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A Column Generation Approach For Graph Coloring
 INFORMS Journal on Computing
, 1995
"... We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approac ..."
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Cited by 94 (2 self)
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We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approach, while requiring the solution of a difficult subproblem as well as needing sophisticated branching rules, solves small to moderate size problems quickly. We have also implemented an exact graph coloring algorithm based on DSATUR for comparison. Implementation details and computational experience are presented. 1 INTRODUCTION The graph coloring problem is one of the most useful models in graph theory. This problem has been used to solve problems in school timetabling [10], computer register allocation [7, 8], electronic bandwidth allocation [11], and many other areas. These applications suggest that effective algorithms for solving the graph coloring problem would be of great importance. D...
CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions
, 2005
"... Combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is NPcomplete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and ..."
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Cited by 58 (9 self)
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Combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is NPcomplete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and lower bounding (also across components), elaborate and dynamically chosen bidordering heuristics, and a host of structural observations. CABOB attempts to capture structure in any instance without making assumptions about the instance distribution. Experiments against the fastest prior algorithm, CPLEX 8.0, show that CABOB is often faster, seldom drastically slower, and in many cases drastically faster—especially in cases with structure. CABOB’s search runs in linear space and has significantly better anytime performance than CPLEX. We also uncover interesting aspects of the problem itself. First, problems with short bids, which were hard for the first generation of specialized algorithms, are easy. Second, almost all of the CATS distributions are easy, and the run time is virtually unaffected by the number of goods. Third, we test several random restart strategies, showing that they do not help on this problem—the runtime distribution does not have a heavy tail.
An Interior Point Approach to the Maximum Independent Set Problem in Dense Random Graphs
 In Proceedings of the XIII Latin American Conference on Informatics
, 1989
"... We present an interior point approach to the zeroone integer programming feasibility problem based on the minimization of an appropriate potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such ..."
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Cited by 6 (3 self)
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We present an interior point approach to the zeroone integer programming feasibility problem based on the minimization of an appropriate potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such that each consecutive point reduces the value of the potential function. An integer solution (not necessarily feasible) is generated at each iteration by a rounding scheme. The direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid. We illustrate the approach by considering a class of difficult NPcomplete problems: finding a maximum independent set of a dense random graph. Some implementation details are discussed and preliminary computational results are presented. We solve several large independent set problems in graphs having up to 1000 vertices and over 250,000 edges. Key words: Integer programming, interior point method, maxim...
Chordal editing is fixedparameter tractable
 In 31st International Symposium on Theoretical Aspects of Computer Science
, 2014
"... Graph modification problems are typically asked as follows: is there a set of k operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly ..."
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Cited by 2 (1 self)
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Graph modification problems are typically asked as follows: is there a set of k operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem which allows all three types of operations: given a graph G and integers k1, k2, and k3, the chordal editing problem asks if G can be transformed into a chordal graph by at most k1 vertex deletions, k2 edge deletions, and k3 edge additions. Clearly, this problem generalizes both chordal vertex/edge deletion and chordal completion (also known as minimum fillin). Our main result is an algorithm for chordal editing in time 2O(k log k) · nO(1), where k: = k1 + k2 + k3; therefore, the problem is fixedparameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case chordal deletion.
Dvi file name: "ccreview.dvi". This PostScript file was produced on host "shakespeare.rutgers.edu".
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GRAPHTHEORETIC STUDIES OF COMBINATORIAL OPTIMIZATION PROBLEMS
, 2013
"... Graphtheoretic studies of combinatorial optimization problems ..."
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