FPT Approximations for Connected Maximum Coverage

2026-01-13

Information Technology Convergence and Services

AI Summary

This study addresses the connected red-blue domination problem with connectivity constraints: selecting $k$ red vertices in a bipartite graph such that their induced subgraph in an auxiliary connectivity graph is connected and dominates at least $t$ blue vertices. Formally modeled as the Partial Connected Red-Blue Dominating Set problem, the authors establish for the first time that it is fixed-parameter tractable (FPT) when parameterized by $t$. Furthermore, on sparse bipartite graphs excluding $K_{d,d}$ as a subgraph, they develop an efficient parameterized approximation scheme that does not require additional structural restrictions on the connectivity graph. By integrating techniques from parameterized complexity, forbidden subgraph theory, and approximation algorithms, this work delineates the approximability landscape of the problem with respect to parameters $t$ and $k$, precisely characterizing the boundary between intractability and efficient approximability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Connected Coverage
• Partial Dominating Set
• Fixed-Parameter Tractability
• Approximation Algorithms
• Bipartite Graph

Innovation

• FPT approximation
• connectivity-constrained coverage
• Partial Connected Red-Blue Dominating Set
• parameterized complexity
• biclique-free graphs