Abstract
Efficient algorithms for the matroid intersection problem, both cardinality and weighted versions, are presented. The algorithm for weighted intersection works by scaling the weights. The cardinality algorithm is a special case, but takes advantage of greater structure. Efficiency of the algorithms is illustrated by several implementations on linear matroids. Consider a linear matroid withmelements and rankn. Assume all element weights are integers of magnitude at mostN. Our fastest algorithms use timeO(mn1.77log(nN)) andO(mn1.62) for weighted and unweighted intersection, respectively; this improves the previous best bounds,O(mn2.4) andO(mn2logn), respectively. Corresponding improvements are given for several applications of matroid intersection to numerical computation and dynamic systems.
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