Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems

Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

9 Scopus citations

Abstract

Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: Given an n-vertex undirected graph G and a weight function w : V (G)→R+, find a minimum weight subset S V (G) such that G-S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(logO(1) n)-A pproximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(logO(1) n)-approximation algorithms for the following vertex deletion problems. Let F be a finite set of graphs containing a planar graph, and F = G(F) be the family of graphs such that every graph H2 G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F = G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log1.5 n) and O(log2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012]. We give an O(log2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. We give an O(log3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one. We believe that our recursive scheme can be applied to obtain O(logO(1) n)-approximation algorithms for many other problems as well.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018
EditorsEric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen
PublisherSchloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Number of pages15
ISBN (Print)9783959770859
DOIs
StatePublished - 1 Aug 2018
Event21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States
Duration: 20 Aug 201822 Aug 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume116
ISSN (Print)1868-8969

Conference

Conference21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018
Country/TerritoryUnited States
CityPrinceton
Period20/08/1822/08/18

Keywords

  • Approximation Algorithms
  • Planar-F-Deletion
  • Separator

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