A cutting plane algorithm for the max-cut problem

Simone, Caterina De; Rinaldi, Giovanni

doi:10.1080/10556789408805564

Cited by 29 publications

(10 citation statements)

References 7 publications

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“…From the above observation, in the case of the stable set problem it might be worthwhile tailoring the separation method to the case in which b i ∈ {0, −1, +1} for all i ∈ V . A separation heuristic for a subclass of these inequalities appears in De Simone & Rinaldi [4].…”

Section: Discussionmentioning

confidence: 99%

“…It is not difficult to show that the odd clique inequalities imply the W (p, q) inequalities with p = 2q + 1 (the odd holes) and p = 2q + 2, and that they do not imply any web inequalities with p > 4q. Using a computer program, we found that the odd clique inequalities also imply a few other web inequalities, such as the W (10, 3), W (13,4) and W (13,5) inequalities. We did not succeed in finding a full characterization, but in any case the odd clique inequalities are less productive than the general rounded psd inequalities.…”

Section: Corollary 2 There Exists a Polynomial-time Separation Algormentioning

confidence: 99%

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Exploring the Relationship Between Max-Cut and Stable Set Relaxations

Giandomenico

Letchford

2005

Math. Program.

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Abstract. The max-cut and stable set problems are two fundamental N P -hard problems in combinatorial optimization. It has been known for a long time that any instance of the stable set problem can be easily transformed into a max-cut instance. Moreover, Laurent, Poljak, Rendl and others have shown that any convex set containing the cut polytope yields, via a suitable projection, a convex set containing the stable set polytope. We review this work, and then extend it in the following ways. We show that the rounded version of certain 'positive semidefinite' inequalities for the cut polytope imply, via the same projection, a surprisingly large variety of strong valid inequalities for the stable set polytope. These include the clique, odd hole, odd antihole, web and antiweb inequalities, and various inequalities obtained from these via sequential lifting. We also examine a less general class of inequalities for the cut polytope, which we call odd clique inequalities, and show that they are, in general, much less useful for generating valid inequalities for the stable set polytope.As well as being of theoretical interest, these results have algorithmic implications. In particular, we obtain as a by-product a polynomial-time separation algorithm for a class of inequalities which includes all web inequalities.

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Section: Discussionmentioning

confidence: 99%

Section: Corollary 2 There Exists a Polynomial-time Separation Algormentioning

confidence: 99%

Exploring the Relationship Between Max-Cut and Stable Set Relaxations

Giandomenico

Letchford

2005

Math. Program.

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“…For some of these inequalities separation procedures have been proposed (see, e.g., [23], [11], and [24]). …”

Section: Reduction and Lifting For Separationmentioning

confidence: 99%

Branch-and-Cut Algorithms for Combinatorial Optimization and Their Implementation in ABACUS

Elf

Gutwenger

Jünger

et al. 2001

Lecture Notes in Computer Science

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Abstract. Branch-and-cut (-and-price) algorithms belong to the most successful techniques for solving mixed integer linear programs and combinatorial optimization problems to optimality (or, at least, with certified quality). In this unit, we concentrate on sequential branch-and-cut for hard combinatorial optimization problems, while branch-and-cut for general mixed integer linear programming is treated in [−→ Martin] and parallel branch-and-cut is treated in [−→ Ladányi/Ralphs/Trotter]. After telling our most recent story of a successful application of branchand-cut in Section 1, we give in Section 2 a brief review of the history, including the contributions of pioneers with an emphasis on the computational aspects of their work. In Section 3, the components of a generic branch-and-cut algorithm are described and illustrated on the traveling salesman problem. In Section 4, we first elaborate a bit on the important separation problem where we use the traveling salesman problem and the maximum cut problem as examples, then we show how branchand-cut can be applied to problems with a very large number of variables (branch-and-cut-and-price). Section 5 is devoted to the design and applications of the ABACUS software framework for the implementation of branch-and-cut algorithms. Finally, in Section 6, we make a few remarks on the solution of the exercise consisting of the design of a simple TSP-solver in ABACUS. Our Most Recent StoryBranch-and-cut has become a widely used method for the solution of hard integer or mixed integer problems. We refer to a recent survey of Caprara and Fischetti [12] for a view on its wide range of applications. In this chapter, the emphasis will be mostly on combinatorial optimization problems. Before going into a brief historical tour through the main algorithmic achievements that lead to or are connected with branch-and-cut, hoping to get the reader's interest before entering into more technical topics, we want to report on our most recent experience [29] with this method. The little story that follows is not only an example where branch-and-cut was quite useful, but shows also how combinatorial optimization can sometimes provide an excellent modeling tool to solve real world problems.

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“…The max cut problem is defined as the problem of finding the cut of maximum weight. It is well-known [1,4] that the Max Cut problem can be modeled as the following Problem PA.…”

Section: Max Cut Exponential Modelsmentioning

confidence: 99%