Algebraic Algorithms for bMatching, Shortest Undirected Paths, and fFactors
Abstract
Let G=(V,E) be a graph with f:V\to Z_+ a function assigning degree bounds to vertices. We present the first efficient algebraic algorithm to find an ffactor. The time is \tilde{O}(f(V)^{\omega}). More generally for graphs with integral edge weights of maximum absolute value W we find a maximum weight ffactor in time \tilde{O}(Wf(V)^{\omega}). (The algorithms are randomized, correct with high probability and Las Vegas; the time bound is worstcase.) We also present three specializations of these algorithms: For maximum weight perfect fmatching the algorithm is considerably simpler (and almost identical to its special case of ordinary weighted matching). For the singlesource shortestpath problem in undirected graphs with conservative edge weights, we present a generalization of the shortestpath tree, and we compute it in \tilde{O(Wn^{\omega}) time. For bipartite graphs, we improve the known complexity bounds for vertex capacitated maxflow and mincost maxflow on a subclass of graphs.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.6740
 Bibcode:
 2013arXiv1304.6740G
 Keywords:

 Computer Science  Data Structures and Algorithms