Improved algorithm for the half-disjoint paths problem

Abstract

In this paper, we consider the half integral disjoint paths packing. For a graph G and k pairs of vertices (s1,t1),(s 2,t2),..., (sk ,tk ) in G, the objective is to find paths P1,...,Pk in G such that P i joins si and ti for i=1,2,...,k, and in addition, each vertex is on at most two of these paths. We give a polynomial-time algorithm to decide the feasibility of this problem with k=O((logn/loglogn)1/12). This improves a result by Kleinberg [12] who proved the same conclusion when k=O((loglogn)2/15). Our algorithm still works for several problems related to the bounded unsplittable flow. These results can all carry over to problems involving edge capacities. Our main technical contribution is to give a "crossbar" of a polynomial size of the tree-width of the graph. © 2010 Springer-Verlag.