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Journal of Combinatorial Optimization

, Volume 22, Issue 1, pp 78–96 | Cite as

Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

  • Yuichi Asahiro
  • Jesper Jansson
  • Eiji Miyano
  • Hirotaka OnoEmail author
  • Kouhei Zenmyo
Article

Abstract

Given a simple, undirected graph G=(V,E) and a weight function w:E→ℤ+, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1,k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1+1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2−1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a ratio of 3/2 for k=2 (note that this matches the inapproximability bound above), and (2−2/(k+1)) for any k≥3, respectively, in polynomial time.

Graph orientation Degree Approximation algorithm Inapproximability Maximum flow Scheduling 

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References

  1. Asahiro Y, Miyano E, Ono H, Zenmyo K (2007) Graph orientation algorithms to minimize the maximum outdegree. Int J Found Comput Sci 18(2):197–216 MathSciNetzbMATHCrossRefGoogle Scholar
  2. Biedl T, Chan T, Ganjali Y, Hajiaghayi MT, Wood DR (2005) Balanced vertex-orderings of graphs. Discrete Appl Math 48(1):27–48 MathSciNetCrossRefGoogle Scholar
  3. Brodal GS, Fagerberg R (1999) Dynamic representations of sparse graphs. In: Proc WADS1999. LNCS, vol 1663, pp 342–351 Google Scholar
  4. Chrobak M, Eppstein D (1991) Planar orientations with low out-degree and compaction of adjacency matrices. Theor Comput Sci 86(2):243–266 MathSciNetzbMATHCrossRefGoogle Scholar
  5. Chv́atal V (1975) A combinatorial theorem in plane geometry. J Comb Theory, Ser B 18:39–41 MathSciNetCrossRefGoogle Scholar
  6. Cormen T, Leiserson C, Rivest R (1990) Introduction to algorithms. MIT Press, Cambridge zbMATHGoogle Scholar
  7. Fomin FV, Matamala M, Rapaport I (2004) Complexity of approximating the oriented diameter of chordal graphs. J Graph Theory 45(4):255–269 MathSciNetzbMATHCrossRefGoogle Scholar
  8. Garey M, Johnson D (1979) Computers and intractability: A guide to the theory of NP-completeness. W H Freeman, New York zbMATHGoogle Scholar
  9. Goldberg AV, Rao S (1998) Beyond the flow decomposition barrier. J ACM 45(5):783–797 MathSciNetzbMATHCrossRefGoogle Scholar
  10. Horowitz E, Sahni S (1976) Exact and approximate algorithms for scheduling nonidentical processors. J ACM 23(2):317–327 MathSciNetzbMATHCrossRefGoogle Scholar
  11. Kára J, Kratochvíl J, Wood DR (2005) On the complexity of the balanced vertex ordering problem. In: Proc COCOON2005. LNCS, vol 3595, pp 849–858 Google Scholar
  12. King V, Rao S, Tarjan R (1994) A faster deterministic maximum flow algorithm. J Algorithms 17:447–474 MathSciNetCrossRefGoogle Scholar
  13. Kowalik L (2006) Approximation scheme for lowest outdegree orientation and graph density measures. In: Proc ISAAC2006. LNCS, vol 4288, pp 557–566 Google Scholar
  14. Lenstra JK, Shmoys DB, Tardos E (1990) Approximation algorithms for scheduling unrelated parallel machines. Math Program 46(3):259–271 MathSciNetzbMATHCrossRefGoogle Scholar
  15. O’Rourke J (1987) Art gallery theorems and algorithms. Oxford University Press, Oxford zbMATHGoogle Scholar
  16. Pinedo M (2002) Scheduling: Theory, algorithms, and systems, 2nd edn. Prentice-Hall, Englewood Cliffs zbMATHGoogle Scholar
  17. Schrijver A (2003) Combinatorial optimization. Springer, Berlin zbMATHGoogle Scholar
  18. Schuurman P, Woeginger GJ (1999) Polynomial time approximation algorithms for machine scheduling: Ten open problems. J Sched 2:203–213 MathSciNetzbMATHCrossRefGoogle Scholar
  19. Venkateswaran V (2004) Minimizing maximum indegree. Discrete Appl Math 143(1–3):374–378 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Yuichi Asahiro
    • 1
  • Jesper Jansson
    • 2
  • Eiji Miyano
    • 3
  • Hirotaka Ono
    • 4
    Email author
  • Kouhei Zenmyo
    • 3
  1. 1.Department of Information ScienceKyushu Sangyo UniversityFukuokaJapan
  2. 2.Ochanomizu UniversityTokyoJapan
  3. 3.Department of Systems Design and InformaticsKyushu Institute of TechnologyFukuokaJapan
  4. 4.Department of InformaticsKyushu UniversityFukuokaJapan

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