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13
On the Separation of Split Cuts and Related Inequalities
 Mathematical Programming
"... The split cuts of Cook, Kannan and Schrijver are generalpurpose valid inequalities for integer programming which include a variety of other wellknown cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was ..."
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The split cuts of Cook, Kannan and Schrijver are generalpurpose valid inequalities for integer programming which include a variety of other wellknown cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was recently cited as an open problem by Cornuejols & Li [10]. In this paper we settle this question by proving strong NPcompleteness of separation for split cuts. As a byproduct we also show NPcompleteness of separation for several other classes of inequalities, including the MIRinequalities of Nemhauser and Wolsey and some new inequalities which we call balanced split cuts and binary split cuts. We also strengthen NPcompleteness results of Caprara & Fischetti [5] (for {0, 1 2 }cuts) and Eisenbrand [12] (for ChvatalGomory cuts). To compensate for this bleak picture, we also give a positive result for the Symmetric Travelling Salesman Problem. We show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle. Key words: Cutting planes, separation, complexity, travelling salesman problem, comb inequalities. 1
Algorithms for Maximum Independent Set Applied to Map Labelling
, 2000
"... We consider the following map labelling problem: given distinct points p 1 , p 2 , . . . , p n in the plane, and given #, find a maximum cardinality set of pairwise disjoint axisparallel # # squares Q 1 , Q 2 , . . . , Q r . This problem reduces to that of finding a maximum cardinality indepe ..."
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Cited by 15 (0 self)
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We consider the following map labelling problem: given distinct points p 1 , p 2 , . . . , p n in the plane, and given #, find a maximum cardinality set of pairwise disjoint axisparallel # # squares Q 1 , Q 2 , . . . , Q r . This problem reduces to that of finding a maximum cardinality independent set in an associated graph called the conflict graph. We describe several heuristics for the maximum cardinality independent set problem, some of which use an LP solution as input. Also, we describe a branchandcut algorithm to solve it to optimality. The standard independent set formulation has an inequality for each edge in the conflict graph which ensures that only one of its endpoints can belong to an independent set. To obtain good starting points for our LPbased heuristics and good upper bounds on the optimal value for our branchandcut algorithm we replace this set of inequalities by the set of inequalities describing all maximal cliques in the conflict graph. For this streng...
On the dominoparity inequalities for the STSP
"... One method which has been used very successfully for finding optimal and provably good solutions for large instances of the Symmetric Travelling Salesman Problem (ST SP) is the branch and cut method. This method requires knowledge of classes of useful valid inequalities for the polytope associated ..."
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Cited by 7 (0 self)
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One method which has been used very successfully for finding optimal and provably good solutions for large instances of the Symmetric Travelling Salesman Problem (ST SP) is the branch and cut method. This method requires knowledge of classes of useful valid inequalities for the polytope associated with the ST SP, as well as efficient separation routines for these classes of inequalities. Recently a new class of valid inequalities called the dominoparity inequalites were introduced for the ST SP. An efficient separation routine is known for these constraints if certain conditions are satisfied by the point to be separated. This separation routine has never been implemented or tested. We present several performance enhancements for this separation routine, and discuss our implementation of this improved algorithm. We test our implementation and provide results which we believe demonstrate the practical usefulness of these constraints and the separation routine for the ST SP within a branch and cut framework.
Binary Clutter Inequalities for Integer Programs
 Mathematical Programming
, 2003
"... We introduce a new class of valid inequalities for general integer linear programs, called binary clutter (BC) inequalities. They include the {0, 1/2}cuts of Caprara and Fischetti as a special case and have some interesting connections to binary matroids, binary clutters and Gomory corner polyhedra ..."
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Cited by 3 (1 self)
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We introduce a new class of valid inequalities for general integer linear programs, called binary clutter (BC) inequalities. They include the {0, 1/2}cuts of Caprara and Fischetti as a special case and have some interesting connections to binary matroids, binary clutters and Gomory corner polyhedra. We show that the separation problem...
Cutting Planes and the Elementary Closure in Fixed Dimension
, 1999
"... The elementary closure P 0 of a polyhedron P is the intersection of P with all its GomoryChvatal cutting planes. P 0 is a rational polyhedron provided that P is rational. The known bounds for the number of inequalities dening P 0 are exponential, even in xed dimension. We show that the numbe ..."
