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The authors have solved the all pairs shortest distances (APSD) problem for graphs with integer edge lengths. Our algorithm is subcubic for edge lengths of small ( M) absolute value. In this paper we show how to transform these algorithms to solve the all pairs shortest paths (APSP), in the same time complexity, up to a polylogarithmic factor. For n=|V| the number of vertices, M the bound on edge length, and | the exponent of matrix multiplication, we get the following results: 1. A directed nonnegative APSP(n, M) algorithm which runs in O (T(n, M)) time, where T(n, m)=

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DenseA b s t r a c t This paper describes a new approach for solving a large subproblem of the subset-sum problem. It is useful for solving other NP-hard integer programming problems. The limits and potential of this approach axe investigated.The approach yields an algorithm for solving the dense version of the subsetsum problem. It runs in time O(£log£), where l is the bound on the size of the elements. But for dense enough inputs and target numbers near the middle sum it runs in time O(m), where m is the number of elements. Consequently, it improves the previously best algorithms by at least one order of magnitude and sometimes by two.The algorithm yields a characterization of the set of subset sums as a collection of arithmetic progressions with the same difference. This characterization is derived by elementary number theoretic and algorithmic techniques. Such a characterization was first obtained by using analytic number theory and yielded inferior algorithms. I n t r o d u c t i o nThere are several ways to cope with the NP-hardness of optimization problems. One is to look for an approximate solution rather than the optimum. There is a vast literature about approximation algorithn~s and approximation schemes. For some problems there are very good approximation algorithms, for others, the problem is still NP-hea'd even ff we settle for an approximate solution. Another way to cope with complexity is to settle for the average case or to allow probabilistic algorithms. There are cases where this approach has paid off and faster algorithms have been discovered. But this has not been the case with NP-hard optimization problems.A third approach of coping with NP-hardness is to try to restrict the problem and design a polynomial-time algorithm, tIere too, there have been mixed results. Some