Improved algorithms for finding length-bounded two vertex-disjoint paths in a planar graph and minmax k vertex-disjoint paths in a directed acyclic graph

  • Authors:

  • Chih-Chiang Yu;Chien-Hsin Lin;Biing-Feng Wang

  • Affiliations:

  • Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan 30013, ROC;Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan 30013, ROC;Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan 30013, ROC

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2010

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Abstract

This paper is composed of two parts. In the first part, an improved algorithm is presented for the problem of finding length-bounded two vertex-disjoint paths in an undirected planar graph. The presented algorithm requires O(n^3b"m"i"n) time and O(n^2b"m"i"n) space, where b"m"i"n is the smaller of the two given length bounds. In the second part of this paper, we consider the minmax k vertex-disjoint paths problem on a directed acyclic graph, where k=2 is a constant. An improved algorithm and a faster approximation scheme are presented. The presented algorithm requires O(n^k^+^1M^k^-^1) time and O(n^kM^k^-^1) space, and the presented approximation scheme requires O((1/@e)^k^-^1n^2^klog^k^-^1M) time and O((1/@e)^k^-^1n^2^k^-^1log^k^-^1M) space, where @e is the given approximation parameter and M is the length of the longest path in an optimal solution.