Unsplittable Flow in Paths and Trees and Column-Restricted Packing Integer Programs

We consider the unsplittable flow problem (UFP) and the closely related column-restricted packing integer programs (CPIPs). In UFP we are given an edge-capacitated graph G = (V,E) and k request pairs R1, . . . , Rk where each Ri consists of a source-destination pair (si, ti), a demand di and a weight wi. The goal is to find a maximum weight subset of requests that can be routed unsplittably in G. Most previous work on UFP has focused on the no-bottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al. [3], we consider UFP on paths as well as trees without the no-bottleneck assumption. We give a simple O(log n) approximation for UFP on trees when all weights are identical; this yields an O(log n) approximation for the weighted case. These are the first non-trivial approximations for UFP on trees. We develop a new LP relaxation for UFP on paths that has an integrality gap of O(log n); previously there was no relaxation with o(n) gap. In contrast we show that the integrality gap of the natural LP has an O(n/t) gap even after applying t rounds of the Sherali-Adams lift-and-project scheme. We also consider UFP in general graphs and CPIPs without the no-bottleneck assumption and obtain new and useful results.