Algebraic Algorithms for B-Matching, Shortest Undirected Paths, and F-Factors

Let G = (V, E) be a graph with f : V → Z₊ a function assigning degree bounds to vertices. We present the first efficient algebraic algorithm to find an f-factor. The time is O(f(V )^ω). More generally for graphs with integral edge weights of maximum absolute value W we find a maximum weight f-factor in time Õ(Wf(V )^ω). (The algorithms are correct with high probability and can be made Las Vegas.) We also present three specializations of these algorithms: For maximum weight perfect f-matching the algorithm is considerably simpler (and almost identical to its special case of ordinary weighted matching). For the single-source shortestpath problem in undirected graphs with conservative edge weights, we define a generalization of the shortest-path tree, and we compute it in ̃Õ(Wn^ω) time. For bipartite graphs, we improve the known complexity bounds for vertex-capacitated max-flow and min-cost max-flow on a subclass of graphs.