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Cited by 2 (0 self)
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The elementary closure P 0 of a polyhedron P is the intersection of P with all its GomoryChvatal cutting planes. P 0 is a rational polyhedron provided that P is rational. The known bounds for the number of inequalities dening P 0 are exponential, even in xed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in xed dimension. If P is a simplicial cone, we construct a polytope Q, whose integral elements correspond to cutting planes of P . The vertices of the integer hull Q I include the facets of P 0 . A polynomial upper bound on their number can be obtained by applying a result of Cook et al. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices of Q I .
Algorithms to Separate {0,1/2}ChvátalGomory Cuts
"... ChvátalGomory cuts are among the most wellknown classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or 1 1 2, such cuts are known as {0, 2}cuts. It has been proven by Caprara and Fischetti [8] that separation of {0, 1 2}cuts is NP ..."
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ChvátalGomory cuts are among the most wellknown classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or 1 1 2, such cuts are known as {0, 2}cuts. It has been proven by Caprara and Fischetti [8] that separation of {0, 1 2}cuts is NPhard. In this paper, we study ways to separate {0, 1 2}cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian eliminationlike procedure. To separate the most violated {0, 1 2}cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of stateoftheart ILPsolvers.
{0, 1/2}CUTS AND THE LINEAR ORDERING PROBLEM: SURFACES THAT DEFINE FACETS
"... We find new facetdefining inequalities for the linear ordering polytope generalizing the wellknown Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1/2}cuts produces triangulations of the Möbius band and of the corr ..."
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We find new facetdefining inequalities for the linear ordering polytope generalizing the wellknown Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1/2}cuts produces triangulations of the Möbius band and of the corresponding (closed) surface, the projective plane. In that sense, Möbius ladder inequalities have the same ‘shape’ as the projective plane. Inspired by the classification of surfaces, a classic result in topology, we prove that a surface has facetdefining {0, 1/2}cuts of the same ‘shape ’ if and only if it is nonorientable.
The Merchant Subtour Problem
, 2000
"... We consider the problem of a travelling merchant that makes money by buying commodities where they are cheap and selling them in other places where he can make a profit. The merchant ships commodities of his own choice in a van of fixed capacity. Given the prices of all the commodities in all of ..."
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We consider the problem of a travelling merchant that makes money by buying commodities where they are cheap and selling them in other places where he can make a profit. The merchant ships commodities of his own choice in a van of fixed capacity. Given the prices of all the commodities in all of the places, and the cost of driving from one place to another, the problem the merchant faces each day is to select a subset of the cities that he can visit in a day, and that allows him to maximise the profit he makes. We call this problem the Merchant Subtour Problem. The MSP models the pricing problem of a rather complex pickup and delivery problem that was given to us by the Dutch logistics company Van Gend & Loos. We show that a special case of the MSP has a totally unimodular constraint matrix. This knowledge enables us to develop a tabusearch algorithm for finding good feasible solutions to the MSP, and a branchandpriceandcut algorithm for solving the MSP to optimality. ...
On the Separation of Maximally Violated modk Cuts (Extended Abstract)
, 1998
"... Abstract Separation is of fundamental importance in cuttingplane based techniques for Integer Linear Programming (ILP). In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of wellstructured cuts. In this paper we add ..."
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Abstract Separation is of fundamental importance in cuttingplane based techniques for Integer Linear Programming (ILP). In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of wellstructured cuts. In this paper we address the separation of Chv'atal rank1 inequalities in the context of general ILP's of the form minfc T x : Ax b; x integerg, where A is an m \Theta n integer matrix and b an mdimensional integer vector. In particular, for any given integer k we study modk cuts of the form T Ax b T bc for any 2 f0; 1=k; : : : ; (k \Gamma 1)=kg m such that T A is integer. Following the line of research recently proposed for mod2 cuts by Applegate, Bixby, Chv'atal and Cook [1] and Fleischer and Tardos [14], we restrict to maximally violated cuts, i.e., to inequalities which are violated by (k \Gamma 1)=k by the given fractional point. We show that, for any k prime, such a separation require...