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37
On the Solution of Traveling Salesman Problems
 DOC. MATH. J. DMV
, 1998
"... Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TS ..."
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Cited by 226 (7 self)
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Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M. Grötschel , S. Hong , M. Jünger , P. Miliotis , D. Naddef , M. Padberg ... some of its refinements that led to the solution of a 13,509city instance.
CHVATAL CLOSURES FOR MIXED INTEGER PROGRAMMING PROBLEMS
, 1990
"... Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. T ..."
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Cited by 86 (0 self)
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Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cuttingplane proofs is that for a polyhedron P and integral vector w, if max(wx]x ~ P, wx integer} = t, then wx ~ t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that #x be integer for some other integral vector #. The cuttingplane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cuttingplane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvátal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.
MINIMUMWEIGHT TWOCONNECTED SPANNING NETWORKS
, 1990
"... We consider the problem of constructing a minimumweight, twoconnected network spanning all the points in a set V. We assume a symmetric, nonnegative distance function d ( ' ) defined on V x V which satisfies the triangle inequality. We obtain a structural characterization of optimal solution ..."
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Cited by 47 (3 self)
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We consider the problem of constructing a minimumweight, twoconnected network spanning all the points in a set V. We assume a symmetric, nonnegative distance function d ( ' ) defined on V x V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal twoconnected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thar any twoconnected graph satisfying these conditions is the unique optimal solution for a particular choice of 'canonical' distances satisfying the triangle inequality. we use these properties to show that the weight of an optimal traveling salesman cycle i, uiort f times the weight of an optimal twoconnected solution; examples are provided which approach this bound arbiirarily closely. In addition, we obtain similar results for the variation of thii problem where the network need only span a prespecified subset of the points.
WorstCase Comparison of Valid Inequalities for the TSP
 MATH. PROG
, 1995
"... We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor gr ..."
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Cited by 40 (1 self)
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We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10/9. The corresponding factor for the class of clique tree inequalities is 8/7, while it is 4/3 for the path configuration inequalities.
SCIP  a framework to integrate Constraint and Mixed Integer Programming
, 2005
"... Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today’s stateoftheart algorithms of both fields have several strategies in common, in particular the branchandbound process to recursively divide the problem in ..."
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Cited by 34 (2 self)
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Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today’s stateoftheart algorithms of both fields have several strategies in common, in particular the branchandbound process to recursively divide the problem into smaller subproblems. On the other hand, the main techniques to process each subproblem are different, and it was observed that they have complementary strengths. We present the programming framework Scip that integrates techniques from both fields in order to exploit the strengths of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, Scip does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, indepth control and high performance.
A New BranchandCut Algorithm for the Capacitated Vehicle Routing Problem
 Mathematical Programming
, 2003
"... We present a new branchandcut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomo ..."
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Cited by 27 (4 self)
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We present a new branchandcut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities, and classical Gomory mixedinteger cuts. For each of these classes of inequalities we descrine our separation algorithms in detail......
SEPARATING A SUPERCLASS OF COMB INEQUALITIES IN PLANAR GRAPHS
, 2000
"... Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedu ..."
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Cited by 24 (8 self)
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Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x∗ , nds one or more inequalities in the class which are violated by x , or proves that none exist. Such algorithms are at the core of the highly successful branchandcut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We de ne a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x ∗ e ¿0 is planar. The separation algorithm runs in O(n³) time, where n is the number of vertices in the graph.
On the Separation of Split Cuts and Related Inequalities
 MATHEMATICAL PROGRAMMING
, 2001
"... 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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Cited by 24 (3 self)
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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.
Separating Maximally Violated Comb Inequalities in Planar Graphs
 Math. Oper. Res
, 1997
"... The Traveling Salesman Problem (TSP) is a benchmark problem in combinatorial optimization. It was one of the very first problems used for developing and testing approaches to solving large integer programs, including cutting plane algorithms and branchandcut algorithms. Much of the research in thi ..."
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Cited by 14 (2 self)
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The Traveling Salesman Problem (TSP) is a benchmark problem in combinatorial optimization. It was one of the very first problems used for developing and testing approaches to solving large integer programs, including cutting plane algorithms and branchandcut algorithms. Much of the research in this area has been focused on finding new classes of facets for the TSP polytope, and much less attention has been paid to algorithms for separating from these classes of facets. In this paper, we consider the problem of finding violated comb inequalities. If there are no violated subtour constraints in a fractional solution of the TSP, a comb inequality may not be violated by more than 1. Given a fractional solution in the subtour elimination polytope whose graph is planar, we either find a violated comb inequality or determine that there are no comb inequalities violated by 1. Our algorithm runs in O(n + MC(n)) time, where MC(n) is the time to compute a cactus representation of all minimum cu...