Valuated Matroid Intersection II: Algorithms

Based on the optimality criteria established in Part I, we show a primal-type cyclecanceling algorithm and a primal-dual-type augmenting algorithm for the valuated independent assignment problem: Given a bipartite graph G = (V , V −;A) with arc weight w : A → R and matroid valuations ω and ω− on V + and V − respectively, find a matching M(⊆ A) that maximizes ∑ {w(a) | a ∈ M}+ ω(∂M) + ω−(∂−M), where ∂M and ∂−M denote the sets of vertices in V + and V − incident to M . The proposed algorithms generalize the previous algorithms for the independent assignment problem as well as for the weighted matroid intersection problem, including those due to Lawler (1975), Iri-Tomizawa (1976), Fujishige (1977), Frank (1981), and Zimmermann (1992).