The cut polytope and the Boolean quadric polytope

Simone, Caterina De

doi:10.1016/0012-365x(90)90056-n

Cited by 114 publications

(68 citation statements)

References 3 publications

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“…First assume that the objective function is quadratic. In case the problem is unconstrained, one can formulate it as a maximum cut problem on an associated graph [8]. Even in the presence of constraints, valid inequalities for the cut polytope remain valid for Q after transformation, and can be separated using the same transformation.…”

Section: Introductionmentioning

confidence: 99%

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Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

2010

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Abstract. In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP). Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming. Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvátal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastical speedup in the solution of constrained quadratic 0-1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions.

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Section: Introductionmentioning

confidence: 99%

“…However, the proposed methods can be adapted to general non-linear functions, as discussed later. The easiest example of a binary quadratic optimization problem is the maximum cut problem, which is equivalent to optimizing a degree-two polynomial over the hyper cube [8].…”

Section: Introductionmentioning

confidence: 99%

Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

2010

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“…The boolean quadric polytope is a fundamental problem in quadratic 0 − 1 optimization, and it is also an affine image of the so-called "cut polytope," which is the polytope associated with the well-known max-cut problem [1,5]. The boolean quadric and cut polytopes have been studied in great depth (see Deza and Laurent [6] for an extensive survey).…”

Section: A Connection With the Boolean Quadric Polytopementioning

confidence: 99%

Generalized network design polyhedra

Feremans¹,

Letchford²,

Salazar-González³

2011

Networks

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In recent years, there has been an increased literature on so-called generalized network design problems (GNDPs), such as the generalized minimum spanning tree problem and the generalized traveling salesman problem. In a GNDP, the node set of a graph is partitioned into "clusters," and the feasible solutions must contain one node from each cluster. Up to now, the polyhedra associated with different GNDPs have been studied independently. The purpose of this article is to show that it is possible, to a certain extent, to derive polyhedral results for all GNDPs simultaneously. Along the way, we point out some interesting connections to other polyhedra, such as the quadratic semiassignment polytope and the boolean quadric polytope.

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“…The latter is an instance of the unconstrained quadratic binary optimization problem, which is well-known to be equivalent to a maximum cut problem in some associated graph with an additional node [3]. In an undirected graph G = (V, E), the cut δ(W ) induced by a set W ⊆ V is defined as the set of edges (u, v) such that u ∈ W and v ∈ W .…”

Section: Related Workmentioning

confidence: 99%

“…Note that (LQLO) is a quadratic binary optimization problem where the feasible solutions need to satisfy further side constraints, namely those restricting the set of feasible solutions to linear orderings. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem [3], the task is to intersect a cut polytope with a set of hyperplanes.…”

Section: Bipartite Crossing Minimizationmentioning

confidence: 99%

Exact Bipartite Crossing Minimization under Tree Constraints

Baumann

Buchheim

Liers

2010

Experimental Algorithms

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Abstract. A tanglegram consists of a pair of (not necessarily binary) trees T1, T2 with leaf sets L1, L2. Additional edges, called tangles, may connect nodes in L1 with those in L2. The task is to draw the tanglegram with a minimum number of tangle edge crossings while making sure that no crossing occurs between edges within each tree. This problem has relevant applications in computational biology, e.g., for the comparison of phylogenetic trees. In this work, we show that the problem can be formulated as a quadratic linear ordering problem (QLO) with additional side constraints. In [2] it was shown that, appropriately reformulated, the QLO polytope is a face of some cut polytope. It turns out that the additional side constraints arising in our application do not destroy this property. Therefore, any polyhedral approach to max-cut can be used in our context. We present experimental results for drawing random and realistic tanglegrams on binary and on general trees. Using both linear and semidefinite programming techniques, we show that our approach is very efficient in practice.

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The cut polytope and the Boolean quadric polytope

Cited by 114 publications

References 3 publications

Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

Generalized network design polyhedra

Exact Bipartite Crossing Minimization under Tree Constraints

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