Improved algorithm for the half-disjoint paths problem

Authors:
Ken-Ichi Kawarabayashi;Yusuke Kobayashi
Affiliations:
National Institute of Informatics, Tokyo, Japan;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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Abstract

In this paper, we consider the half integral disjoint paths packing. For a graph G and k pairs of vertices (s1, t1), (s2, t2), ..., (sk, tk) in G, the objective is to find paths P1, ..., Pk in G such that Pi 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((log n/ log logn)1/12). This improves a result by Kleinberg [12] who proved the same conclusion when k = O((log log n)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.