Single source multiroute flows and cuts on uniform capacity networks

Abstract

For an integer h ≥ 1, an elementary h-route flow is a flow along h edge disjoint paths between a source and a sink, each path carrying a unit of flow, and a single commodity h-route flow is a non-negative linear combination of elementary h-flows. An instance of a single source multicommodity flow problem for a graph G = (V, E) consists of a source vertex s e V and k sinks t1,..., tk e V; we denote it I = (s; t1,..., tk). In the single source multicommodity multiroute flow problem, we are given an instance I = (s; t1,..., tk) and an integer h ≥ 1, and the objective is to maximize the total amount of flow that is transferred from the source to the sinks so that the capacity constraints are obeyed and, moreover, the flow of each commodity is an h-route flow.We study the relation between classical and multiroute single source flows on networks with uniform capacities and we provide a tight bound. In particular, we prove the following result. Given an instance ≥ = (s; t1,...,tk) such that each s - ti pair is h-connected, the maximum classical flow between s and the ti's is at most 2(1 - 1/h)-times larger than the maximum h-route flow between s and the ti's and this is the best possible bound for h ≥ 2. This, as we show, is in contrast to the situation of general multicommodity multiroute flows that are up to k(1 - 1/h)-times smaller than their classical counterparts.As a corollary, we establish a max-flow min-cut theorem for the single source multicommodity multiroute flow and cut. An h-disconnecting cut for I is a set of edges F ⊆ E such that for each i, the maximum h-flow between s - ti is zero. We show that the maximum h-flow is within 2(h-1) of the mininimum h-disconnecting cut, independently of the number of commodities; we also describe a 2(h - 1)-approximation algorithm for the minimum h-disconnecting cut problem.