Showing papers on "Vertex cover published in 1973"

Approximation algorithms for combinatorial problems

Abstract: Simple, polynomial-time, heuristic algorithms for finding approximate solutions to various polynomial complete optimization problems are analyzed with respect to their worst case behavior, measured by the ratio of the worst solution value that can be chosen by the algorithm to the optimal value. For certain problems, such as a simple form of the knapsack problem and an optimization problem based on satisfiability testing, there are algorithms for which this ratio is bounded by a constant, independent of the problem size. For a number of set covering problems, simple algorithms yield worst case ratios which can grow with the log of the problem size. And for the problem of finding the maximum clique in a graph, no algorithm has been found for which the ratio does not grow at least as fast as 0(ne), where n is the problem size and e> 0 depends on the algorithm.

Finding a Minimum Node Cover in an Arbitrary Graph.

Abstract: : The paper describes a node covering algorithm, i.e., a procedure for finding a smallest set of nodes covering all edges of an arbitrary graph. The algorithm is based on the concept of a dual node-clique set which allows one to identify partial covers associated with integer dual feasible solutions to the linear programming equivalent of the node covering problem. An initial partial cover with the above property is first found by a labeling procedure. Another labeling procedure then successively modifies the dual node-clique set, so that more and more edges are covered, i.e., the (primal) infeasibility of the solution is gradually reduced, while integrality and dual feasibility are preserved. When this cannot be continued, the problem is partitioned and the procedure applied to the resulting subproblems. While the steps of the algorithm correspond to sequences of dual simplex pivots, these are carried out implicitly, by labeling. The procedure is illustrated on examples, and some early computational experience is reported. The authors conclude with a discussion of potential improvements and extensions. (Author)