вб Given a number ¥ with жизйе § , a graph - Semantic Scholar
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вб Given a number ¥ with жизйе § , a graph - Semantic Scholar
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FINDING TWO DISJOINT PATHS IN A NETWORK WITH NORMALIZED -MIN-SUM OBJECTIVE FUNCTION
A
ferent lengths. The objective of this problem is to find disjoint paths such that the total length of the paths is minimized, where the th edge-length is associated with the th path. They showed that all four versions of the G-MINSUM -path problem are strongly NP-complete even for [10]. In [12], we considered the problem of finding two disjoint paths such that the length of the shorter path is minimized (named the MIN-MIN 2-path problem). We showed that all four versions of the MIN-MIN 2-path problem are strongly NP-complete[12]. In the same paper, we also showed there does not exist any polynomial-time approximation algorithm with a constant approximation bound for any of these four versions of the MIN-MIN 2-path problem . In this paper we consider a generalized unless and be two disjoint weighted 2-path problem. Let paths from to in a given graph , and a non-negative value. Define
ABSTRACT , a graph Given a number with and two nodes and in , we consider the problem and from to such of finding two disjoint paths with that is minimized. The paths may be node-disjoint or edge-disjoint, and the network may be directed or undirected. This problem has applications in reliable communication. We show that all four versions of this problem are NP-complete. Then we give an approximation of , and show that this bound is best possible for directed graphs. For acyclic directed graphs, we also give a pseudopolynomial-time algorithm for finding optimal solutions.
