Document

# A (2 + ε)-Factor Approximation Algorithm for Split Vertex Deletion

## File

LIPIcs.ICALP.2020.80.pdf
• Filesize: 0.66 MB
• 16 pages

## Cite As

Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. A (2 + ε)-Factor Approximation Algorithm for Split Vertex Deletion. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 80:1-80:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.80

## Abstract

In the Split Vertex Deletion (SVD) problem, the input is an n-vertex undirected graph G and a weight function w: V(G) → ℕ, and the objective is to find a minimum weight subset S of vertices such that G-S is a split graph (i.e., there is bipartition of V(G-S) = C ⊎ I such that C is a clique and I is an independent set in G-S). This problem is a special case of 5-Hitting Set and consequently, there is a simple factor 5-approximation algorithm for this. On the negative side, it is easy to show that the problem does not admit a polynomial time (2-δ)-approximation algorithm, for any fixed δ > 0, unless the Unique Games Conjecture fails. We start by giving a simple quasipolynomial time (n^O(log n)) factor 2-approximation algorithm for SVD using the notion of clique-independent set separating collection. Thus, on the one hand SVD admits a factor 2-approximation in quasipolynomial time, and on the other hand this approximation factor cannot be improved assuming UGC. It naturally leads to the following question: Can SVD be 2-approximated in polynomial time? In this work we almost close this gap and prove that for any ε > 0, there is a n^O(log 1/(ε))-time 2(1+ε)-approximation algorithm.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Approximation algorithms analysis
##### Keywords
• Approximation Algorithms
• Graph Algorithms
• Split Vertex Deletion

## Metrics

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

## References

1. R Bar-Yehuda and S Even. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2(2):198-203, 1981. URL: https://doi.org/10.1016/0196-6774(81)90020-1.
2. 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.
3. Mao-cheng Cai, Xiaotie Deng, and Wenan Zang. An approximation algorithm for feedback vertex sets in tournaments. SIAM J. Comput., 30(6):1993-2007, 2000.
4. Marek Cygan and Marcin Pilipczuk. Split vertex deletion meets vertex cover: New fixed-parameter and exact exponential-time algorithms. Inf. Process. Lett., 113(5-6):179-182, 2013.
5. Samuel Fiorini, Gwenaël Joret, and Oliver Schaudt. Improved approximation algorithms for hitting 3-vertex paths. CoRR, abs/1808.10370, 2018. URL: http://arxiv.org/abs/1808.10370.
6. Stephane Foldes and Peter L. Hammer. Split graphs. Proceedings of the 8th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pages 311-315, 1977.
7. 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. Computational Complexity 2003. URL: https://doi.org/10.1016/j.jcss.2007.06.019.
8. John M Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is np-complete. Journal of Computer and System Sciences, 20(2):219-230, 1980.
9. Daniel Lokshtanov, Pranabendu Misra, Joydeep Mukherjee, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. 2-approximating feedback vertex set in tournaments. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1010-1018. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.61.
10. Matthias Mnich, Virginia Vassilevska Williams, and Lászlo A Végh. A 7/3-approximation for feedback vertex sets in tournaments. In 24th Annual European Symposium on Algorithms (ESA 2016). Schloss Dagstuhl, 2016.
11. Jelani Nelson. A note on set cover inapproximability independent of universe size. Electronic Colloquium on Computational Complexity (ECCC), 14(105), 2007. URL: http://eccc.hpi-web.de/eccc-reports/2007/TR07-105/index.html.
12. Ewald Speckenmeyer. On feedback problems in diagraphs. In Graph-Theoretic Concepts in Computer Science, 15th International Workshop, WG '89, Castle Rolduc, The Netherlands, June 14-16, 1989, Proceedings, volume 411 of Lecture Notes in Computer Science, pages 218-231. Springer, 1989. URL: https://doi.org/10.1007/3-540-52292-1_16.
13. Jie You, Jianxin Wang, and Yixin Cao. Approximate association via dissociation. Discrete Applied Mathematics, 219:202-209, 2017. URL: https://doi.org/10.1016/j.dam.2016.11.007.
X

Feedback for Dagstuhl Publishing