Approximation Algorithms for Connected Maximum Coverage, Minimum Connected Set Cover, and Node-Weighted Group Steiner Tree

TL;DR

This work studies CBC and its directed rooted variant DCBC, along with related problems such as UCBC, CSC, GST, and DST. It introduces polynomial-time bicriteria approximation algorithms that deliver budget-violation guarantees of at most $1+\\epsilon$ while achieving sublinear approximation ratios, notably $O\left(\frac{\log^2|X|}{\\epsilon^2}\right)$ for UCBC and $O\left(\frac{\sqrt{|V|}\log^2|X|}{\\epsilon^2}\right)$ for DCBC. A key technical contribution is a node-weighted Steiner-tree subroutine for directed graphs, with a bound $O\left((1+\\epsilon)\sqrt{|V\\setminus R|}\log|R|\right)$ relative to the fractional relaxation, and a special bidirected-sink-terminal case achieving $O(\log|R|)$-optimality. These results imply improved bicriteria bounds for DBNS, CSC, and GST, demonstrating substantial progress over prior linear-in-|V| or budget-dependent guarantees. The findings have broad impact on network design and coverage problems, offering practical algorithms with provable guarantees under budget constraints and enabling tighter reductions to related combinatorial optimization tasks.

Abstract

In the Connected Budgeted maximum Coverage problem (CBC), we are given a collection of subsets $\mathcal{S}$, defined over a ground set $X$, and an undirected graph $G=(V,E)$, where each node is associated with a set of $\mathcal{S}$. Each set in $\mathcal{S}$ has a different cost and each element of $X$ gives a different prize. The goal is to find a subcollection $\mathcal{S}'\subseteq \mathcal{S}$ such that $\mathcal{S}'$ induces a connected subgraph in $G$, the total cost of the sets in $\mathcal{S}'$ does not exceed a budget $B$, and the total prize of the elements covered by $\mathcal{S}'$ is maximized. The Directed rooted Connected Budgeted maximum Coverage problem (DCBC) is a generalization of CBC where the underlying graph $G$ is directed and in the subgraph induced by $\mathcal{S}'$ in $G$ must be an out-tree rooted at a given node. The current best algorithms achieve approximation ratios that are linear in the size of $G$ or depend on $B$. In this paper, we provide two algorithms for CBC and DCBC that guarantee approximation ratios of $O\left(\frac{\log^2|X|}{ε^2}\right)$ and $O\left(\frac{\sqrt{|V|}\log^2|X|}{ε^2}\right)$, resp., with a budget violation of a factor $1+ε$, where $ε\in (0,1]$. Our algorithms imply improved approximation factors of other related problems. For the particular case of DCBC where the prize function is additive, we improve from $O\left(\frac{1}{ε^2}|V|^{2/3}\log|V|\right)$ to $O\left(\frac{1}{ε^2}|V|^{1/2}\log^2|V|\right)$. For the minimum connected set cover, a minimization version of CBC, and its directed variant, we obtain approximation factors of $O(\log^3|X|)$ and $O(\sqrt{|V|}\log^3|X|)$, resp. For the Node-Weighted Group Steiner Tree and and its directed variant, we obtain approximation factors of $O(\log^3k)$ and $O(\sqrt{|V|}\log^3k)$, resp., where $k$ is the number of groups.

Theorems & Definitions (25)

• Lemma 3.1
• proof
• Theorem 3.2
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['th:ratiotree']}
• Lemma 3.4: Lemma 3 in bateni2018improved
• Theorem 3.5
• Corollary 3.6
• Theorem 4.1