Development of Algorithm Theory Based on Mathematical Programming and Probability Tyeory

·After introducing a generalization of the vertex cover problem on graphs with vertex and edge constraints，we show it to be polynomially approximable within a factor of2，using an extended version of the submodular set cover algorithm。·The tree cover problem is known approximable within a factor of2only when all the edge costs are uniform，whereas some related problems such as vertex cover and edge dominating set are2-approximable under general costs.We develop a primal-dual algorithm for tree cover and show that its approximation factor is2when only two kinds of edge weights，differing by a multiplicative factor of at least2，are allowed.·While several2-approximation NC algorithms are known for the vertex cover problem on graphs，no such algorithm is known for the connected vertex cover problem.We develop2-approximation NC and RNC algorithms for tree cover and connected vertex cover。·It is shown that the set multicover problem can be approximated within a factor of H(K)-1/6 by a modified greedy algorithm newly developed for set multicover.·We develop an efficient and purely combinatorial algorithm for the covering0-1 integer program problem，and show its performance is in general as good as those of the rounding algorithms。·Extending a local search heuristic for the unweighted set packing problem，it is shown that the k-set packing problem with weights1and w such that w[greater than or equal]2canbe approximated within a factor of k/2ε。·We introduce a production planning problem called the capacitated supply-demand(CSD)problem，and，to analyze its structural properties，extend the submodular set cover problem to the one，called submodular integer cover(SIC)，with submodular constraints on integer vectors instead of on subsets.By applying the primal-dual heuristic for SIC to CSD，it is shown that CSD can be approximated by a factor dependent on the network structure but not on any numerical value.

・After introducing a generalization of the vertex cover problem on graphs with vertex and edge constraints, we show it to be polynomially approximable within a factor of 2,using an extended version of the submodular set cover algorithm.・The tree cover problem is known approximable within a factor of 2 only when all the edge costs are uniform, whereas some related problems such as vertex cover and edge dominating set are 2-approximable under general costs. We develop a primal-dual algorithm for tree cover and show that its approximation factor is 2 when only two kinds of edge weights, differing by a multiplicative factor of at least 2,are allowed.・While several 2-approximation NC algorithms are known for the vertex cover problem on graphs, no such algorithm is known for the connected vertex cover problem. We develop 2-approximation NC and RNC algorithms for tree cover and connected vertex cover.・It is shown that the set multicover problem can be approximated within a factor of H(k)-1/6 by a modified greedy algorithm newly developed for set multicover.・We develop an efficient and purely combinatorial algorithm for the covering 0-1 integer program problem, and show its performance is in general as good as those of the rounding algorithms.・Extending a local search heuristic for the unweighted set packing problem, it is shown that the k-set packing problem with weights 1 and w such that w【greater than or equal】2 canbe approximated within a factor of k/2+ε.・We introduce a production planning problem called the capacitated supply-demand (CSD) problem, and, to analyze its structural properties, extend the submodular set cover problem to the one, called submodular integer cover (SIC), with submodular constraints on integer vectors instead of on subsets. By applying the primal-dual heuristic for SIC to CSD, it is shown that CSD can be approximated by a factor dependent on the network structure but not on any numerical value.