The k-Splittable Flow Problem

Abstract

In traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, no restriction is imposed on the number of paths used for delivering each commodity; it is thus feasible to spread the flow over a large number of different paths. Motivated by routing problems arising in real-life applications, e.g., telecommunication, unsplittable flows have moved into the focus of research. Here, the demand of each commodity may not be split but has to be sent along a single path.In this paper a generalization of this problem is studied. In the considered flow model, a commodity can be split into a bounded number of chunks which can then be routed on different paths. In contrast to classical (splittable) flows and unsplittable flows, the single-commodity case of this problem is already NP-hard and even hard to approximate. We present approximation algorithms for the single-and multi-commodity case and point out strong connections to unsplittable flows. Moreover, results on the hardness of approximation are presented. In particular, we show that some of our approximation results are in fact best possible, unless P = NP.

Maximum Flows on Disjoint Paths

“…These side-constraints may arise from the application itself, but they can also arise due to restrictions induced by available technology or by the choice of routing protocol; see [13] for a survey illustrating some of these issues. The work most closely related to our own, though, concerns k-splittable flows introduced by Baier, Köhler and Skutella [8]. A k-splittable flow is a flow that can be routed along k paths -note that these paths are not required to be disjoint.…”

k‐Splittable delay constrained routing problem: A branch‐and‐price approach

“…Such an approach is well suited when considering a limit on the number of active paths in the routing solution [4]. This constraint is known to define the k-splittable flow [2]. Although these authors showed that it turns most network flow problems NP-hard, a few references were dedicated to exact algorithms.…”

“…Both these difficulties have been treated in the recent literature, but separately. The k-splittable flow problem has been first presented by Baier et al [2] and then analyzed by Martens and Skutella [12]. The basic idea is to consider a compromise situation between the splittable and © 2009 Wiley Periodicals, Inc. unsplittable cases.…”

Maximizing Throughput in Queueing Networks with Limited Flexibility

“…We will denote this first algorithm as Algorithm 1. It is the combination of the algorithm of [6] with the algorithm of [10], and goes through the following stages:…”

“…The algorithms of [6] and [25] used in Section 4.2.1 are cost preserving, in the sense that the cost of the integral solution produced by rounding a fractional solution is not bigger than the cost of that fractional solution. When in the first step we transform the budget-constrained ksplittable flow problem into a budget-constrained exactly-k-splittable flow problem, [6] proves that the optimal solution of the latter is not only an 1/2 approximation of the former, but it also respects the initial budget constraint. Also the algorithm of [25] we use in Section 4.2.1 produces an assignment that always respects the budgetary constraint (although it may not produce the optimal makespan).…”