A note on the 2-circulant inequalities for the max-cut problem

Abstract The max-cut problem is a much-studied N P -hard combinatorial optimisation problem. Poljak and Turzik found some facet-defining inequalities for this problem, which we call 2-circulant inequalities. Two polynomial-time separation algorithms have been found for these inequalities, but one is very slow and the other is very complicated. We present a third algorithm, which is as fast as the faster of the existing two, but much simpler.

[1]  Bert Gerards,et al.  Matrices with the edmonds—Johnson property , 1986, Comb..

[2]  Michael Jünger,et al.  Lifting and separation procedures for the cut polytope , 2014, Math. Program..

[3]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[4]  G. Rinaldi,et al.  Computational approaches to Max-Cut , 2012 .

[5]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[6]  Bert Gerards.  Testing the Odd Bicycle Wheel Inequalities for the Bipartite Subgraph Polytope , 1985, Math. Oper. Res..

[7]  Manfred W. Padberg,et al.  The boolean quadric polytope: Some characteristics, facets and relatives , 1989, Math. Program..

[8]  Caterina De Simone,et al.  The cut polytope and the Boolean quadric polytope , 1990, Discret. Math..

[9]  C. Helmberg,et al.  Solving quadratic (0,1)-problems by semidefinite programs and cutting planes , 1998 .

[10]  Adam N. Letchford,et al.  Gap Inequalities for the Max-Cut Problem: A Cutting-Plane Algorithm , 2012, ISCO.

[11]  S. Poljak,et al.  On a positive semidefinite relaxation of the cut polytope , 1995 .

[12]  Ali Ridha Mahjoub,et al.  On the cut polytope , 1986, Math. Program..

[13]  Monique Laurent,et al.  Max-cut problem , 1997 .

[14]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[15]  Gérard Cornuéjols,et al.  Integer programming , 2014, Math. Program..

[16]  Sahar Karimi,et al.  Max-cut Problem , 2007 .

[17]  M. Grötschel,et al.  Combinatorial optimization , 1996 .

[18]  Adam N. Letchford,et al.  A new separation algorithm for the Boolean quadric and cut polytopes , 2014, Discret. Optim..

[19]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[20]  Adam N. Letchford.  On Disjunctive Cuts for Combinatorial Optimization , 2001, J. Comb. Optim..

[21]  Michael Jünger,et al.  Experiments in quadratic 0–1 programming , 1989, Math. Program..

[22]  Svatopluk Poljak,et al.  Max-cut in circulant graphs , 1992, Discret. Math..

[23]  Matteo Fischetti,et al.  {0, 1/2}-Chvátal-Gomory cuts , 1996, Math. Program..

[24]  Eddie Cheng.  Separating subdivision of bicycle wheel inequalities over cut polytopes , 1998, Oper. Res. Lett..

[25]  G. Rinaldi,et al.  A cutting plane algorithm for the max-cut problem. , 1992 .

[26]  Franz Rendl,et al.  Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations , 2009, Math. Program..