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24
On the complexity of the parity argument and other inefficient proofs of existence
 JCSS
, 1994
"... We define several new complexity classes of search problems, "between " the classes FP and FNP. These new classes are contained, along with factoring, and the class PLS, in the class TFNP of search problems in FNP that always have a witness. A problem in each of these new classes is defined in terms ..."
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Cited by 155 (8 self)
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We define several new complexity classes of search problems, "between " the classes FP and FNP. These new classes are contained, along with factoring, and the class PLS, in the class TFNP of search problems in FNP that always have a witness. A problem in each of these new classes is defined in terms of an implicitly given, exponentially large graph. The existence of the solution sought is established via a simple graphtheoretic argument with an inefficiently constructive proof; for example, PLS can be thought of as corresponding to the lemma "every dag has a sink. " The new classes are based on lemmata such as "every graph has an even number of odddegree nodes. " They contain several important problems for which no polynomial time algorithm is presently known, including the computational versions of Sperner's lemma, Brouwer's fixpoint theorem, Chfvalley's theorem, and the BorsukUlam theorem, the linear complementarity problem for Pmatrices, finding a mixed equilibrium in a nonzero sum game, finding a second Hamilton circuit in a Hamiltonian cubic graph, a second Hamiltonian decomposition in a quartic graph, and others. Some of these problems are shown to be complete. © 1994 Academic Press, Inc. 1.
LargeStep Markov Chains for the TSP Incorporating Local Search Heuristics
 Operations Research Letters
, 1992
"... We consider a new class of optimization heuristics which combine local searches with stochastic sampling methods, allowing one to iterate local optimization heuristics. We have tested this on the Euclidean Traveling Salesman Problem, improving 3opt by over 1.6% and LinKernighan by 1.3%. This wo ..."
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Cited by 58 (5 self)
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We consider a new class of optimization heuristics which combine local searches with stochastic sampling methods, allowing one to iterate local optimization heuristics. We have tested this on the Euclidean Traveling Salesman Problem, improving 3opt by over 1.6% and LinKernighan by 1.3%. This work was supported in part by the grants DOEFG0385ER25009 and NSFECS8909127, and by a grant from the PSCCUNY Research Award Program. Correspondence regarding this work should be addressed to S. Otto. y This manuscript was published in Operation Research Letters, v. 11, pp. 21924, 1992. 1 Introduction Given N cities labeled by i = 1; N , separated by distances d ij , the Traveling Salesman Problem (TSP) consists in finding the shortest tour, i.e., the shortest closed path visiting every city exactly once. To be specific, we will consider the symmetric TSP where d ij = d ji , but our method generalizes to the asymmetric case also. The problem of finding the optimal tour is a difficult...
Computational Study of a Family of MixedInteger Quadratic Programming Problems
 Mathematical programming
, 1995
"... . We present computational experience with a branchandcut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables. Such problems arise in financial applications. The algorithm solves the largest reallife problems in a few minutes of run ..."
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Cited by 46 (6 self)
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. We present computational experience with a branchandcut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables. Such problems arise in financial applications. The algorithm solves the largest reallife problems in a few minutes of runtime. 1 Introduction. We are interested in optimization problems QMIP of the form: min x T Qx + c T x s.t. Ax b (1) jsupp(x)j K (2) 0 x j u j ; all j (3) where x is an nvector, Q is a symmetric positivesemidefinite matrix, supp(x) = fj : x j ? 0g and K is a positive integer. Problems of this type are of interest in portfolio optimization. Briefly, variables in the problem correspond to commodities to be bought, the objective is a measure of "risk", the constraints (1) prescribe levels of "performance", and constraint (2) specifies that not too many 1 different types of commodities can be chosen. All data is derived from statistical information. A good deal of previous work ha...
Kernelization Algorithms for the Vertex Cover Problem: Theory and Experiments (Extended Abstract)
"... Faisal N. AbuKhzam + , Rebecca L. Collins + , Michael R. Fellows # , Michael A. Langston + , W. Henry Suters + and Chris T. Symons + 1 ..."
