Document Open Access Logo

Round and Bipartize for Vertex Cover Approximation

Authors Danish Kashaev, Guido Schäfer

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2023.20.pdf
  • Filesize: 0.93 MB
  • 20 pages

Document Identifiers

Author Details

Danish Kashaev
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Guido Schäfer
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
  • ILLC, University of Amsterdam, The Netherlands

Cite AsGet BibTex

Danish Kashaev and Guido Schäfer. Round and Bipartize for Vertex Cover Approximation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.20

Abstract

The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (𝒢, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/ρ, where 2ρ -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph 𝒢̃ : = 𝒢/S and satisfies ρ ∈ [2,∞], with ρ = ∞ corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/ρ) (1 - α) + 2 α, where α ∈ [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph 𝒢̃, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ρ and α, which are ρ = 2 and α = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Rounding techniques
Keywords
  • Combinatorial optimization
  • approximation algorithms
  • rounding algorithms
  • beyond the worst-case analysis

Metrics

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

Related Versions

References

  1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. o(√log(n)) approximation algorithms for min uncut, min 2cnf deletion, and directed cut problems. In Symposium on the Theory of Computing, 2005. Google Scholar
  2. Antonios Antoniadis, Christian Coester, Marek Elias, Adam Polak, and Bertrand Simon. Online metric algorithms with untrusted predictions. In International Conference on Machine Learning, pages 345-355. PMLR, 2020. Google Scholar
  3. Antonios Antoniadis, Themis Gouleakis, Pieter Kleer, and Pavel Kolev. Secretary and online matching problems with machine learned advice. Advances in Neural Information Processing Systems, 33:7933-7944, 2020. Google Scholar
  4. Sanjeev Arora, Béla Bollobás, and László Lovász. Proving integrality gaps without knowing the linear program. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pages 313-322. IEEE, 2002. Google Scholar
  5. Stephan Artmann, Robert Weismantel, and Rico Zenklusen. A strongly polynomial algorithm for bimodular integer linear programming. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1206-1219, 2017. Google Scholar
  6. R Balasubramanian, Michael R Fellows, and Venkatesh Raman. An improved fixed-parameter algorithm for vertex cover. Information Processing Letters, 65(3):163-168, 1998. Google Scholar
  7. Étienne Bamas, Andreas Maggiori, and Ola Svensson. The primal-dual method for learning augmented algorithms. Advances in Neural Information Processing Systems, 33:20083-20094, 2020. Google Scholar
  8. Reuven Bar-Yehuda and Shimon Even. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2(2):198-203, 1981. Google Scholar
  9. Reuven Bar-Yehuda and Shimon Even. A local-ratio theorm for approximating the weighted vertex cover problem. Technical report, Computer Science Department, Technion, 1983. Google Scholar
  10. Abbas Bazzi, Samuel Fiorini, Sebastian Pokutta, and Ola Svensson. No small linear program approximates vertex cover within a factor 2- ε. Mathematics of Operations Research, 44(1):147-172, 2019. Google Scholar
  11. Adrian Bock, Yuri Faenza, Carsten Moldenhauer, and Andres Jacinto Ruiz-Vargas. Solving the stable set problem in terms of the odd cycle packing number. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014. Google Scholar
  12. Jonathan F Buss and Judy Goldsmith. Nondeterminism within p. SIAM Journal on Computing, 22(3):560-572, 1993. Google Scholar
  13. Jianer Chen, Iyad A Kanj, and Weijia Jia. Vertex cover: further observations and further improvements. Journal of Algorithms, 41(2):280-301, 2001. Google Scholar
  14. Jianer Chen, Lihua Liu, and Weijia Jia. Improvement on vertex cover for low-degree graphs. Networks: An International Journal, 35(4):253-259, 2000. Google Scholar
  15. Michele Conforti, Samuel Fiorini, Tony Huynh, Gwenaël Joret, and Stefan Weltge. The stable set problem in graphs with bounded genus and bounded odd cycle packing number. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2896-2915. SIAM, 2020. Google Scholar
  16. Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals of mathematics, pages 439-485, 2005. Google Scholar
