In an earlier paper (Mathematical Programming 43 (1989) 57–69) we characterized the class of facets of the set covering polytope defined by inequalities with coefficients equal to 0, 1 or 2. In this paper we connect ...
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In an earlier paper (Mathematical Programming 43 (1989) 57–69) we characterized the class of facets of the set covering polytope defined by inequalities with coefficients equal to 0, 1 or 2. In this paper we connect that characterization to the theory of facet lifting. In particular, we introduce a family of lower dimensional polytopes and associated inequalities having only three nonzero coefficients, whose lifting yields all the valid inequalities in the above class, with the lifting coefficients given by closed form expressions.
The simple graph partitioning problem is to partition an edge-weighted graph into mutually node-disjoint subgraphs, each containing at most b nodes, such that the sum of the weights of all edges in the subgraphs is ma...
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The simple graph partitioning problem is to partition an edge-weighted graph into mutually node-disjoint subgraphs, each containing at most b nodes, such that the sum of the weights of all edges in the subgraphs is maximal. In this paper we provide several classes of facet-defining inequalities for the associated simple graph partitioning polytope. (c) 2006 Elsevier B.V. All rights reserved.
We study the generalization to bipartite and 2-connected plane graphs of the Clar number, an optimization model proposed by Clar [E. Clar, The Aromatic Sextet, John Wiley & Sons, London, 1972] to compute indices o...
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We study the generalization to bipartite and 2-connected plane graphs of the Clar number, an optimization model proposed by Clar [E. Clar, The Aromatic Sextet, John Wiley & Sons, London, 1972] to compute indices of benzenoid hydrocarbons. Hansen and Zheng [P. Hansen, M. Zheng, The Clar number of a benzenoid hydrocarbon and linear programming, J. Math. Chem. 15 (1994) 93-107] formulated the Clar problem as an integer program and conjectured that solving the linear programming relaxation always yields integral solutions. We establish their conjecture by proving that the constraint matrix of the Clar integer program is always unimodular. Interestingly, in general these matrices are not totally unimodular. Similar results hold for the Fries number, an alternative index for benzenoids proposed earlier by Fries [K. Fries, Uber Byclische Verbindungen and ihren Vergleich mit dem Naphtalin, Ann. Chem. 454 (1927) 121-324]. (c) 2006 Elsevier Inc. All rights reserved.
The Graphical Traveling Salesman Polyhedron (GTSP) has been proposed by Naddef and Rinaldi to be viewed as a relaxation of the Symmetric Traveling Salesman Polytope (STSP). It has also been employed by Applegate, Bixb...
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The Graphical Traveling Salesman Polyhedron (GTSP) has been proposed by Naddef and Rinaldi to be viewed as a relaxation of the Symmetric Traveling Salesman Polytope (STSP). It has also been employed by Applegate, Bixby, Chvatal, and Cook for solving the latter to optimality by the branch-and-cut method. There is a close natural connection between the two polyhedra. Until now, it was not known whether there are facets in TT-form of the GTSP polyhedron which are not facets of the STSP polytope as well. In this paper we give an affirmative answer to this question for n >= 9. We provide a general method for proving the existence of such facets, at the core of which lies the construction of a continuous curve on a polyhedron. This curve starts in a vertex, walks along edges, and ends in a vertex not adjacent to the starting vertex. Thus there must have been a third vertex on the way.
Variable lower bounds in Mixed Integer Programs are constraints with the general form x greater than or equal to Ly, where x is a continuous variable and y is a binary or an integer variable. This type of constraints ...
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Variable lower bounds in Mixed Integer Programs are constraints with the general form x greater than or equal to Ly, where x is a continuous variable and y is a binary or an integer variable. This type of constraints is present in some Lot-Sizing models, where x represents the amount of some article produced by a machine and y is an element of {0, 1} indicates whether the machine is set-up for that article (y = 1) or not (y = 0). In these models, production below some level is not allowed, in order to make full use of resources. In this paper we study, from the polyhedral viewpoint, the mixed integer models whose feasible set includes an additive constraint Sigma x(j) less than or equal to (greater than or equal to, =) D, and constraints Ly(j) less than or equal to x(j) less than or equal to Ky(j), where x(j) are continuous variables, y(j) are integer or binary variables, and K is a large constant. We derive families of strong valid inequalities and show they are sufficient to describe completely the convex hulls of the sets of feasible solutions. Moreover, we develop polynomial algorithms to solve the separation problem associated to each of the families obtained. Finally, we incorporate the inequalities obtained in a cutting plane algorithm, and test their ability to solve some multi-item lot-sizing problems.
We study a variant of the weighted consecutive ones property problem. Here, a 0/1-matrix is given with a cost associated to each of its entries and one has to find a minimum cost set of zero entries to be turned to on...
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We study a variant of the weighted consecutive ones property problem. Here, a 0/1-matrix is given with a cost associated to each of its entries and one has to find a minimum cost set of zero entries to be turned to ones in order to make the matrix have the consecutive ones property for rows. We investigate polyhedral and combinatorial properties of the problem and we exploit them in a branch-and-cut algorithm. In particular, we devise preprocessing rules and investigate variants of "local cuts". We test the resulting algorithm on a number of instances, and we report on these computational experiments.
The max-cut and stable set problems are two fundamental NP-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 ma...
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The max-cut and stable set problems are two fundamental NP-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.
A (convex) polytope P is said to be 2-level if for each hyperplane H that supports a facet of P, the vertices of P can be covered with H and exactly one other translate of H. The study of these polytopes is motivated ...
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A (convex) polytope P is said to be 2-level if for each hyperplane H that supports a facet of P, the vertices of P can be covered with H and exactly one other translate of H. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d <= 7. Our approach is inductive: for each fixed (d - 1)-dimensional 2-level polytope P-0, we enumerate all d-dimensional 2-level polytopes P that have P-0 as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet P-0, we obtain all 2-level polytopes in dimension d.
This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constr...
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This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some other well-studied convex sets. Several classes of valid and facet-inducing inequalities are also derived.
To every linear extension L of a poset P = (P, <) we associate a 0, 1-vector x = x(L) with x(e) = 1 if and only if e is preceded by a jump in L or e is the first element in L. Let the setup polyhedron S = conv{x(L)...
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To every linear extension L of a poset P = (P, <) we associate a 0, 1-vector x = x(L) with x(e) = 1 if and only if e is preceded by a jump in L or e is the first element in L. Let the setup polyhedron S = conv{x(L): L is an element of (S)} be the convex hull of the incidence vectors of all linear extensions of P. For the case of series-parallel posets we solve the optimization problem over S and give a linear description of S.
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