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

TL;DR

This paper presents a polylogarithmic approximation algorithm for the Edge-Disjoint Paths with Congestion problem at congestion level 2, improving previous bounds and approaching theoretical hardness limits.

Contribution

It introduces the first polylogarithmic approximation algorithm for EDP with congestion 2, using flow relaxation rounding, surpassing prior polynomial bounds.

Findings

Achieves an $O(poly log k)$-approximation for EDP with congestion 2.

Improves previous approximation bounds from $ ilde O(n^{3/7})$ to polylogarithmic.

Approaches the known hardness limits for this problem.

Abstract

In the Edge-Disjoint Paths with Congestion problem (EDPwC), we are given an undirected n-vertex graph G, a collection M={(s_1,t_1),...,(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 $O(\sqrt n)$-approximation, by rounding the standard multi-commodity flow relaxation of the problem. This matches the $\Omega(\sqrt n)$ 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 flow 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 $\tilde{\Omega}((\log n)^{1/(c+1)})$ for any constant congestion c. Prior to our work, the best approximation factor for EDPwC with congestion 2 was $\tilde O(n^{3/7})$, and the best algorithm achieving a polylogarithmic approximation required congestion 14.