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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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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...
A BRANCHANDPRICE APPROACH FOR Graph Multicoloring
"... We present a branchandprice framework for solving the graph multicoloring problem. We propose column generation to implicitly optimize the linear programming relaxation of an independent set formulation (where there is one variable for each independent set in the graph) for graph multicoloring. T ..."
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Cited by 3 (0 self)
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We present a branchandprice framework for solving the graph multicoloring problem. We propose column generation to implicitly optimize the linear programming relaxation of an independent set formulation (where there is one variable for each independent set in the graph) for graph multicoloring. This approach, while requiring the solution of a difficult subproblem, is a promising method to obtain good solutions for small to moderate size problems quickly. Some implementation details and initial computational experience are presented.
Analysis of an exhaustive search algorithm in random graphs and the n c log n asymptotics. Submitted for publication
"... We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual Gn;pmodel where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n c log ..."
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We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual Gn;pmodel where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n c log n, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms. Key words. Asymptotic expansion, random graphs, graph algorithms, generating functions, Laplace transform, saddlepoint method, Pantograph equation. AMS subject classifications. 60C05 05A16 05C69 05C80 68W40 68R10 1. Introduction. An independent set or stable set of a graph G is a subset of vertices in G no two of which are adjacent. The Maximum Independent Set (MIS) Problem consists in finding an independent set with the largest cardinality; it is among the first known NPhard problems and has become a fundamental, representative, prototype instance of combinatorial optimization and computational complexity; see [27]. A large number of algorithms (exact or
RAINBOW ARBORESCENCE IN RANDOM DIGRAPHS
"... Abstract. We consider the ErdősRényi random directed graph process, which is a stochastic process that starts with n vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let D(n,m) be a graph with m edges obtained after m st ..."
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Abstract. We consider the ErdősRényi random directed graph process, which is a stochastic process that starts with n vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let D(n,m) be a graph with m edges obtained after m steps of this process. Each edge ei (i = 1, 2,...,m) of D(n,m) independently chooses a colour, taken uniformly at random from a given set of n(1 + O(log log n / log n)) = n(1 + o(1)) colours. We stop the process prematurely at time M when the following two events hold: D(n,M) has at most one vertex that has indegree zero and there are at least n − 1 distinct colours introduced (M = n(n − 1) if at the time when all edges are present there are still less than n − 1 colours introduced; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether D(n,M) has a rainbow arborescence (that is, a directed, rooted tree on n vertices in which all edges point away from the root and all the edges are different colours). Clearly, both properties are necessary for the desired tree to exist and we show that, asymptotically almost surely, the answer to this question is “yes”.
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, 2012
"... Analysis of an exhaustive search algorithm in random graphs and the n c log nasymptotics ..."
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Analysis of an exhaustive search algorithm in random graphs and the n c log nasymptotics
Dvi file name: "ccreview.dvi". This PostScript file was produced on host "shakespeare.rutgers.edu".
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Average Case Analysis of NPcomplete Problems: Maximum Independent Set and Exhaustive Search Algorithms
"... In this article, we deal with rigorous average case analysis of NP complete problems, relying on some mathematical tools from complex analysis and probability theory. We consider here one of the historical prototype of such problems: Maximum Independent Set (MIS). For a large class of exhaustive alg ..."
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In this article, we deal with rigorous average case analysis of NP complete problems, relying on some mathematical tools from complex analysis and probability theory. We consider here one of the historical prototype of such problems: Maximum Independent Set (MIS). For a large class of exhaustive algorithms which always find exactly a MIS, their complexity is directly related to their number of iterations. Under the Γ(n, m) and G(n, p) distribution for graphs, we give some fascinating phase transitions between exponential (A n), superpolynomial (n ln n), and polynomial (n d) average complexities. Our approach gives a precise picture of the “complexity landscape ” of these algorithms, depending on the average degree (or the ratio vertices/edges) of the graph, and gives access to the location of the “hard regions ” where people could then sample their inputs if they want to make some benchmarks (may it be for the worst or average case). The challenging associated mathematical aspects force us to introduce new analyses (for a large class of recurrences), which will clearly be of interest for many other NP hard problems (typically, optimization problems on graphs). Direct applications cover graph 3coloring (which is also a deep motivation for considering our class of exhaustive TarjanChvàtal like algorithms), and numerous constraint satisfaction problems.