On 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 reallife applications, such as, 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, already the single-commodity case of this problem is 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. It particular, we show that some of our approximation results are in fact best possible, unless P=NP.

Demonstration of OpenFlow-Controlled Network Orchestration for Adaptive SVC Video Manycast

“…The network is still the directed graph in which the bandwidth capacity of each link is bounded. Then, in order to serve and minimize the possibility of network congestion in the future (i.e., maximizing the revenue gain from subsequent requests), we need to solve the maximum 2-splittable flow problem, which is known to be -hard [24]. Due to the complexity, we will propose several time-efficient heuristics for MRLM in the next section.…”

A graph-based approach for proteoform identification and quantification using top-down homogeneous multiplexed tandem mass spectra

“…To our best knowledge, the ME k SF problem has not been studied. However, the maximum k -splittable flow (M k SF) problem, which is related to the ME k SF problem, has been extensively studied and has various applications in commodity transportation and telecommunication network optimization [13–18]. …”

“…Baier et al [13, 14] first investigated the M k SF problem and proved the NP-hardness of the problem on directed graphs for k =2. They proposed approximation algorithms with a performance ratio for the maximum 2 and 3-splittable flow problem and presented a -approximation algorithm for the general M k SF problem.…”

The k-splittable flow model and a heuristic algorithm for minimizing congestion in the MPLS networks

“…A set of commodities is denoted by L, each commodity l ∈ L has a source node s l , a destination node t l , an amount d l to be delivered and a maximal number of directed paths that the commodity can use k l . Limiting the number of paths that a commodity can use is in fact the ksplittable multi-commodity flow problem which is introduced by Baier [1]. The k-splittable flow problem can be described as the above.…”

“…Baier et al [1], who solved the Maximum Budget-Constrained Singe-and Multi-commodity k-splittable flow problem using approximation algorithms. The authors proved that the maximum single-commodity k-splittable flow problem is NP-hard in the strong sense for directed graphs.…”