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NETWORKS 2004年第1期44卷 1-11页

作者： Caprara, A Panconesi, A Rizzi, R Univ Bologna DEIS I-40136 Bologna Italy Univ Bologna Dipartimento Informat I-40127 Bologna Italy Aarhus Univ Dept Comp Sci BRICS DK-8000 Aarhus Denmark

We address the problem of finding the largest collection of edge-disjoint cuts in an undirected graph, dubbed CUT PACKING, focusing on its complexity, about which very little is known. We show a very close relationship with INDEPENDENT SET, namely, for the same graph G, the size of the largest cut packing of G is at least the independence number of G, and at most twice that number. This implies that any approximation guarantee for INDEPENDENT SET immediately extends to CUT PACKING within a factor of 2. In particular, this yields a 2-approximation algorithm for CUT PACKING in perfect graphs. We then present polynomial-time algorithms for several classes of perfect (and related) graphs, including triangulated graphs and their complements, bipartite graphs and their complements, and Seymour graphs. Finally, we discuss various linear programming relaxations for the problem, finding combinatorial dual problems Of CUT PACKING and characterizing the cases in which duality is strong. (C) 2004 Wiley Periodicals, Inc.

关键词： packing edge-disjoint cuts independent set complexity approximation linear programming relaxation

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Minimizing the stabbing number of matchings, trees, and triangulations 04

Minimizing the stabbing number of matchings, trees, and tria...

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Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms

作者： Sándor P. Fekete Marco E. Lübbecke Henk Meijer Braunschweig University of Technology Braunschweig Germany Technische Universität Berlin Berlin Germany Queen's University Kingston Ontario Canada

ISBN: (纸本)9780898715583

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. We investigate problems of finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open problem; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'*** show that minimum stabbing problems are NP-complete. We also show that an iterated rounding technique is applicable for matchings and spanning trees of minimum stabbing number by showing that there is a polynomially solvable LP-relaxation that has fractional solutions with at least one heavy edge. This suggests constant-factor approximations. Our approach uses polyhedral methods that are related to another open problem (from a combinatorial optimization list), in combination with geometric properties. We also demonstrate that the resulting techniques are practical for actually solving problems with up to several hundred points optimally or near-optimally.

关键词： spanning tree matching linear programming relaxation iterated rounding stabbing number crossing number complexity triangulation

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Solving project scheduling problems by minimum cut computations

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MANAGEMENT SCIENCE 2003年第3期49卷 330-350页

作者： Möhring, RH Schulz, AS Stork, F Uetz, M Tech Univ Berlin Fak 2 Inst Math D-10623 Berlin Germany MIT Alfred P Sloan Sch Management Cambridge MA 02139 USA ILOG Deutschland GmbH D-61352 Bad Homburg Germany Univ Limburg Fac Econ & Business Adm NL-6200 MD Maastricht Netherlands

In project scheduling, a set of precedence-constrained jobs has to be scheduled so as to minimize a given objective. In resource-constrained project scheduling, the jobs additionally compete for scarce resources. Due to its universality, the latter problem has a variety of applications in manufacturing, production planning, project management, and elsewhere. It is one of the most intractable problems in operations research, and has therefore become a popular playground for the latest optimization techniques, including virtually all local search paradigms. We show that a somewhat more classical mathematical programming approach leads to both competitive feasible solutions and strong lower bounds, within reasonable computation times. The basic ingredients of our approach are the Lagrangian relaxation of a time-indexed integer programming formulation and relaxation-based list scheduling, enriched with a useful idea from recent approximation algorithms for machine scheduling problems. The efficiency of the algorithm results from the insight that the relaxed problem can be solved by computing a minimum cut in an appropriately defined directed graph. Our computational study covers different types of resource-constrained project scheduling problems, based on several notoriously hard test sets, including practical problem instances from chemical production planning.

关键词： project scheduling resource constraints linear programming relaxation Lagrangian relaxation minimum cut

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An approximation algorithm for the minimum-cost k-vertex connected subgraph

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SIAM JOURNAL ON COMPUTING 2003年第4期32卷 1050-1055页

作者： Cheriyan, J Vempala, S Vetta, A Univ Waterloo Dept Combinator & Optimizat Waterloo ON N2L 3G1 Canada MIT Dept Math Cambridge MA 02139 USA McGill Univ Dept Comp Sci Montreal PQ H3A 2A7 Canada

We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k(2). The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation.

关键词： approximation algorithm l-critically k-connected graph linear programming relaxation network design k-outconnected graph vertex connectivity

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Packing cycles in undirected graphs

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JOURNAL OF ALGORITHMS 2003年第1期48卷 239-256页

作者： Caprara, A Panconesi, A Rizzi, R Univ Bologna DEIS I-40136 Bologna Italy Univ Bologna Dipartimento Informat I-40127 Bologna Italy Aarhus Univ Dept Comp Sci BRICS DK-8000 Aarhus Denmark

Given an undirected graph G with n nodes and m edges, we address the problem of finding a largest collection of edge-disjoint cycles in G. The problem, dubbed CYCLE PACKING, is very closely related to a few genome rearrangement problems in computational biology. In this paper, we study the complexity and approximability of CYCLE PACKING, about which very little is known although the problem is natural and has practical applications. We show that the problem is APX-hard but can be approximated within a factor of 0 (log n) by a simple greedy approach. We do not know whether the O(log n) factor is tight, but we give a nontrivial example for which the ratio achieved by greedy is not constant, namely Omega(rootlog n/(log log n)). We also show that, for "not too sparse" graphs, i.e., graphs for which m = Omega (n(l+l/t+delta)) for some positive integer t and for any fixed 8 > 0, we can achieve an approximation arbitrarily close to 2t/3 in polynomial time. In particular, for any epsilon > 0, this yields a 4/3 + epsilon approximation when m = Omega(n(3/2+delta)), therefore also for dense graphs. Finally, we briefly discuss a natural linear programming relaxation for the problem. (C) 2003 Elsevier Inc. All rights reserved.

