FPT Approximation using Treewidth: Capacitated Vertex Cover, Target Set Selection and Vector Dominating Set
Huairui Chu, Bingkai Lin
TL;DR
This paper investigates three W[1]-hard problems on graphs with bounded treewidth: Capacitated Vertex Cover, Target Set Selection, and Vector Dominating Set. It presents two FPT approximation strategies that blend tree-decomposition DP with discretization and a repair step to enforce capacity/threshold constraints, exploiting monotone and splittable structure where applicable. The main contributions include a $(1+o(1))$-approximation FPT for CVC and VDS and a tunable FPT-approximation for TSS with a tradeoff between running time and approximation ratio, along with explicit running-time bounds. The methods rely on Lampis-style rounding to control DP state sizes and bag-based separator decompositions, offering a framework for FPT approximations on treewidth-bounded graphs with practical implications for related network-design and influence-propagation problems.
Abstract
Treewidth is a useful tool in designing graph algorithms. Although many NP-hard graph problems can be solved in linear time when the input graphs have small treewidth, there are problems which remain hard on graphs of bounded treewidth. In this paper, we consider three vertex selection problems that are W[1]-hard when parameterized by the treewidth of the input graph, namely the capacitated vertex cover problem, the target set selection problem and the vector dominating set problem. We provide two new methods to obtain FPT approximation algorithms for these problems. For the capacitated vertex cover problem and the vector dominating set problem, we obtain $(1+o(1))$-approximation FPT algorithms. For the target set selection problem, we give an FPT algorithm providing a tradeoff between its running time and the approximation ratio.