Computing the maximum degree of minors in mixed polynomial matrices via combinatorial relaxation
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
ESA'11 Proceedings of the 19th European conference on Algorithms
On the Kronecker Canonical Form of Mixed Matrix Pencils
SIAM Journal on Matrix Analysis and Applications
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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].