边不相交且拥塞为2的路径的多项式近似算法
A Polylogarithimic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2
摘要：
在边不相交的拥塞路径问题（EDPwC）中，给定一个无向n-顶点图G，一个集合M={（s_1，t_1），.，（s_k，t_k）}和整数c。目标是通过路径连接最大可能数量的需求对，使得最大边缘拥塞-共享任何边缘的路径数量-以c为界。当最大允许拥塞为c=1时，这是经典的边不相交路径问题（EDP）。EDP的最佳电流近似算法实现了$O(\sqrt n)$近似，通过舍入标准的多商品流松弛的问题。这与该弛豫的完整性间隙的$Ω(\sqrt n)$下限相匹配。通过舍入相同的多商品流松弛，我们给出了拥塞c=2的EDPwC的$O(poly log k)$ -近似算法。这给出了经由该松弛的EDPwC的子多项式近似的最佳可能拥塞。我们的结果也接近最佳的路由对的数量，因为EDPwC是已知的是很难近似到任何恒定的拥塞c的$\tildeΩ((\log n)^{1/(c+1)})$的一个因素。在我们的工作之前，拥塞2的EDPwC的最佳近似因子是$\tilde O(n^{3/7})$，并且实现多对数近似的最佳算法需要拥塞14。
英文摘要：
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 $Ω(\sqrt n)$ 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 $\tildeΩ((\log n)^{1/(c+1)})$ 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.
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