On Linear Programming Relaxations for Unsplittable Flow in Trees

We study some linear programming relaxations for the Unsplittable Flow problem on trees (UFPtree). Inspired by results obtained by Chekuri, Ene, and Korula for Unsplittable Flow on paths (UFP-path), we present a relaxation with polynomially many constraints that has an integrality gap bound of O(logn ·min{logm, logn}) where n denotes the number of tasks and m denotes the number of edges in the tree. This matches the approximation guarantee of their combinatorial algorithm and is the first demonstration of an efficiently-solvable relaxation for UFP-tree with a sub-linear integrality gap. The new constraints in our LP relaxation are just a few of the (exponentially many) rank constraints that can be added to strengthen the natural relaxation. A side effect of how we prove our upper bound is an efficient O(1)-approximation for solving the rank LP. We also show that our techniques can be used to prove integrality gap bounds for similar LP relaxations for packing demand-weighted subtrees of an edge-capacitated tree. On the other hand, we show that the inclusion of all rank constraints does not reduce the integrality gap for UFP-tree to a constant. Specifically, we show the integrality gap is Ω( √ logn) even in cases where all tasks share a common endpoint. In contrast, intersecting instances of UFP-path are known to have an integrality gap of O(1) even if just a few of the rank 1 constraints are included. We also observe that applying two rounds of the Lovász-Schrijver SDP procedure to the natural LP for UFP-tree derives an SDP whose integrality gap is also O(logn ·min{logm, logn}). 1998 ACM Subject Classification G.1.6 Optimization, G.2.2 Graph Theory, I.1.2 Algorithms