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Optimal 3-Terminal cuts and linear programming 1

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7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Cunningham, William H. Tang, Lawrence Department of Combinatorics and Optimization University of Waterloo WaterlooONN2L 3G1 Canada Department of Mathematics University of British Columbia VancouverBCV6T 1Y8 Canada

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

Given an undirected graph G = (V, E) and three specified terminal nodes t1, t2, t3, a 3-cutis a subset A of Esuch that no two terminals are in the same component of G\A. If a non-negative edge weight ce is specified for each e ∈ E, the optimal 3-cut problemis to find a 3-cut of minimum total weight. this problem is NP-hard, and in fact, is max-SNP-hard. An approximation algorithm having performance guarantee has recently been given by Călinescu, Karloff, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3-cut has weight at most times the optimal LP value. It is proved here that can be improved to, and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3-cut problem having performance guarantee 12/11. © Springer-Verlag Berlin Heidelberg 1999.

关键词： Approximation algorithms

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Parity constrained κ-Edge-connected orientations 7th

引用

7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Frank, András Király, Zoltán Department of Operations Research Eötvös University Rákóczi út 5 BudapestH-1088 Hungary Ericsson Traffic Laboratory Laborc u. 1 BudapestH-1037 Hungary Department of Computer Science Eötvös University Rákóczi út 5 BudapestH-1088 Hungary

ISBN: (纸本)3540660194

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. the main result is a characterization of undirected graphs G = (V, E) having a κ-edge-connected T-odd orientation for every subset T ⊆ V with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2κ + 2)-edge-connected graph with |V| + |E| even has a κ-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having κ + 1 edge-disjoint paths from s to every node and κ edge-disjoint paths from every node to s. © Springer-Verlag Berlin Heidelberg 1999.

关键词： combinatorial optimization

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Valid inequalities for problems with additive variable upper bounds 1

引用

7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Atamtürk, Alper Nemhauser, George L. Savelsbergh, Martin W. P. Department of Industrial Engineering and Operations Research University of California BerkeleyCA94720-1777 United States School of Industrial and Systems Engineering Georgia Institute of Technology AtlantaGA30332-0205 United States

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

We study the facial structure of a polyhedron associated with the single node relaxation of network flow problems with additive variable upper bounds. this type of structure arises, for example, in network design/expansion problems and in production planning problems with setup times. We first derive two classes of valid inequalities for this polyhedron and give the conditions under which they are facet-defining. then we generalize our results through sequence independent lifting of valid inequalities for lower-dimensional projections. Our computational experience with large network expansion problems indicates that these inequalities are very effective in improving the quality of the linear programming relaxations. © Springer-Verlag Berlin Heidelberg 1999.

关键词： Linear programming

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Solving the convex cost integer dual network flow problem 1

引用

7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Ahuja, Ravindra K. Hochbaum, Dorit S. Orlin, James B. Industrial and Systems Engineering University of Florida GainesvilleFL32611 United States Department of IE and OR and Haas School of Management University of California BerkeleyCA94720 United States Sloan School of Management MIT CambridgeMA02139 United States

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

In this paper, we consider a convex optimization problem where the objective function is the sum of separable convex functions, the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the form π(i) - π(j) ≤ wij), and the variables are required to take integer values within a specified range bounded by an integer U. Let m denote the number of constraints and (n+m) denote the number of variables. We call this problem the convex cost integer dual network flow problem. In this paper, we develop network flow based algorithms to solve the convex cost integer dual network flow problem efficiently. We show that using the Lagrangian relaxation technique, the convex cost integer dual network flow problem can be reduced to a convex cost primal network flow problem where each cost function is a piecewise linear convex function with integer slopes. We next show that the cost scaling algorithm for the minimum cost flow problem can be adapted to solve the convex cost integer dual network flow problem in O(nm log(n2/m) log(nU)) time. this algorithm improves the best currently available algorithm and is also likely to yield algorithms with attractive empirical performance. © Springer-Verlag Berlin Heidelberg 1999.

关键词： Cost functions

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Search strategies for hybrid search spaces

Search strategies for hybrid search spaces

引用

international conference on Tools for Artificial Intelligence (ICTAI)

作者： C. Gomes B. Selman Department of Computer Science Cornell University Ithaca NY USA

Recently, there has been much interest in enhancing purely combinatorial formalisms with numerical information. For example, planning formalisms can be enriched by taking resource constraints and probabilistic information into account. the mixed integer programming (MIP) paradigm from operations research provides a natural tool for solving optimization problems that combine such numeric and non-numeric information. the MIP approach relies heavily on linear program relaxations and branch-and-bound search. this is in contrast with depth-first or iterative deepening strategies generally used in AI. We provide a detailed characterization of the structure of the underlying search spaces as explored by these search strategies. Our analysis indicates that the traditional approach of identifying dominating search strategies for a given problem domain is inadequate. We show that much can be gained from combining search strategies for solving hard MIP problems, thereby leveraging the strength of different search strategies regarding both the combinatorial and numeric components of the problem.

关键词： Artificial intelligence Linear programming Read only memory Constraint optimization Computer science Vehicles combinatorial mathematics Operations research Ear Relaxation methods

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Consecutive ones and a betweenness problem in computational biology 6th

Consecutive ones and a betweenness problem in computational ...

引用

6th international conference on integer programming and combinatorial optimization (ipco VI)

作者： Christof, T Oswald, M Reinelt, G Heidelberg Univ Inst Angew Math D-69120 Heidelberg Germany

ISBN: (纸本)354064590X

In this paper we consider a variant of the betweenness problem occurring in computational biology. We present a new polyhedral approach which incorporates the solution of consecutive ones problems and show that it supersedes an earlier one. A particular feature of this new branch-and-cut algorithm is that it is not based on an explicit integer programming formulation of the problem and makes use of automatically generated facet-defining inequalities.

