We study the maximum edge-disjoint path problem ( medp ) in planar graphs $$G=(V,E)$$ G = ( V , E ) with edge capacities u ( e ). We are given a set of terminal pairs $$s_it_i$$ s i t i , $$i=1,2 \ldots , k$$ i = 1 , 2 … , k and wish to find a maximum routable subset of demands. That is, a subset of demands that can be connected by a family of paths that use each edge at most u ( e ) times. It is well-known that there is an integrality gap of $$\Omega (\sqrt{n})$$ Ω ( n ) for the natural LP relaxation, even in planar graphs (Garg–Vazirani–Yannakakis). We show that if every edge has capacity at least 2, then the integrality gap drops to a constant. This result is tight also in a complexity-theoretic sense: recent results of Chuzhoy–Kim–Nimavat show that it is unlikely that there is any polytime-solvable LP formulation for medp which has a constant integrality gap for planar graphs. Along the way, we introduce the concept of rooted clustering which we believe is of independent interest.

中文翻译：

---

拥塞为 2 的平面图中的最大边不相交路径

我们研究了边容量为 u ( e ) 的平面图 $$G=(V,E)$$ G = ( V , E ) 中的最大边不相交路径问题 ( medp )。给定一组终端对 $$s_it_i$$ siti , $$i=1,2 \ldots , k$$ i = 1 , 2 ... , k 并希望找到最大可路由的需求子集。也就是说，可以通过最多使用每条边 u ( e ) 次的路径族连接的需求子集。众所周知，即使在平面图 (Garg-Vazirani-Yannakakis) 中，自然 LP 松弛也存在 $$\Omega (\sqrt{n})$$ Ω ( n ) 的完整性差距。我们表明，如果每条边的容量至少为 2，那么完整性差距就会下降到一个常数。这个结果在复杂性理论意义上也很严格：Chuzhoy-Kim-Nimavat 的最新结果表明，medp 不可能有任何多时间可解的 LP 公式，它对平面图具有恒定的完整性间隙。在此过程中，我们引入了我们认为具有独立兴趣的有根聚类的概念。