A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2

In the Edge-Disjoint Paths with Congestion problem (EDPwC), we are given an undirected n-vertex graph G, a collection M = {(s₁, t₁),..., (s_k, t_k)} of demand pairs and an integer c. The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion - the number of paths sharing any edge - is bounded by c. When the maximum allowed congestion is c = 1, this is the classical Edge-Disjoint Paths problem (EDP). The best current approximation algorithm for EDP achieves an O(√n)-approximation, by rounding the standard multicommodity How relaxation of the problem. This matches the Ω(√n) lower bound on the integrality gap of this relaxation. We show an O(poly log k)-approximation algorithm for EDPwC with congestion c = 2, by rounding the same multi-commodity How relaxation. This gives the best possible congestion for a sub-polynomial approximation of EDPwC via this relaxation. Our results are also close to optimal in terms of the number of pairs routed, since EDPwC is known to be hard to approximate to within a factor of Ω̅(log n)^1/(c+1)) for any constant congestion c. Prior to our work, the best approximation factor for EDPwC with congestion 2 was O̅(n^3/7), and the best algorithm achieving a polylogarithmic approximation required congestion 14.