Flow-Cut Gaps and Face Covers in Planar Graphs

Abstract

The relationship between the sparsest cut and the maximum concurrent multi-flow in graphs has been studied extensively. For general graphs, the worst-case gap between these two quantities is now settled: When there are k terminal pairs, the flow-cut gap is O(log k), and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. In particular, it has been conjectured that the flow-cut gap in planar networks is O(1), while the known bounds place the gap somewhere between 2 (Lee and Raghavendra, 2003) and O( √ log k) (Rao, 1999).A seminal result of Okamura and Seymour (1981) shows that when all the terminals of a planar network lie on a single face, the flow-cut gap is exactly 1. This setting can be generalized by considering planar networks where the terminals lie on one of γ > 1 faces in some fixed planar drawing. Lee and Sidiropoulos (2009) proved that the flow-cut gap is bounded by a function of γ, and Chekuri, Shepherd, and Weibel (2013) showed that the gap is at most 3γ. We significantly improve these asymptotics by establishing that the flow-cut gap is O(log γ). This is achieved by showing that the edge-weighted shortest-path metric induced on the terminals admits a stochastic embedding into trees with distortion O(log γ). The latter result is tight, e.g., for a square planar lattice on Θ(γ) vertices.The preceding results refer to the setting of edge-capacitated networks. For vertex-capacitated networks, it can be significantly more challenging to control flow-cut gaps. While there is no exact vertex-capacitated version of the Okamura-Seymour Theorem, an approximate version holds; Lee, Mendel, and Moharrami (2015) showed that the vertex-capacitated flow-cut gap is O(1) on planar networks whose terminals lie on a single face. We prove that the flow-cut gap is O(γ) for vertex-capacitated instances when the terminals lie on at most γ faces. In fact, this result holds in the more general setting of submodular vertex capacities.

A face cover perspective to ℓ1 embeddings of planar graphs

“…Chekuri, Shepherd and Weibel [CSW13] constructed an embedding with distortion of 3γ. Recently, Krauthgamer, Lee and Rika [KLR19] managed to construct an embedding into 1 with O(log γ) distortion by first applying a stochastic embedding into trees. This method has benefits, since trees are very simple to work with.…”

Nearly ETH-tight Algorithms for Planar Steiner Tree with Terminals on Few Faces

“…Of particular interest is the case when all terminals lie on k given faces of the planeembedded input graph G. This parameter has a long history in the study of cuts and (multicommodity) flows (e.g. [35,10,28,27]) and shortest paths (e.g. [24,11]).…”

“…[24,11]). Krauthgamer et al [27] (in this SODA) dubbed it the terminal face cover number γ(G). The case γ(G) = 1 is known as an Okamura-Seymour graph [37].…”

Multicommodity Flows in Planar Graphs with Demands on Faces

“…Chekuri et al [3] showed a flow cut gap of 2 O(k) for k-outerplanar graphs. [6] made progress towards this conjecture by showing an O(log h) bound on the flow-cut gap, where h is the number of faces having source/sink vertices. This was subsequently improved to O( √ log h) by Filster [4].…”