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

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 $O(n^3{b}_{\min })$ time and $O(n^2{b}_{\min })$ space, where $b_{\min }$ 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 $O(n^{k+1}{M}^{k-1})$ time and $O(n^k{M}^{k-1})$ space, and the presented approximation scheme requires $O({(1/ϵ)}^{k-1}{n}^{2k}{\log }^{k-1}M)$ time and $O({(1/ϵ)}^{k-1}{n}^{2k-1}{\log }^{k-1}M)$ space, where ϵ is the given approximation parameter and M is the length of the longest path in an optimal solution.