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

Authors Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Geevarghese Philip, Saket Saurabh

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Author Details

Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Pranabendu Misra
  • Max Planck Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany
Fahad Panolan
  • IIT Hyderabad, India
Geevarghese Philip
  • Chennai Mathematical Institute, UMI ReLaX, Chennai, India
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
  • University of Bergen, Norway

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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)


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
  • Approximation Algorithms
  • Graph Algorithms
  • Split Vertex Deletion


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