Valuated Matroid Intersection II: Algorithms

Based on the optimality criteria established in part I [{\em SIAM J. Discrete Math.}, 9 (1996), pp. 545--561] we show a primal-type cycle-canceling algorithm and a primal--dual-type augmenting algorithm for the valuated independent assignment problem: given a bipartite graph $G=(V\sp{+}, V\sp{-}; A)$ with arc weight $w: A \to \hbox{{\RR}}$ and matroid valuations $\omega\sp{+}$ and $\omega\sp{-}$ on $V\sp{+}$ and $V\sp{-}$, respectively, find a matching $M (\subseteq A)$ that maximizes $\sum \{w(a) \mid a \in M \} + \omega\sp{+}(\partial\sp{+} M) + \omega\sp{-}(\partial\sp{-} M)$, where $\partial\sp{+} M$ and $\partial\sp{-} M$ denote the sets of vertices in $V\sp{+}$ and $V\sp{-}$ 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 [{\em Math. Prog.}, 9 (1975), pp. 31--56], Iri and Tomizawa [{\em J. Oper. Res. Soc. Japan}, 19 (1976), pp. 32--57], Fujishige [{\em J. Oper. Res. Soc. Japan}, 20 (1977), pp. 1--15], Frank [{\em J. Algorithms}, 2 (1981), pp. 328--336], and Zimmermann [{\em Discrete Appl. Math.}, 36 (1992), pp. 179--189].