关键词： packing edge-disjoint cycles complexity approximation linear programming relaxation

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Packing cycles in undirected graphs

Packing cycles in undirected graphs

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12th Annual ACM-SIAM Symposium on Discrete Algorithms

作者： Caprara, A Panconesi, A Rizzi, R Univ Bologna DEIS I-40136 Bologna Italy Univ Bologna Dipartimento Informat I-40127 Bologna Italy Aarhus Univ Dept Comp Sci BRICS DK-8000 Aarhus Denmark

关键词： packing edge-disjoint cycles complexity approximation linear programming relaxation

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Extending Dantzig's bound to the bounded multiple-class binary Knapsack problem

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MATHEMATICAL programming 2002年第1期94卷 125-136页

作者： Vanderbeck, F Univ Bordeaux 1 MAB F-33405 Talence France

The bounded multiple-class binary knapsack problem is a variant of the knapsack problem where the items are partitioned into classes and the item weights in each class are a multiple of a class weight. Thus, each item has an associated multiplicity. The constraints consists of an upper bound on the total item weight that can be selected and upper bounds on the total multiplicity of items that can be selected in each class. The objective is to maximize the sum of the profits associated with the selected items. This problem arises as a sub-problem in a column generation approach to the cutting stock problem. A special case of this model, where item profits are restricted to be multiples of a class profit, corresponds to the problem obtained by transforming an integer knapsack problem into a 0-1 form. However, the transformation proposed here does not involve a duplication of solutions as the standard transformation typically does. The paper shows that the LP-relaxation of this model can be solved by a greedy algorithm in linear time, a result that extends those of Dantzig (1957) and Balas and Zemel (1980) for the 0-1 knapsack problem. Hence, one can derive exact algorithms for the multi-class binary knapsack problem by adapting existing algorithms for the 0-1 knapsack problem. Computational results are reported that compare solving a bounded integer knapsack problem by transforming it into a standard binary knapsack problem versus using the multiple-class model as a 0-1 form.

关键词： Knapsack problem integer programming linear programming relaxation

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2002年第4期15卷 450-469页

作者： Schulz, AS Skutella, M MIT Alfred P Sloan Sch Management Cambridge MA 02139 USA Tech Univ Berlin Fak 2 Inst Math D-10623 Berlin Germany

We present a new class of randomized approximation algorithms for unrelated parallel machine scheduling problems with the average weighted completion time objective. The key idea is to assign jobs randomly to machines with probabilities derived from an optimal solution to a linear programming (LP) relaxation in time-indexed variables. Our main results are a (2+epsilon)-approximation algorithm for the model with individual job release dates and a (3/2+epsilon)-approximation algorithm if all jobs are released simultaneously. We obtain corresponding bounds on the quality of the LP relaxation. It is an interesting implication for identical parallel machine scheduling that jobs are randomly assigned to machines;in which each machine is equally likely. In addition, in this case the algorithm has running time O(n log n) and performance guarantee 2. Moreover, the approximation result for identical parallel machine scheduling applies to the on-line setting in which jobs arrive over time as well, with no difference in performance guarantee.

关键词： approximation algorithm randomized rounding linear programming relaxation scheduling on-line algorithm

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Designing high-quality approximation algorithms for combinatorial optimization problems

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IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS 2000年第3期E83D卷 462-479页

作者： Asano, T Iwama, K Takada, H Yamashita, Y Chuo Univ Dept Informat & Syst Engn Tokyo 1128551 Japan

For NP-hard combinatorial optimization problems, approximation algorithms with high performances have been proposed. In many of these algorithms, mathematical programming techniques have been used and proved to be very useful. In this survey, we present recent mathematical programming techniques as well as classic fundamental techniques, by showing how these techniques are used in designing high-quality approximation algorithms for NP-hard combinatorial optimization problems.

关键词： dynamic programming linear programming relaxation primal dual method scaling semidefinite programming

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Tighter relaxations for the cutting stock problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1999年第3期112卷 654-663页

作者： Nitsche, C Scheithauer, G Terno, J Dresden Univ Technol Inst Numer Math D-01062 Dresden Germany

In the cutting stock problem (CSP) a given order for smaller pieces has to be cut from larger stock material in such a way that the number of stock material needed is minimal. Based on the classical integer linear programming model the common solution technique consists of solving the corresponding continuous relaxation problem followed by several heuristics which construct integer solutions. In many cases an optimal solution can be obtained quickly in this way. But for instances which do not possess the integer round-up property the optimality of the solution obtained cannot be verified by means of the LP bound. In order to overcome this non-satisfactory situation, two tighter relaxations of the CSP are proposed, and results of theoretical and numerical investigations are presented. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： cutting stock problem modified integer round-up property linear programming relaxation

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