A Constant-factor Approximation for Weighted Bond Cover

2021-05-03

International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

AI Summary

This work studies the weighted $mathcal{F}$-vertex deletion problem, focusing on the prototypical case of weighted $c$-Bond Cover (i.e., weighted $ heta_c$-minor deletion). We present the first constant-factor approximation algorithm for the class of $ heta_c$-minor-free graphs, resolving a long-standing gap in the design of $O(1)$-approximations for weighted vertex deletion problems on minor-closed graph families. Methodologically, we introduce a novel integration of structural graph theory—specifically, the graph structure theorem—with the primal-dual framework, enabling unified handling of three canonical structural cases. Our approach further incorporates two-terminal protrusion reduction and localized weighted contraction techniques. The resulting algorithm achieves an $O(1)$ approximation ratio and constitutes the first generic design paradigm for weighted vertex deletion on minor-closed families—strictly improving upon prior methods, which only applied to three isolated special cases.

Abstract

The Weighted $mathcal{F}$-Vertex Deletion for a class ${cal F}$ of graphs asks, weighted graph $G$, for a minimum weight vertex set $S$ such that $G-Sin{cal F}.$ The case when ${cal F}$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted $mathcal{F}$-Vertex Deletion. Only three cases of minor-closed ${cal F}$ are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class ${cal F}$ of $ heta_c$-minor-free graphs, under the equivalent setting of the Weighted $c$-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret, Paul, Sau, Saurabh, and Thomass'{e}, SIDMA'14] which states the following: any graph $G$ containing a $ heta_c$-minor-model either contains a large two-terminal protrusion, or contains a constant-size $ heta_c$-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted $mathcal{F}$-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families.

Problem

• Weighted F-Vertex Deletion Problem
• Approximation Algorithm
• Graph Theory

Innovation

• Approximation Algorithm
• Weighted F-Vertex Deletion
• Graph Theory