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Cited by 41 (13 self)
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Faisal N. AbuKhzam + , Rebecca L. Collins + , Michael R. Fellows # , Michael A. Langston + , W. Henry Suters + and Chris T. Symons + 1
WHEN DOES THE POSITIVE SEMIDEFINITENESS CONSTRAINT HELP IN LIFTING PROCEDURES?
, 2001
"... We study the liftandproject procedures of Lovász and Schrijver for 01 integer programming problems. We prove that the procedure using the positive semidefiniteness constraint is not better than the one without it, in the worst case. Various examples are considered. We also provide geometric condi ..."
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Cited by 29 (3 self)
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We study the liftandproject procedures of Lovász and Schrijver for 01 integer programming problems. We prove that the procedure using the positive semidefiniteness constraint is not better than the one without it, in the worst case. Various examples are considered. We also provide geometric conditions characterizing when the positive semidefiniteness constraint does not help.
Covering arrays and intersecting codes
 Journal of Combinatorial Designs
, 1993
"... A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rkn ..."
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Cited by 26 (0 self)
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A tcovering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once. Such arrays are used for example in circuit testing, and one wishes to minimize k for given values of n and t. The case t = 2 was solved by Rknyi, Katona, and Kleitman and Spencer. The present article is concerned with the case t = 3, where important (but unpublished) contributions were made by Busschbach and Roux in the 1980s. One of the principal constructions makes use of intersecting codes (linear codes with the property that any two nonzero codewords meet). This article studies the properties of 3covering arrays and intersecting codes, and gives a table of the best 3covering arrays presently known. For large n the minimal k satisfies 3.21256 < k / log n < 7.56444. 01993
Computational Experience with a Difficult MixedInteger Multicommodity Flow Problem
 MATHEMATICAL PROGRAMMING
, 1994
"... The following problem arises in the study of lightwave networks. Given a demand matrix containing amounts to be routed between corresponding nodes, we wish to design a network with certain topological features, and in this network, route all the demands, so that the maximum load (total flow) on any ..."
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Cited by 25 (1 self)
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The following problem arises in the study of lightwave networks. Given a demand matrix containing amounts to be routed between corresponding nodes, we wish to design a network with certain topological features, and in this network, route all the demands, so that the maximum load (total flow) on any edge is minimized. As we show, even small instances of this combined design/routing problem are extremely intractable. We describe computational experience with a cutting plane algorithm for this problem.
Scalable parallel algorithms for fpt problems
 Algorithmica
, 2006
"... Algorithmic methods based on the theory of fixedparameter tractability are combined with powerful computational platforms to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the NPhard vertex cover problem are comput ..."
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Cited by 22 (8 self)
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Algorithmic methods based on the theory of fixedparameter tractability are combined with powerful computational platforms to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the NPhard vertex cover problem are computed. To accomplish this, an efficient sequential algorithm and various forms of parallel algorithms are devised, implemented and compared. The importance of maintaining a balanced decomposition of the search space is shown to be critical to achieving scalability. Target problems need only be amenable to reduction and decomposition. Applications in high throughput computational biology are also discussed.
Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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Cited by 19 (9 self)
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
The Constraint Equations
 50 Years of the Cauchy Problem
, 2004
"... Initial data for solutions of Einstein’s gravitational field equations cannot be chosen freely: the data must satisfy the four Einstein constraint equations. We first discuss the geometric origins of the Einstein constraints and the role the constraint equations play in generating solutions of the f ..."
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Cited by 18 (1 self)
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Initial data for solutions of Einstein’s gravitational field equations cannot be chosen freely: the data must satisfy the four Einstein constraint equations. We first discuss the geometric origins of the Einstein constraints and the role the constraint equations play in generating solutions of the full system. We then discuss various ways of obtaining solutions of the Einstein constraint equations, and the nature of the space of solutions.