On fixed cost k-flow problems

Mohammadtaghi Hajiaghayi, Rohit Khandekar, Guy Kortsarz, Zeev Nutov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In the Fixed Cost k -Flow problem, we are given a graph G = (V,E) with edge-capacities {u e |e ∈E} and edge-costs {c e |e ∈E}, source-sink pair s,t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an st-cut in H is at least k. We show that Group Steiner is a special case of Fixed Cost k -Flow, thus obtaining the first polylogarithmic lower bound for the problem; this also implies the first non constant lower bounds for the Capacitated Steiner Network and Capacitated Multicommodity Flow problems. We then consider two special cases of Fixed Cost k -Flow. In the Bipartite Fixed-Cost k -Flow problem, we are given a bipartite graph G = (A ∪ B,E) and an integer k > 0. The goal is to find a node subset S ⊆ A ∪ B of minimum size |S| such G has k pairwise edge-disjoint paths between S ∩ A and S ∩ B. We give an O(√k log k) approximation for this problem. We also show that we can compute a solution of optimum size with Ω(k/polylog(n)) paths, where n = |A| + |B|. In the Generalized-P2P problem we are given an undirected graph G = (V,E) with edge-costs and integer charges {b v :v ∈ V}. The goal is to find a minimum-cost spanning subgraph H of G such that every connected component of H has non-negative charge. This problem originated in a practical project for shift design [10]. Besides that, it generalizes many problems such as Steiner Forest, k -Steiner Tree, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a related problem called Connected Rent or Buy Multicommodity Flow and give a log3+ε n approximation scheme for it using Group Steiner techniques.

Original languageEnglish
Title of host publicationApproximation and Online Algorithms - 11th International Workshop, WAOA 2013, Revised Selected Papers
PublisherSpringer Verlag
Pages49-60
Number of pages12
ISBN (Print)9783319080000
DOIs
StatePublished - 2014
Event11th International Workshop on Approximation and Online Algorithms, WAOA 2013 - Sophia Antipolis, France
Duration: 5 Sep 20136 Sep 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8447 LNCS

Conference

Conference11th International Workshop on Approximation and Online Algorithms, WAOA 2013
Country/TerritoryFrance
CitySophia Antipolis
Period5/09/136/09/13

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

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