Two NP-Complete Augmentation Problems
Two NP-Complete Augmentation Problems
Matroid partitioning.
Arc-disjoint in-trees in directed graphs
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Rooted k-connections in digraphs
Discrete Applied Mathematics
Approximations for minimum and min-max vehicle routing problems
Journal of Algorithms
Operations Research Letters
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Given a directed graph D= (V,A) with a set of dspecified vertices S= {s1,...,sd} 茂戮驴 Vand a function $f\colon S \to \mathbb{Z}_+$ where 茂戮驴+denotes the set of non-negative integers, we consider the problem which asks whether there exist $\sum_{i=1}^d f(s_i)$ in-trees denoted by $T_{i,1},T_{i,2},\ldots, T_{i,f(s_i)}$ for every i= 1,...,dsuch that $T_{i,1},\ldots,T_{i,f(s_i)}$ are rooted at si, each Ti,jspans vertices from which siis reachable and the union of all arc sets of Ti,jfor i= 1,...,dand j= 1,...,f(si) covers A. In this paper, we prove that such set of in-trees covering Acan be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in $\sum_{i=1}^df(s_i)$ and the size of D. Furthermore, for the case where Dis acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering Acan be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.