Maximum flows on disjoint paths
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
An O(log n)-Approximation Algorithm for the Edge-Disjoint Paths Problem in Eulerian Planar Graphs
ACM Transactions on Algorithms (TALG)
A node-capacitated okamura-seymour theorem
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Maximum edge-disjoint paths in k-sums of graphs
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We study the maximum edge-disjoint paths problem in undirected planar graphs: given a graph $G$ and node pairs (demands) $s_1t_1$, $s_2t_2$, $\dots$, $s_kt_k$, the goal is to maximize the number of demands that can be connected (routed) by edge-disjoint paths. The natural multicommodity flow relaxation has an $\Omega(\sqrt{n})$ integrality gap, where $n$ is the number of nodes in $G$. Motivated by this, we consider solutions with small constant congestion $c1$, that is, solutions in which up to $c$ paths are allowed to use an edge (alternatively, each edge has a capacity of $c$). In previous work we obtained an $O(\log n)$ approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into well-linked subproblems. In this paper we obtain an $O(1)$ approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura-Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation.