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

An implementation of a combinatorial approximation algorithm...

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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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integer programs with prescribed number of solutions and a weighted version of Doignon-bell-scarf's theorem

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17th international conference on integer programming and combinatorial optimization, ipco 2014

作者： Aliev, Iskander De Loera, Jesús A. Louveaux, Quentin Cardiff University United Kingdom University of California Davis United States Université de Liège Belgium

ISBN: (纸本)9783319075563

In this paper we study a generalization of the classical feasibility problem in integer linear programming, where an ILP needs to have a prescribed number of solutions to be considered solved. We first provide a generalization of the famous Doignon-Bell-Scarf theorem: Given an integer k, we prove that there exists a constant c(k,n), depending only on the dimension n and k, such that if a polyhedron {x: Ax≤b} contains exactly k integer solutions, then there exists a subset of the rows of cardinality no more than c(k,n), defining a polyhedron that contains exactly the same k integer solutions. the second contribution of the article presents a structure theory that characterizes precisely the set sg ≥ k (A) of all vectors b such that the problem Ax = b, x ≥ 0, x Ε Zn, has at least k-solutions. We demonstrate that this set is finitely generated, a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computation. Similar results can be derived for those right-hand-side vectors that have exactly k solutions or fewer than k solutions. Finally we show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of sg ≥ (A) as a generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors that have exactly k solutions (similarly for at least k or less than k solutions). Under the same assumptions we prove that the k-Frobenius number can be computed in polynomial time. © 2014 Springer international Publishing Switzerland.

关键词： Bells

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1-sparsity Approximation Bounds for Packing integer Programs 20th

1-sparsity Approximation Bounds for Packing Integer Programs

引用

20th international conference on integer programming and combinatorial optimization, ipco 2019

作者： Chekuri, Chandra Quanrud, Kent Torres, Manuel R. University of Illinois at Urbana-Champaign UrbanaIL United States

ISBN: (纸本)9783030179526

We consider approximation algorithms for packing integer programs (PIPs) of the form where c, A, and b are nonnegative. We let denote the width of A which is at least 1. Previous work by Bansal et al. [1] obtained an -approximation ratio where is the maximum number of nonzeroes in any column of A (in other words the -column sparsity of A). they raised the question of obtaining approximation ratios based on the -column sparsity of A (denoted by) which can be much smaller than Motivated by recent work on covering integer programs (CIPs) [4, 7] we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. [1] (but with a twist), yield approximation ratios for PIPs based on First, following an integrality gap example from [1], we observe that the case of is as hard as maximum independent set even when In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width where we obtain an -approximation. In the large width regime, when we obtain an -approximation. We also obtain a -approximation when. © 2019, Springer Nature Switzerland AG.

关键词： Approximation algorithms

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Solving the convex cost integer dual network flow problem 1

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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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Generating cuts from multiple-term disjunctions

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8th international integer programming and combinatorial optimization conference, ipco 2001

作者： Perregaard, Michael Balas, Egon Carnegie Mellon University PittsburghPA United States

ISBN: (纸本)3540422250

the traditional approach towards generating lift-and-project cuts involves solving a cut generating linear program (CGLP) that grows prohibitively large if a multiple-term disjunction is used instead of the classical dichotomy. In this paper we present an alternative approach towards generating lift-and-project cuts that solves the cut generation problem in the original space (i.e. does not introduce any extra variables). We present computational results that show our method to be superior to the classical approach. As soon as the number of terms in the disjunction grows larger than 4, the benefit of not solving the large CGLP directly is clear, and the gap grows larger as more terms are included in the disjunction. © Springer-Verlag Berlin Heidelberg 2001.

关键词： Linear programming

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On preemptive resource constrained scheduling: Polynomial-time approximation schemes

On preemptive resource constrained scheduling: Polynomial-ti...

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9th international integer programming and combinatorial optimization conference, ipco 2002

作者： Jansen, Klaus Porkolab, Lorant Institut für Informatik und praktische Mathematik Christian-Albrechts-Universität zu Kiel Kiel24 098 Germany PricewaterhouseCoopers London United Kingdom

ISBN: (纸本)9783540478676

We study resource constrained scheduling problems where the objective is to compute feasible preemptive schedules minimizing the makespan and using no more resources than what are available. We present approximation schemes along with some inapproximibility results showing how the approximability of the problem changes in terms of the number of resources. the results are based on linear programming formulations (though with exponentially many variables) and some interesting connections between resource constrained scheduling and (multidimensional, multiple-choice, and cardinality constrained) variants of the classical knapsack problem. In order to prove the results we generalize a method by Grigoriadis et al. for the max-min resource sharing problem to the case with weak approximate block solvers (i.e. with only constant, logarithmic, or even worse approximation ratios). Finally we present applications of the above results in fractional graph coloring and multiprocessor task scheduling. © Springer-Verlag Berlin Heidelberg 2002.

