Results 1  10
of
24
Edmonds polytopes and a hierarchy of combinatorial problems
, 2006
"... Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integ ..."
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Cited by 143 (0 self)
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Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integers, by the inequality Σaj xj ≤ a with a ≥[a0]. Obviously, if integers x1,x2,...,xn satisfy all the inequalities in S, then they satisfy also all inequalities in the closure of S. Conversely, let Σcj xj ≤ c0 hold for all choices of integers x1,x2,...,xn, that satisfy all the inequalities in S. Then we prove that Σcj xj ≤ c0 belongs to the closure of S. To each integer linear programming problem, we assign a nonnegative integer, called its rank. (The rank is the minimum number of iterations of the operation (ii) that are required in order to eliminate the integrality constraint.) We prove that there is no upper bound on the rank of problems arising from the search for largest independent sets in graphs.
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 47 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
A constantfactor approximation algorithm for packet routing, and balancing local vs. global criteria
 In Proceedings of the ACM Symposium on the Theory of Computing (STOC
, 1997
"... Abstract. We present the first constantfactor approximation algorithm for a fundamental problem: the storeandforward packet routing problem on arbitrary networks. Furthermore, the queue sizes required at the edges are bounded by an absolute constant. Thus, this algorithmbalances a global criterio ..."
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Cited by 45 (4 self)
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Abstract. We present the first constantfactor approximation algorithm for a fundamental problem: the storeandforward packet routing problem on arbitrary networks. Furthermore, the queue sizes required at the edges are bounded by an absolute constant. Thus, this algorithmbalances a global criterion (routing time) with a local criterion (maximum queue size) and shows how to get simultaneous good bounds for both. For this particular problem, approximating the routing time well, even without considering the queue sizes, was open. We then consider a class of such local vs. global problems in the context of covering integer programs and show how to improve the local criterion by a logarithmic factor by losing a constant factor in the global criterion.
A FixedPoint Approach to Stable Matchings and Some Applications
, 2001
"... We describe a fixedpoint based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley [11], the MenelsohnDulmage theorem [21], the KunduLawler theorem [19], Ta ..."
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Cited by 30 (5 self)
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We describe a fixedpoint based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley [11], the MenelsohnDulmage theorem [21], the KunduLawler theorem [19], Tarski's fixed point theorem [32], the CantorBernstein theorem, Pym's linking theorem [22, 23] or the monochromatic path theorem of Sands et al. [29]. In this framework, we formulate a matroidgeneralization of the stable marriage theorem and study the lattice structure of generalized stable matchings. Based on the theory of lattice polyhedra and blocking polyhedra, we extend results of Vande Vate [33] and Rothblum [28] on the bipartite stable matching polytope.
Decomposition of Balanced Matrices
 J. COMBINATORIAL THEORY, SER. B
, 1999
"... A 0,1 matrix is balanced if it does not contain a square submatrix of odd order with two ones per row and per column. We show that a balanced 0,1 matrix is either totally unimodular or its bipartite representation has a cutset consisting of two adjacent nodes and some of their neighbors. This resul ..."
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Cited by 29 (5 self)
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A 0,1 matrix is balanced if it does not contain a square submatrix of odd order with two ones per row and per column. We show that a balanced 0,1 matrix is either totally unimodular or its bipartite representation has a cutset consisting of two adjacent nodes and some of their neighbors. This result yields a polytime recognition algorithm for balancedness. To prove the result, we first prove a decomposition theorem for balanced 0,1 matrices that are not strongly balanced.
On Wireless Spectrum Estimation and Generalized Graph Coloring
, 1998
"... We address the problem of estimating the spectrum required in a wireless network for a given demand and Graph Coloring interference pattern. This problem can be abstracted as a generalization of the graph coloring problem, which typically presents additional degree of hardness compared to the standa ..."
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Cited by 12 (0 self)
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We address the problem of estimating the spectrum required in a wireless network for a given demand and Graph Coloring interference pattern. This problem can be abstracted as a generalization of the graph coloring problem, which typically presents additional degree of hardness compared to the standard coloring problem. It is worthwhile to note that the question of estimating the spectrum requirement differs markedly from that of allocating channels. The main focus
The Generalized Minimum Spanning Tree Problem
, 2002
"... We consider the Generalized Minimum Spanning Tree Problem denoted by GMSTP. It is known that GMSTP is NPhard and even finding a near optimal solution is NPhard. We introduce a new mixed integer programming formulation of the problem which contains a polynomial number of constraints and a polynomia ..."
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Cited by 11 (3 self)
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We consider the Generalized Minimum Spanning Tree Problem denoted by GMSTP. It is known that GMSTP is NPhard and even finding a near optimal solution is NPhard. We introduce a new mixed integer programming formulation of the problem which contains a polynomial number of constraints and a polynomial number of variables. Based on this formulation we give an heuristic solution, a lower bound procedure and an upper bound procedure and present the advantages of our approach in comparison with an earlier method. We present a solution procedure for solving GMST problem using cutting planes.
A generalization of the perfect graph theorem under the disjunctive index
 Mathematics of Operations Research
"... In this paper we relate antiblocker duality between polyhedra, graph theory and the disjunctive procedure. In particular, we analyze the behavior of the disjunctive procedure over the clique relaxation, K(G), of the stable set polytope in a graph G and the one associated to its complementary graph ..."
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Cited by 6 (1 self)
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In this paper we relate antiblocker duality between polyhedra, graph theory and the disjunctive procedure. In particular, we analyze the behavior of the disjunctive procedure over the clique relaxation, K(G), of the stable set polytope in a graph G and the one associated to its complementary graph, K ( ¯ G). We obtain a generalization of the Perfect Graph Theorem proving that the disjunctive indices of K(G) andK ( ¯ G)always coincide.
Progress on Perfect Graphs
, 2003
"... A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restr ..."
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Cited by 6 (3 self)
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A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first
Duality, ainvariants and canonical modules of rings arising from linear optimization problems
 Bull. Math. Soc. Sci. Math. Roumanie (N.S
"... Abstract. The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and Gorenstein property—as well as the canonical module a ..."
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Cited by 3 (0 self)
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Abstract. The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and Gorenstein property—as well as the canonical module and the ainvariant—of Rees algebras and subrings arising from systems with the integer rounding property. We relate the algebraic properties of Rees algebras and monomial subrings with integer rounding properties and present a duality theorem. 1.