A simple (2+ε)-approximation algorithm for Split Vertex Deletion

09/23/2020 ∙ by Matthew Drescher, et al. ∙ Université Libre de Bruxelles 0

A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph G and weight function w: V(G) →ℚ_≥ 0, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices X such that G-X is a split graph. It is easy to show that a graph is a split graph if and only it it does not contain a 4-cycle, 5-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy 5-approximation algorithm. On the other hand, for every δ >0, SVD does not admit a (2-δ)-approximation algorithm, unless P=NP or the Unique Games Conjecture fails. For every ϵ >0, Lokshtanov, Misra, Panolan, Philip, and Saurabh recently gave a randomized (2+ϵ)-approximation algorithm for SVD. In this work we give an extremely simple deterministic (2+ϵ)-approximation algorithm for SVD.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

References

  • [1] N. Bousquet, A. Lagoutte, and S. Thomassé. Clique versus independent set. European Journal of Combinatorics, 40:73 – 92, 2014.
  • [2] N. Bousquet, A. Lagoutte, and S. Thomassé. The Erdős-Hajnal conjecture for paths and antipaths. J. Combin. Theory Ser. B, 113:261–264, 2015.
  • [3] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The strong perfect graph theorem. Ann. of Math. (2), 164(1):51–229, 2006.
  • [4] P. Erdős and A. Hajnal. Ramsey-type theorems. volume 25, pages 37–52. 1989. Combinatorics and complexity (Chicago, IL, 1987).
  • [5] J. Fox and B. Sudakov. Induced Ramsey-type theorems. Adv. Math., 219(6):1771–1800, 2008.
  • [6] A. Freund, R. Bar-Yehuda, and K. Bendel. Local ratio: a unified framework for approximation algorithms. ACM Computing Surveys, 36:422–463, 01 2005.
  • [7] S. Khot and O. Regev. Vertex cover might be hard to approximate to within . J. Comput. System Sci., 74(3):335–349, 2008.
  • [8] D. Lokshtanov, P. Misra, J. Mukherjee, F. Panolan, G. Philip, and S. Saurabh. -approximating feedback vertex set in tournaments. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1010–1018. SIAM, 2020.
  • [9] D. Lokshtanov, P. Misra, F. Panolan, G. Philip, and S. Saurabh. A (2+ )-factor approximation algorithm for split vertex deletion. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020.
  • [10] V. Rödl.

    On universality of graphs with uniformly distributed edges.

    Discrete Math., 59(1-2):125–134, 1986.
  • [11] E. Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399–401. CNRS, Paris, 1978.