Many heuristic algorithms have been proposed for graph coloring. The simplest is perhaps the greedy algorithm. Many variations have been proposed for this algorithm at various levels of sophistication, but it is generally assumed that the coloring will occur in a single attempt. We note that if a new permutation of the vertices is chosen which respects the independent sets of a previous coloring, then applying the greedy algorithm will result in a new coloring in which the number of colors used does not increase, yet may decrease. We introduce several heuristics for generating new permutations that are fast when implemented and effective in reducing the coloring number. The resulting Iterated Greedy algorithm(IG) can obtain colorings in the range 100 to 103 on graphs in G 1000; 1 2 . More interestingly, it can optimally color k-colorable graphs with k up to 60 and n = 1000, exceeding results of anything in the literature for these graphs. We couple this algorithm with several other c...

We report on experiments with a new hybrid graph coloring algorithm, which combines a parallel version of Morgenstern's S-Impasse algorithm [20], with exhaustive search. We contribute new test data arising in five different application domains, including register allocation and class scheduling. We test our algorithms both on this test data and on several types of randomly generated graphs. We compare our parallel implementation, which is done on the CM-5, with two simple heuristics, the Saturation algorithm of Br'elaz [4] and the Recursive Largest First (RLF) algorithm of Leighton [18]. We also compare our results with previous work reported by Morgenstern [20] and Johnson et al. [13]. Our main results are as follows. ffl On the randomly generated graphs, the performance of Hybrid is consistently better than the sequential algorithms, both in terms of speed and number of colorings produced. However, on large random graphs, our algorithms do not come close to the best colorings found ...

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.

We study the complexity of the problem 3-Colorability when restricted to those input graphs on which a given graph coloring heuristic is able to solve the problem. The heuristics we consider include the sequential algorithm traversing the vertices of the graph in various orderings (e.g., by decreasing degree or in the recursive smallest-last order) as well as Wood's algorithm. For each heuristic considered here, we prove that the corresponding restriction of 3-Colorability remains NP-complete. 1 Introduction Graph coloring problems are of great importance in both theory and applications and have been intensely studied during the past century. Applications of constructing a graph coloring with as few colors as possible arise, for instance, in scheduling and partitioning problems (see Garey and Johnson [GJ79]). Unfortunately, the (optimization) problem of finding the chromatic number of a given graph is very complex, and even the (decision) problem of determining whether or not a given...

Local Search meta-heuristics are an emerging class of methods for tackling combinatorial search and optimization problems, which recently have been shown to be very effective for a large number of combinatorial problems. The Local Search techniques are based on the iterative exploration of a solution space: at each iteration, a Local Search algorithm steps from one solution to one of its “neighbors”, i.e., solutions that are (in some sense) close to the starting one. One major drawback of this family of techniques is the lack of robustness on a wide variety of problem instances. In fact, in many cases, these methods assure finding good results in reasonable running times, whereas in other cases Local Search techniques are trapped in the so-called local minima. Several approaches to the solution of this problem recently appeared in the literature. These approaches range from the employment of statistical properties (e.g., random explorations of the solution space), to the application of learning methods or hybrid techniques. In this thesis we propose an alternative approach to cope with local minima, which is based

The proposed research aims to provide the technical framework which will enable multiple Wireless LANs to provide the assured network service while being in communication range of each other. The Access Points involved compete and cooperate while keeping the overall goal of maximum system utilization above individual optimization. We use channel access time as the means of resource allocation, while keeping the system modular so that any other unit can also be used without changing the protocol. We provide a secondary protocol to ensure robustness of our main algorithm for Access Allocation. We also

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