Akanksha Agrawal
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Abstract
For a family of graphs ℱ, the W<scp;>eighted</scp;> ℱ V<scp;>ertex</scp;> D<scp;>eletion</scp;> problem, 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. We devise a recursive scheme to obtain O(logO(1) n)-approximation algorithms for such problems, building upon the classical technique of finding balanced separators. We obtain the first O(logO(1) n)-approximation algorithms for the following problems.
• Let F be a finite set of graphs containing a planar graph, and ℱ=G(F) be the maximal family of graphs such that every graph H∈ 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 (WP F-MFD) problem. We give a randomized and a deterministic approximation algorithms for WP F-MFD with ratios O(log1.5 n) and O(log2 n), respectively. Prior to our work, a randomized constant factor approximation algorithm for the unweighted version was known [FOCS 2012]. After our work, a deterministic constant factor approximation algorithm for the unweighted version was also obtained [SODA 2019].
• We give an O(log2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion, 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.
We believe that our recursive scheme can be applied to obtain O(logO(1) n)-approximation algorithms for many other problems as well.
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Index Terms
- Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems
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