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

Separation for the Max-Cut Problem on 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) 14 th Combinatorial Optimization Workshop Aussois, January 6, 2010 Thorsten Bonato Separation for Max-Cut on General Graphs 1 / 20

Outline Max-Cut Problem 1 Separation using Graph Contraction 2 Computational Results 3 Thorsten Bonato Separation for Max-Cut on General Graphs 2 / 20

Outline Max-Cut Problem 1 Separation using Graph Contraction 2 Computational Results 3 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. S Any S ⊆ V induces a set δ ( S ) of edges with exactly one end node in S . The set δ ( S ) is δ ( S ) called a cut of G with shores S and V \ S . V \ 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. S Any S ⊆ V induces a set δ ( S ) of edges with exactly one end node in S . The set δ ( S ) is δ ( S ) called a cut of G with shores S and V \ S . V \ S Finding a cut with maximum aggregate edge weight is known as max-cut problem. 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( K 3 ) 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 CUT( K 3 ) described by two inequality classes: Odd-cycle: x ( F ) − x ( C \ F ) ≤ | F | − 1 , for each cycle C of G , for all F ⊆ C , | F | odd . Trivial: 0 ≤ x e ≤ 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 Max-Cut Problem 1 Separation using Graph Contraction 2 Computational Results 3 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 ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate 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 ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate 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 ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Contract 0-edges. z ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Contract 0-edges. z ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Project Extend ( a ′ , α ′ ) ′ z ′ G Separate 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 ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Project Extend ( a ′ , α ′ ) ′ z ′ G Separate 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 ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Separate extended LP solution. z ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Separate extended LP solution. z ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Project Extend ( a ′ , α ′ ) ′ z ′ G Separate 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 ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Extend Project ( a ′ , α ′ ) ′ z ′ G Separate 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 ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Extend Project ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Lift inequality. z ( c , γ ) ab Switch Un-switch (˜ c , ˜ γ ) ˜ z G dg e cf Contract Lift ( c , γ ) z G hi Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Lift inequality. z ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Switch lifted inequality. z ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction Switch lifted inequality. z ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate Thorsten Bonato Separation for Max-Cut on General Graphs 8 / 20

Outline of the Separation using Graph Contraction z ( c , γ ) a c b Switch Un-switch (˜ c , ˜ γ ) ˜ z G e d f Contract Lift ( c , γ ) z G g h i Project Extend ( a ′ , α ′ ) ′ z ′ G Separate 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 . uv Contraction of uv merges the edges uw and vw . 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 . uv Contraction of uv merges the edges uw and vw . If the LP values of the merged edges differ, w e. g., z uw > z vw Thorsten Bonato Separation for Max-Cut on General Graphs 9 / 20