Network Problems with Non-Polynomial Weights and Applications

The most efficient algorithms for several network problems like minimum cost flow and the maximum weight matching problem follow the primal-dual paradigm. These algorithms perform arithmetic (additions and subtractions) on numbers of magnitude O(nC) when the edge weights (also called costs) are integers bounded by C and n denotes the number of vertices. Under the standard assumption that arithmetic on numbers of magnitude O(n) has constant cost, arithmetic on numbers of this size has cost O((logC)/ log n). We show for the scaling versions of these algorithms that arithmetic on numbers of size polynomial in n suffices without increasing the asymptotic number of arithmetic operations. In this way, we obtain an O(T + √ nm log(nC)) time and O(S+m) space algorithm for the maximum weight matching problem on bipartite graphs where T and S are the time and space bound of an algorithm to sequentially enumerate the sets Ei, 1 ≤ i ≤ ⌈logC⌉, of all edges having a one in the i-th bit of their weight. The previously best algorithm had running time O( √ nm(log(nC))/ log n). We obtain similar improvements for the capacitated transshipment problem with polynomially bounded