Document Open Access Logo

Approximating Upper Degree-Constrained Partial Orientations

Authors Marek Cygan, Tomasz Kociumaka



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2015.212.pdf
  • Filesize: 478 kB
  • 13 pages

Document Identifiers

Author Details

Marek Cygan
Tomasz Kociumaka

Cite AsGet BibTex

Marek Cygan and Tomasz Kociumaka. Approximating Upper Degree-Constrained Partial Orientations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 212-224, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.212

Abstract

In the Upper Degree-Constrained Partial Orientation (UDPO) problem we are given an undirected graph G=(V,E), together with two degree constraint functions d^-,d^+:V -> N. The goal is to orient as many edges as possible, in such a way that for each vertex v in V the number of arcs entering v is at most d^-(v), whereas the number of arcs leaving v is at most d^+(v). This problem was introduced by Gabow [SODA'06], who proved it to be MAXSNP-hard (and thus APX-hard). In the same paper Gabow presented an LP-based iterative rounding 4/3-approximation algorithm. As already observed by Gabow, the problem in question is a special case of the classic 3-Dimensional Matching, which in turn is a special case of the k-Set Packing problem. Back in 2006 the best known polynomial time approximation algorithm for 3-Dimensional Matching was a simple local search by Hurkens and Schrijver [SIDMA'89], the approximation ratio of which is (3+epsilon)/2; hence the algorithm of Gabow was an improvement over the approach brought from the more general problems. In this paper we show that the UDPO problem when cast as 3-Dimensional Matching admits a special structure, which is obliviously exploited by the known approximation algorithms for k-Set Packing. In fact, we show that already the local-search routine of Hurkens and Schrijver gives (4+epsilon)/3-approximation when used for the instances coming from UDPO. Moreover, the recent approximation algorithm for 3-Set Packing [Cygan, FOCS'13] turns out to be a (5+epsilon)/4-approximation for UDPO. This improves over 4/3 as the best ratio known up to date for UDPO.
Keywords
  • graph orientations
  • degree-constrained orientations
  • approximation algorithm
  • local search

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Esther M. Arkin and Refael Hassin. On local search for weighted k-set packing. In Rainer Burkard and Gerhard Woeginger, editors, Algorithms - ESA 1997, volume 1284 of LNCS, pages 13-22. Springer Berlin Heidelberg, 1997. Google Scholar
  2. Jørgen Bang-Jensen and Gregory Z. Gutin. Digraphs: theory, algorithms and applications. Springer Monographs in Mathematics. Springer, second edition, 2009. Google Scholar
  3. Piotr Berman. A d/2 approximation for maximum weight independent set in d-claw free graphs. In Magnús M. Halldórsson, editor, Algorithm Theory - SWAT 2000, volume 1851 of LNCS, pages 214-219. Springer Berlin Heidelberg, 2000. Google Scholar
  4. Piotr Berman and Marek Karpinski. Improved approximation lower bounds on small occurrence optimization. Electronic Colloquium on Computational Complexity (ECCC), 10(008), 2003. Google Scholar
  5. Yuk Hei Chan and Lap Chi Lau. On linear and semidefinite programming relaxations for hypergraph matching. Mathematical Programming, 135(1-2):123-148, 2012. Google Scholar
  6. Barun Chandra and Magnús M. Halldórsson. Greedy local improvement and weighted set packing approximation. Journal of Algorithms, 39(2):223-240, 2001. Google Scholar
  7. Marek Cygan. Improved approximation for 3-dimensional matching via bounded pathwidth local search. In 54th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 509-518. IEEE Computer Society, 2013. Google Scholar
  8. Marek Cygan, Fabrizio Grandoni, and Monaldo Mastrolilli. How to sell hyperedges: The hypermatching assignment problem. In 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 342-351. SIAM, 2013. Google Scholar
  9. Harold N. Gabow. Upper degree-constrained partial orientations. In 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 554-563. SIAM, 2006. Google Scholar
  10. Magnús M. Halldórsson. Approximating discrete collections via local improvements. In 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 160-169. SIAM, 1995. Google Scholar
  11. Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-dimensional matching. In Sanjeev Arora, Klaus Jansen, José D. P. Rolim, and Amit Sahai, editors, Approximation, Randomization, and Combinatorial Optimization - APPROX-RANDOM 2003, volume 2764 of LNCS, pages 83-97. Springer Berlin Heidelberg, 2003. Google Scholar
  12. Cor A. J. Hurkens and Alexander Schrijver. On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathematics, 2(1):68-72, 1989. Google Scholar
  13. Alexander Schrijver. Combinatorial Optimization - Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer, 2003. Google Scholar
  14. Maxim Sviridenko and Justin Ward. Large neighborhood local search for the maximum set packing problem. In Fedor V. Fomin, Rūsiņš Freivalds, Marta Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - ICALP 2013, volume 7965 of LNCS, pages 792-803. Springer Berlin Heidelberg, 2013. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail