All-or-Nothing Multicommodity Flow Problem with Bounded Fractionality in Planar Graphs

Abstract

We study the following all-or-nothing multicommodity flow problem in planar graphs: Input: A graph $G$ with $n$ vertices and $k$ pairs of vertices $(s_1,t_1),(s_2,t_2),\dots, (s_k,t_k)$ in $G$. Find: A largest subset $W$ of $\{1,\dots,k\}$ such that for every $i$ in $W$, we can send one unit of flow between $s_i$ and $t_i$. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri, Khanna, and Shepherd [SIAM J. Comput., 42 (2013), pp. 1467--1493] give a polylogarithmic factor approximation algorithm and show that a natural LP-relaxation has a polylogarithmic integrality gap. This result is in contrast with the integrality gap $\Omega(\sqrt{n})$ for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an $O(1)$-approximation algorithm and show that the integrality gap is $O(1)$. In particular, in polynomial time, we can find an index set $W$ with $|W| = \Omega({OPT})$ and eight $s_i$-$t_i$ paths for each $i \in W$ such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the result by Séguin-Charbonneau and Shepherd [Proceedings of FOCS, 2011, pp. 200--209], who give an $O(1)$-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).