Graph orientation to maximize the minimum weighted outdegree

Yuichi Asahiro, Jesper Andreas Jansson, Eiji Miyano, Hirotaka Ono

Research output: Chapter in book / Conference proceedingConference article published in proceeding or bookAcademic researchpeer-review

Abstract

We study a new variant of the graph orientation problem called MAXMINO where the input is an undirected, edge-weighted graph and the objective is to assign a direction to each edge so that the minimum weighted outdegree (taken over all vertices in the resulting directed graph) is maximized. All edge weights are assumed to be positive integers. This problem is closely related to the job scheduling on parallel machines, called the machine covering problem, where its goal is to assign jobs to parallel machines such that each machine is covered as much as possible. First, we prove that MAXMINO is strongly NP-hard and cannot be approximated within a ratio of 2 = ¿ for constant ¿ > 0 in polynomial time unless P = NP, even if all edge weights belong to {2}, every vertex has degree at most three, and the input graph is bipartite or planar. Next, we show how to solve MAXMINO exactly in polynomial time for the special case in which all edge weights are equal to 1. This technique gives us a simple polynomial-time w max /w min - approximation algorithm for MAXMINO where w max and w min denote the maximum and minimum weights among all the input edges. Furthermore we also observe that this approach yields an exact algorithm for the general case of MAXMINO whose running time is polynomial whenever the number of edges having weight larger than w min is at most logarithmic in the number of vertices. Finally, we, show that MAXMINO is solvable in polynomial time if the input is a cactus graph.
Original languageEnglish
Title of host publicationProceedings of the Eleventh Workshop on Advances in Parallel and Distributed Computational Models (APDCM 2009), part of the Twenty-Third IEEE International Parallel and Distributed Processing Symposium (IPDPS 2009)
PublisherIEEE
DOIs
Publication statusPublished - 2009
Externally publishedYes
EventThe Eleventh Workshop on Advances in Parallel and Distributed
Computational Models, APDCM 2009
- Rome, Italy
Duration: 23 May 200929 May 2009

Conference

ConferenceThe Eleventh Workshop on Advances in Parallel and Distributed
Computational Models, APDCM 2009
CountryItaly
CityRome
Period23/05/0929/05/09

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