  17. Rodney G Downey and Michael R Fellows. Fixed-parameter tractability and completeness. In Complexity Theory: Current Research, pages 191-225, 1992. Google Scholar
  18. Rodney G Downey and Michael Ralph Fellows. Parameterized complexity. Springer Science & Business Media, 2012. Google Scholar
  19. Samuel Fiorini, Gwenael Joret, Stefan Weltge, and Yelena Yuditsky. Integer programs with bounded subdeterminants and two nonzeros per row. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 13-24. IEEE, 2022. Google Scholar
  20. Wayne Goddard and Honghai Xu. Fractional, circular, and defective coloring of series-parallel graphs. Journal of Graph Theory, 81(2):146-153, 2016. Google Scholar
  21. Ervin Györi, Alexandr V Kostochka, and Tomasz Łuczak. Graphs without short odd cycles are nearly bipartite. Discrete Mathematics, 163(1-3):279-284, 1997. Google Scholar
  22. Eran Halperin. Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. SIAM Journal on Computing, 31(5):1608-1623, 2002. Google Scholar
  23. Johan Håstad. Some optimal inapproximability results. Journal of the ACM (JACM), 48(4):798-859, 2001. Google Scholar
  24. Dorit S Hochbaum. Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on computing, 11(3):555-556, 1982. Google Scholar
  25. Dorit S Hochbaum. Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics, 6(3):243-254, 1983. Google Scholar
  26. George Karakostas. A better approximation ratio for the vertex cover problem. In International Colloquium on Automata, Languages, and Programming, pages 1043-1050. Springer, 2005. Google Scholar
  27. Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972. Google Scholar
  28. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2- ε. Journal of Computer and System Sciences, 74(3):335-349, 2008. Google Scholar
  29. Stefan Kratsch and Magnus Wahlström. Compression via matroids: a randomized polynomial kernel for odd cycle transversal. ACM Transactions on Algorithms (TALG), 10(4):1-15, 2014. Google Scholar
  30. Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83-97, 1955. Google Scholar
  31. Silvio Lattanzi, Thomas Lavastida, Benjamin Moseley, and Sergei Vassilvitskii. Online scheduling via learned weights. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1859-1877. SIAM, 2020. Google Scholar
  32. Lap Chi Lau, Ramamoorthi Ravi, and Mohit Singh. Iterative methods in combinatorial optimization, volume 46. Cambridge University Press, 2011. Google Scholar
  33. Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. Journal of the ACM (JACM), 68(4):1-25, 2021. Google Scholar
  34. Burkhard Monien and Ewald Speckenmeyer. Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica, 22(1):115-123, 1985. Google Scholar
  35. George L Nemhauser and Leslie Earl Trotter. Vertex packings: structural properties and algorithms. Mathematical Programming, 8(1):232-248, 1975. Google Scholar
  36. Rolf Niedermeier and Peter Rossmanith. Upper bounds for vertex cover further improved. In Annual Symposium on Theoretical Aspects of Computer Science, pages 561-570. Springer, 1999. Google Scholar
  37. Rolf Niedermeier and Peter Rossmanith. On efficient fixed-parameter algorithms for weighted vertex cover. Journal of Algorithms, 47(2):63-77, 2003. Google Scholar
  38. Christos Papadimitriou and Mihalis Yannakakis. Optimization, approximation, and complexity classes. In Proceedings of the twentieth annual ACM symposium on Theory of computing, pages 229-234, 1988. Google Scholar
  39. Manish Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ml predictions. Advances in Neural Information Processing Systems, 31, 2018. Google Scholar
  40. Bruce Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Operations Research Letters, 32(4):299-301, 2004. Google Scholar
  41. Tim Roughgarden. Beyond the worst-case analysis of algorithms. Cambridge University Press, 2021. Google Scholar
  42. Edward R Scheinerman and Daniel H Ullman. Fractional graph theory: a rational approach to the theory of graphs. Courier Corporation, 2011. Google Scholar
  43. Alexander Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1998. Google Scholar
  44. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  45. Mohit Singh. Integrality gap of the vertex cover linear programming relaxation. Operations Research Letters, 47(4):288-290, 2019. Google Scholar
  46. Ulrike Stege and Michael Ralph Fellows. An improved fixed parameter tractable algorithm for vertex cover. Technical report/Departement Informatik, ETH Zürich, 318, 1999. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact