Multicommodity flow, well-linked terminals, and routing problems

Abstract

We study multicommodity routing problems in both edge and node capacitated undirected graphs. The input to each problem is a capacitated graph G = (V, E) and a set T of node pairs. In the simplest setting, the goal is to route a unit of flow for as many pairs as possible subject to the edge (node) capacity constraints. If the flow for a routed pair is required to be along a single path, it is the well-studied disjoint paths problem. If we allow fractional routings of the flow, it is known as the all-or-nothing flow problem. The nodes in T are referred to as terminals.In recent work [8,9], the authors obtained the first polylogarithmic approximation algorithms for some edge routing problems. A key idea in these algorithms is to decompose an instance into a collection of instances in which the terminals are well-linked. Informally speaking, a set of nodes is well-linked in a graph if it does not have small separators. A decomposition into well-linked instances was previously achieved in [8] via Räcke's hierarchical graph decomposition for oblivious routing [32]. In this paper, we design a simple new decomposition algorithm that is based on computing sparse cuts in a graph. Our new algorithm improves the earlier results for edge routing problems. Another important advantage of the algorithm is that it also applies to node-capacitated problems. We note that for oblivious routing with node capacities, an Ω( √ n) lower bound is known on the congestion [18], and hence the oblivious routing approach cannot yield poly-logarithmic bounds for well-linked decompositions. Using the new decomposition, we obtain a poly-logarithmic approximation for the node capacitated allor-nothing flow problem in general graphs and node-disjoint path problem in planar graphs with O(1) congestion. We also show that the flow-cut gap for product multicommodity *

Maximum Edge-Disjoint Paths in k-Sums of Graphs

“…We start with a grouping technique from [5] that boosts the well-linkedness. Using the preceding theorem we assume that we are working with an instance (G, X, M ) where X is welllinked.…”

“…Following the scheme from [5], one can use the grid minor as a cross bar to route a large number of pairs from M provided we can route Ω(|X|/g) terminals to the "interface" of the grid-minor of size Ω(|X|/g) that is guaranteed to exist in G. We can view the grid minor as rows and columns. In our current context we take every other node in the first row of the grid as the interface of the grid-minor.…”

“…We refer the reader to some recent papers [11,10,23,1]. A framework based on well-linked decompositions [5] has played an important role in understanding the integrality gap of the flow relaxation in undirected graphs. It is based on recursively cutting the input graph along sparse cuts until the given instance is well-linked.…”

A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2

“…Given a graph G = (V, E) and a subset T of vertices called terminals, we say that G is α-well linked for the terminals, iff for any partition (A, B) of V , |E(A, B)| ≥ α·min {|A ∩ T |, |B ∩ T |}. Chekuri, Khanna and Shepherd [CKS04b,CKS05] have shown an efficient algorithm, that, given any EDP instance (G, M), partitions it into a number of sub-instances (G 1 , M 1 ), . .…”

Polynomial bounds for the grid-minor theorem

“…Given a cluster X i , we would like to find a partition W i of X i into clusters that have the α BW -bandwidth property, such that the number of edges connecting different clusters is bounded by O(|Γ i |/ 0 ). There are by now standard algorithms for finding such a decomposition, where we repeatedly select a cluster in W i that does not have the desired bandwidth property, and partition it along a sparse cut [Räc02,CKS05]. Unfortunately, since our bandwidth parameter α BW is independent of n, such an approach can only work when |Γ i | is bounded by poly(k), which is not necessarily true in our case.…”