We present a new graph composition that produces a graph G from a given graph H and a fixed graph B called gear and we study its polyhedral properties. This composition yields counterexamples to a conjecture on the fa...
详细信息
We present a new graph composition that produces a graph G from a given graph H and a fixed graph B called gear and we study its polyhedral properties. This composition yields counterexamples to a conjecture on the facial structure of STAB(G) when G is claw-free. (C) 2008 Elsevier B.V. All rights reserved.
This paper studies two polytopes: the complete set packing and set partitioning polytopes, which are both associated with a binary n-row matrix having all possible columns. Cuts of rank 1 for the latter polytope play ...
详细信息
This paper studies two polytopes: the complete set packing and set partitioning polytopes, which are both associated with a binary n-row matrix having all possible columns. Cuts of rank 1 for the latter polytope play a central role in recent exact algorithms for many combinatorial problems, such as vehicle routing. We show the precise relation between the two polytopes studied, characterize the multipliers that induce rank 1 clique facets and give several families of multipliers that yield other facets. (C) 2018 Elsevier B.V. All rights reserved.
The stable set polytopeis a fundamental object in combinatorial optimization. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pecher and Wagler showed th...
详细信息
The stable set polytopeis a fundamental object in combinatorial optimization. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pecher and Wagler showed that the clique-family inequalities can be strengthened under certain conditions. We show that they can be strengthened even further, using a surprisingly simple mixed-integer rounding argument. (C) 2021 Elsevier B.V. All rights reserved.
We describe an implementation of a cutting plane algorithm for the minimum weight perfect 2-matching problem. This algorithm is based on Edmonds' complete description of the perfect 2-matching polytope and uses th...
详细信息
We describe an implementation of a cutting plane algorithm for the minimum weight perfect 2-matching problem. This algorithm is based on Edmonds' complete description of the perfect 2-matching polytope and uses the simplex algorithm for solving the LP-relaxations coming up. Cutting planes are determined by fast heuristics, or, if these fail, by an efficient implementation of the Padberg-Rao procedure, specialized for 2-matching constraints. With this algorithm 2-matching problems on complete graphs with up to 1000 nodes (i.e., 499,500 variables) have been solved in less than 1 hour CPU time on a medium speed computer.
We consider the problem of finding upper and lower bounds for the probability of the union of events when the probabilities of the single events and the probabilities of the intersections of up to m events are given. ...
详细信息
We consider the problem of finding upper and lower bounds for the probability of the union of events when the probabilities of the single events and the probabilities of the intersections of up to m events are given. It is known that the best possible bounds can be obtained by solving linear programming problems with a number of variables that is exponential in the number of events. Because of their size and structure, these large linear programs are known to be very hard to solve. In the literature simpler, polynomially sized aggregations are considered and numerous closed form or polynomially computable bounds are derived from those. We present here a new approach that introduces additional constraints to the dual linear programming problems in such a way that those become polynomially solvable. By using different sets of additional constraints, we introduce three new classes of polynomially computable upper and lower bounds. We show that they dominate almost all efficiently computable bounds known in the literature. Furthermore, by characterizing the vertices of two new classes of polyhedra, we can show that in two cases our bounds coincide with classical bounds, proving new extremal properties for those well-known bounds. Finally, we provide extensive numerical results comparing the average tightness of the various bounds on a large number of instances.
The sequential ordering problem (SOP) is the generalisation of the asymmetric travelling salesman problem in which there are precedence relations between pairs of nodes. Hernandez & Salazar introduced a multi comm...
详细信息
The sequential ordering problem (SOP) is the generalisation of the asymmetric travelling salesman problem in which there are precedence relations between pairs of nodes. Hernandez & Salazar introduced a multi commodity flow (MCF) formulation for a generalisation of the SOP in which the vehicle has a limited capacity. We strengthen this MCF formulation by fixing variables and adding valid equations. We then use polyhedral projection, together with some known results on flows, cuts and metrics, to derive new families of strong valid inequalities for both problems. Finally, we give computational results, which show that our findings yield good lower bounds in practice. (C) 2015 Elsevier B.V. All rights reserved.
The 1-wheel inequalities for the stable set polytope were introduced by Cheng and Cunningham. In general, there is an exponential number of these inequalities. We present a new polynomial size extended formulation of ...
详细信息
The 1-wheel inequalities for the stable set polytope were introduced by Cheng and Cunningham. In general, there is an exponential number of these inequalities. We present a new polynomial size extended formulation of the stable set relaxation that includes the odd cycle and 1-wheel inequalities. This compact formulation allows one to polynomially optimize over a polyhedron instead of handling the separation problem for 1-wheel inequalities by solving many shortest walk problems and relying on the ellipsoid method.
Recently, Buchheim and Klein (Discrete Appl Math 177:34-52, 2014) suggested to study polynomial-time solvable optimisation problems with linear objective functions combined with exactly one additional quadratic monomi...
详细信息
Recently, Buchheim and Klein (Discrete Appl Math 177:34-52, 2014) suggested to study polynomial-time solvable optimisation problems with linear objective functions combined with exactly one additional quadratic monomial. They concentrated on special quadratic spanning tree or forest problems. We extend their results to general matroid optimisation problems with a set of nested monomials in the objective function. We study polytopes arising from the standard linearisation of the monomials. Our results provide insight on the polyhedral structure of matroid optimisation problems with arbitrary polynomial objective function, with a focus on separation algorithms and strengthened cutting planes. Extending results by Edmonds (Comb Struct Appl, 69-87, 1970) for the matroid polytope we present a complete description for the linearised polytope. Indeed, apart from the standard linearisation one needs appropriately strengthened rank inequalities satisfying certain non-separability conditions. The separation problem of these extended rank inequalities reduces to a submodular function minimisation problem. In the case of exactly one additional non-linear monomial we completely characterise the facetial structure of the associated polytope.
This study describes certain facet classes for the planar subgraph polytope. These facets are extensions of Kuratowski facets and are of the form 2x(U) + x(E(G)\U) < 2 vertical bar U vertical bar+vertical bar E(G)\...
详细信息
This study describes certain facet classes for the planar subgraph polytope. These facets are extensions of Kuratowski facets and are of the form 2x(U) + x(E(G)\U) < 2 vertical bar U vertical bar+vertical bar E(G)\U vertical bar-2 where the edge set U varies and can be empty. Two of the new types of facets complete the class of extended subdivision facets, explored by Junger and Mutzel. In addition, the other types of facets consist of a new class of facets for the polytope called 3-star subdivisions. It is also shown that the extended and 3-star subdivision facets are also equivalent to members of the class of facets with coefficients in (0, 1, 2) for the set covering polytope. Computational results displaying the effectiveness of the facets in a branch-and-cut scheme for the maximum planar subgraph problem are presented. (c) 2007 Wiley Periodicals, Inc.
暂无评论