Decision algorithms for unsplittable flow and the half-disjoint paths problem

TLDR: A polynomial-time algorithm for the bounded unsplittable flow problem, in an arbitrary graph, when the number of terminal pairs is a ilxed constant, which is conceptually much simpler than Robert,son and Seymour’s corresponding algorithm for t.son-Seymour algorithm, together with some new algorithmic components.

Abstract: We consider t.he bounded unsplittable flow problem: given t,erminal pairs in a network, with associated real-valued demands in bhe range [0, 41, find a single flow path for each pair so that no more than 1 unit of demand is routed t.hrough any vertex. Thus, the setting is not directly comparable to that, of 6he classical disjoint paths problem (when all demands are equal to 1) we must deal with connect.ions having varied, real-valued amounts of demand, but we impose the boundedness restriction t.hat each connection can consume at most half the capacity of any vertex Our main result is a polynomial-time algorithm for t,he bounded unsplittable flow problem, in an arbitrary graph, when the number of terminal pairs is a ilxed constant. Our algorithm is conceptually much simpler than Robert,son and Seymour’s corresponding algorithm for t.he disjoint paths problem witch a constant number of terminal pairs; and we can decide the routability of a non-t,rivially super-constant number of terminal pairs (up t,o Q((log log n)2/15)) in polynomial time. We also obtain polynomial-time algorithms for several natural opt.imizat$ion problems derived from the bounded unsplitt,able flow problem, when the number of terminal pairs is sufficiently small, and algorithms with better bounds for the case of planar graphs. The results all carry over to problems involving edge capacities. Our approach makes use of several of the ideas underlying bhe Robert.son-Seymour algorithm, together with some new algorithmic components. The resu1t.s add to a growing body of work suggest*Department of Computer Science, Cornell University, Ithaca NY 14863. Email: ldeinber&s.comell,edu. Supported in part by an Alfred P. Sloan Research Fellowship and by NSF Faculty Early Career Development Award CCR-9701399. ing that versions of our boundedness restriction while often relatively mild from the point of view of the underlying motivation can have very interesting qualitative effects on the tractability of basic routing problems.