Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems

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(log O(1) n)approximation 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(log O(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 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 (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log 1.5 n) and O(log 2 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(log 2 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(log 3 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(log O(1) n)-approximation algorithms for many other problems as well.

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…Initially motivated by efficient kernels, approximation algorithms for CHORDAL DELETION have been developed recently. The current best results are a poly(OPT)-approximation [140,266] and a O(log 2 n)-approximation [260].…”

“…While the unweighted versions of TREEWIDTH k-DELETION and PLANAR H-DELETION admit an approximation algorithm whose approximation ratio only depends on k not n, such an algorithm is not known for WEIGHTED TREEWIDTH k-DELETION or WEIGHTED PLANAR H-DELETION. Agrawal et al [260] gave a randomized O(log 1.5 n)-approximation algorithm and a deterministic O(log 2 n)-approximation algorithm that run in polynomial time for fixed k, i.e., the degree of the polynomial depends on k. Bansal et al [89] gave an O(log n log log n)-approximation algorithm for the edge deletion version. The only graphs H whose weighted minor deletion problem is known to admit a constant factor approximation algorithm are single edge (WEIGHTED VERTEX COVER), triangle (WEIGHTED FEEDBACK VERTEX SET), and diamond [261].…”

“…Algorithms for TREEWIDTH k-DELETION. Here we present high-level ideals of [255,260] for TREEWIDTH k-DELETION and WEIGHTED TREEWIDTH k-DELETION respectively. These two algorithms share the following two important ingredients:…”

“…WEIGHTED TREEWIDTH k-DELETION. Agrawal et al [260] achieves an O(log 1.5 n)-approximation for WEIGHTED TREEWIDTH k-DELETION in time n O(k) . It would be interesting to see whether the running time can be made FPT with parameter k.…”

Efficient Constant-Factor Approximate Enumeration of Minimal Subsets for Monotone Properties with Weight Constraints

“…However, there is no known polynomial-time constant factor approximation algorithm for finding a minimum vertex set of this property. The current best approximation factor is polylogarithmic in the input size due to Agrawal et al [2].…”