separator.

Future work Develop special branching rules. Determine good parameter settings. Further computational experiments. Thank you for your attention!

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