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 2 k log k − 1 M ) time and O ( ( 1 / ϵ ) k − 1 n 2 k − 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.