On the independence number of random graphs
Discrete Mathematics
A lower bound on the independence number of a graph
Discrete Mathematics
LEDA: a platform for combinatorial and geometric computing
LEDA: a platform for combinatorial and geometric computing
Load Balancing for Distributed Branch and Bound Algorithms
IPPS '92 Proceedings of the 6th International Parallel Processing Symposium
Statistical mechanics methods and phase transitions in optimizationproblems
Theoretical Computer Science - Phase transitions in combinatorial problems
Greedy Local Search and Vertex Cover in Sparse Random Graphs
TAMC '09 Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation
Phase Transition in the Bandwidth Minimization Problem
MICAI '09 Proceedings of the 8th Mexican International Conference on Artificial Intelligence
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Artificial Intelligence
Analysis of an iterated local search algorithm for vertex cover in sparse random graphs
Theoretical Computer Science
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The vertex-cover problem is studied for random graphs GN,cN having N vertices and cN edges. Exact numerical results are obtained by a branch-and-bound algorithm. It is found that a transition in the coverability at a c-dependent threshold x =xc(c) appears, where xN is the cardinality of the vertex cover. This transition coincides with a sharp peak of the typical numerical e1ort, which is needed to decide whether there exists a cover with xN vertices or not. For small edge concentrations c(two less than arrows, one inside the other)0:5, a cluster expansion is performed, giving very accurate results in this regime. These results are extended using methods developed in statistical physics. The so-called annealed approximation reproduces a rigorous bound on xc(c) which was known previously. The main part of the paper contains an application of the replica method. Within the replicasymmetric ansatz the threshold xc(c) and various statistical properties of minimal vertex covers can be calculated. For c ??? e=2 the results show an excellent agreement with the numerical findings. At average vertex degree 2c=e, an instability of the simple replica symmetric solution occurs.