FPT Approximations for Connected Maximum Coverage

TL;DR

This paper introduces a unifying model for connectivity-constrained coverage problems, establishes their computational complexity, and develops fixed-parameter tractable approximation schemes under certain graph restrictions.

Contribution

It unifies various connectivity-constrained coverage problems, proves their parameterized inapproximability, and provides FPT approximation schemes for specific graph classes.

Findings

Problem is fixed-parameter tractable by parameter t.

Efficient approximation schemes exist for bipartite graphs excluding K_{d,d}.

FPT approximations work without restrictions on the connectivity graph.

Abstract

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set. Given a red-blue bipartite graph $G$ and an auxiliary connectivity graph $G_{conn}$ on red vertices, and integers $k, t$, the task is to find a $k$-sized subset of red vertices that dominates at least $t$ blue vertices, and that induces a connected subgraph in $G_{conn}$. This formulation captures connected variants of Max Coverage, Partial Dominating Set, and Partial Vertex Cover studied in prior literature. After identifying (parameterized) inapproximability results inherited from known problems, we first show that the problem is fixed-parameter tractable by $t$. Furthermore, when the bipartite graph excludes $K_{d,d}$ as a subgraph, we design (resp. efficient) parameterized approximation schemes for approximating $t$ (resp. $k$). Notably, these FPT approximations do not impose any restrictions on $G_{conn}$. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.