关键词： integer programming

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Solving integer and disjunctive programs by lift and project 6th

Solving integer and disjunctive programs by lift and project

引用

6th international conference on integer programming and combinatorial optimization (ipco VI)

作者： Ceria, S Pataki, G Columbia Univ Grad Sch Business New York NY 10027 USA Columbia Univ Comp Optimizat Res Ctr New York NY 10027 USA Columbia Univ Dept Ind Engn & Operat Res New York NY 10027 USA

ISBN: (纸本)354064590X

We extend the theoretical foundations of the branch-and-cut method using lift-and-project cuts for a broader class of disjunctive constraints, and also present a new, substantially improved disjunctive cut generator. Employed together with an efficient commercial MIP solver, our code is a robust, general purpose method for solving mixed integer programs. We present extensive computational experience with the most difficult problems in the MIPLIB library.

关键词： integer programming

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Approximating disjoint-path problems using greedy algorithms and packing integer programs 6th

Approximating disjoint-path problems using greedy algorithms...

引用

6th international integer programming and combinatorial optimization conference, ipco 1998

作者： Kolliopoulos, Stavros G. Stein, Clifford Dartmouth College Department of Computer Science HanoverNH03755-3510 United States

ISBN: (纸本)354064590X

the edge and vertex-disjoint path problems together with their unsplittable flow generalization are NP-hard problems with a multitude of applications in areas such as routing, scheduling and bin packing. Given the hardness of the problems, we study polynomial-time approximation algorithms with bounded performance guarantees. We introduce techniques which yield new algorithms for a wide range of disjoint-path problems. We use two basic techniques. First, we propose simple greedy algorithms for edge- and vertex-disjoint paths and second, we propose the use of a framework based on packing integer programs for more general problems such as unsplittable flow. As part of our tools we develop improved approximation algorithms for a class of packing integer programs, a result that we believe is of independent interest. © Springer-Verlag Berlin Heidelberg 1998.

关键词： Approximation algorithms

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On the relationship between combinatorial and LP-based approaches to NP-hard scheduling problems 6th

On the relationship between combinatorial and LP-based appro...

引用

6th international integer programming and combinatorial optimization conference, ipco 1998

作者： Uma, R.N. Wein, Joel Department of Computer Science Polytechnic University BrooklynNY11201 United States

ISBN: (纸本)354064590X

Enumerative approaches, such as branch-and-bound, to solv- ing optimization problems require a subroutine that produces a lower bound on the value of the optimal solution. In the domain of scheduling problems the requisite lower bound has typically been derived from either the solution to a linear-programming relaxation of the problem or the solution of a combinatorial relaxation. In this paper we investigate, from both a theoretical and practical perspective, the relationship between several linear-programming based lower bounds and combinatorial lower bounds for two scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled. We establish a number of facts about the relationship between these dif- ferent sorts of lower bounds, including the equivalence of certain linear- programming-based lower bounds for both of these problems to combi- natorial lower bounds used in successful branch-and-bound algorithms. As a result we obtain the first worst-case analysis of the quality of the lower bound delivered by these combinatorial relaxations. We then give an empirical evaluation of the strength of the various lower bounds and heuristics. this extends and puts in a broader context a recent experimental evaluation by Savelsbergh and the authors of the empirical strength of both heuristics and lower bounds based on different LP-relaxations of a single-machine scheduling problem. We observe that on most kinds of synthetic data used in experimental studies a simple heuristic, used in successful combinatorial branch-and-bound algorithms for the problem, outperforms on average all of the LP-based heuristics. However, we identify other classes of problems on which the LP-based heuristics are superior, and report on experiments that give a qualitative sense of the range of dominance of each. Finally, we consider the impact of local improvement on the solutions. © Springer-Verlag Berlin Heidelberg 1998.

关键词： Linear programming

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An implementation of a combinatorial approximation algorithm for minimum-cost multicommodity flow 6th

An implementation of a combinatorial approximation algorithm...

引用

6th international integer programming and combinatorial optimization conference, ipco 1998

作者： Goldberg, Andrew V. Oldham, Jeffrey D. Plotkin, Serge Stein, Cliff NEC Research Institute Inc PrincetonNJ08540 United States Department of Computer Science Stanford University StanfordCA94305-9045 United States Department of Computer Science Dartmouth College HanoverNH03755 United States

ISBN: (纸本)354064590X

the minimum-cost multicommodity flow problem involves simultaneously shipping multiple commodities through a single network so that the total flow obeys arc capacity constraints and has minimum cost. Multicommodity flow problems can be expressed as linear programs, and most theoretical and practical algorithms use linear-programming algorithms specialized for the problems' structures. combinatorial approximation algorithms in [GK96, KP95b, PST95] yield flows with costs slightly larger than the minimum cost and use capacities slightly larger than the given capacities. theoretically, the running times of these algorithms are much less than that of linear-programming-based algorithms. We combine and modify the theoretical ideas in these approximation algorithms to yield a fast, practical implementation solving the minimumcost multicommodity flow problem. Experimentally, the algorithm solved our problem instances (to 1% accuracy) two to three orders of magnitude faster than the linear-programming package CPLEX [CPL95] and the linear-programming based multicommodity flow program PPRN [CN96]. © Springer-Verlag Berlin Heidelberg 1998.

关键词： Approximation algorithms

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