外平面图中基于单树割近似的最大权不交路
Maximum Weight Disjoint Paths in Outerplanar Graphs via Single-Tree Cut Approximators
摘要：
自1997年以来，最大不相交路径问题取得了稳步的进展。实现易于处理的结果通常需要关注松弛，例如：（i）允许解决方案中的一些有界边缘拥塞，（ii）仅考虑单位权重（基数）设置，（iii）仅要求所选需求的分数可路由性（全有或全无流设置）。对于边不相交路径（{ \sc edp}）的一般形式（无拥塞、一般权重、积分路由），即使单位容量树是星的情况也推广了埃德蒙兹提供的精确算法的最大匹配问题。对于一般的能力树，Garg，Vazirani，Yannakakis证明了问题是APX-困难的，Chekuri，Mydlarz，Shepherd提供了一个$4$ -近似。这基本上是唯一一个已知\textsc{edp}一般形式的常数近似的设置。我们推广了他们的结果，给出了外平面图中一般形式\textsc{edp}的常数因子近似算法。该算法的一个关键部分是寻找外平面图的{ \em单树} $O(1)$割逼近器。以前的$O(1)$切逼近器只知道通过分布在树上，这些都是基于隐式的结果古普塔，纽曼，Rabinovich和辛克莱的距离树嵌入结合安德森和Feige的结果。
英文摘要：
Since 1997 there has been a steady stream of advances for the maximum disjoint paths problem. Achieving tractable results has usually required focusing on relaxations such as: (i) to allow some bounded edge congestion in solutions, (ii) to only consider the unit weight (cardinality) setting, (iii) to only require fractional routability of the selected demands (the all-or-nothing flow setting). For the general form (no congestion, general weights, integral routing) of edge-disjoint paths ({\sc edp}) even the case of unit capacity trees which are stars generalizes the maximum matching problem for which Edmonds provided an exact algorithm. For general capacitated trees, Garg, Vazirani, Yannakakis showed the problem is APX-Hard and Chekuri, Mydlarz, Shepherd provided a $4$-approximation. This is essentially the only setting where a constant approximation is known for the general form of \textsc{edp}. We extend their result by giving a constant-factor approximation algorithm for general-form \textsc{edp} in outerplanar graphs. A key component for the algorithm is to find a {\em single-tree} $O(1)$ cut approximator for outerplanar graphs. Previously $O(1)$ cut approximators were only known via distributions on trees and these were based implicitly on the results of Gupta, Newman, Rabinovich and Sinclair for distance tree embeddings combined with results of Anderson and Feige.
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