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 branch-and-bound 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...

We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. From the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and almost 7000 edges.

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 3-coloring (which is also a deep motivation for considering our class of exhaustive Tarjan-Chvàtal like algorithms), and numerous constraint satisfaction problems.

We present a branch-and-price 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 multi-coloring. 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.

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Analysis of an exhaustive search algorithm in random graphs and the n c log n-asymptotics