关键词： Linear programming

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An Approximate Solution of a GNSS Satellite Selection Problem Using Semidefinite programming 1

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10th international conference on optimization and Applications (OPTIMA)

作者： Rapoport, Lev Tormagov, Timofey Russian Acad Sci VA Trapeznikov Inst Control Sci Moscow Russia Moscow Inst Phys & Technol Dolgoprudnyi Moscow Region Russia

ISBN: (数字)9783030386030

ISBN: (纸本)9783030386030;9783030386023

When processing multiple navigation satellite systems, including GPS, GLONASS, Galileo, Beidou, QZSS, the overall number of the pseudorange and carrier phase signals can exceed several tens. On the other hand, a much smaller number of them is usually sufficient to achieve necessary precision of positioning. Also, some parts of precise positioning algorithms, like carrier phase ambiguity resolution, are very sensitive to the problem dimension as they include the integer search. To reduce computational cost of positioning, the optimal choice of signals involved in computations should be performed. optimization is constrained by a given number of satellite signals to be chosen for processing. this optimization problem falls into the class of binary optimization problems which are hard for precise solution. In this paper, we present approaches to an approximate solution of the optimal selection problem. After the linear relaxation of binary constraints, the relaxed problem is convex and can be transformed to semidefinite programming or second-order cone programming problems. the optimal solution of the relaxed problem can be considered as a lower bound of a combinatorial optimization problem. After rounding non-integer variables the approximate solution is obtained. As a result, two-sided bounds of the optimum are obtained. In practice, the approximate solution is very close to precise solution for most real world cases. Because the relaxed problem is convex, it can be solved efficiently.

关键词： GNSS navigation Second-order cone programming Semidefinite programming Satellite selection GDOP

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A polyhedral study of the cardinality constrained knapsack problem

A polyhedral study of the cardinality constrained knapsack p...

引用

9th international conference on integer programming and combinatorial optimization, ipco 2002

作者： De Farias Jr., Ismael R. Nemhauser, George L. CORE 34 Voie du Roman Pays 1348 Louvain-la-Neuve Belgium School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta GA 30332 United States

ISBN: (纸本)9783540478676

A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. this structure occurs, for example, in areas such as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems. © 2002 Springer-Verlag Berlin Heidelberg.

关键词： combinatorial optimization

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Separable concave optimization approximately equals piecewise linear optimization

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10th international conference on integer programming and combinatorial optimization, ipco 2004

作者： Magnanti, thomas L. Stratila, Dan School of Engineering and Sloan School of Management Massachusetts Institute of Technology 77 Massachusetts Avenue CambridgeMA02139-4307 United States Operations Research Center Massachusetts Institute of Technology 77 Massachusetts Avenue CambridgeMA02139-1309 United States

ISBN: (纸本)3540221131

We show how to approximate a separable concave minimization problem over a general closed ground set by a single piecewise linear minimization problem. the approximation is to arbitrary 1+ϵ precision in optimal cost. For polyhedral ground sets in n+ and nondecreasing cost functions, the number of pieces is polynomial in the input size and proportional to 1/ log(1+ ϵ). For general polyhedra, the number of pieces is polynomial in the input size and the size of the zeroes of the concave objective components. We illustrate our approach on the concave-cost uncapacitated multicommodity flow problem. By formulating the resulting piecewise linear approximation problem as a fixed charge, mixed integer model and using a dual ascent solution procedure, we solve randomly generated instances to within five to twenty percent of guaranteed optimality. the problem instances contain up to 50 nodes, 500 edges, 1,225 commodities, and 1,250,000 flow variables. © Springer-Verlag Berlin Heidelberg 2004.

关键词： Cost functions

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Robust branch-and-cut-and-price for the capacitated vehicle routing problem

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10th international conference on integer programming and combinatorial optimization, ipco 2004

作者： Fukasawa, Ricardo Lysgaard, Jens De Aragão, Marcus Poggi Reis, Marcelo Uchoa, Eduardo Werneck, Renato F. School of Industrial and Systems Engineering GeorgiaTech United States Department of Management Science and Logistics Aarhus School of Business Denmark Departamento de Informática PUC Rio de Janeiro Brazil Departamento de Engenharia de Produção Universidade Federal Fluminense Brazil Department of Computer Science Princeton University United States

ISBN: (纸本)3540221131

the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) have been based on either branch-and-cut or Lagrangean relaxation/column generation. this paper presents an algorithm that combines both approaches: it works over the intersection of two polytopes, one associated with a traditional Lagrangean relaxation over q-routes, the other defined by bound, degree and capacity constraints. this is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods. the resulting branch-andcut- and-price algorithm can solve to optimality all instances from the literature with up to 135 vertices. this doubles the size of the instances that can be consistently solved. © Springer-Verlag Berlin Heidelberg 2004.

关键词： integer programming

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