Technology Category
Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Combinatorial OptimizationReasoning under Uncertainty: Stochastic Optimization
Approximation Algorithms for Connected Maximum Coverage, Minimum Connected Set Cover, and Node-Weighted Group Steiner Tree
🤖 AI Summary
This paper studies budget-constrained connected coverage optimization problems under graph constraints, including Connected Budgeted Maximum Coverage (CBC), Directed Rooted Connected Budgeted Coverage (DCBC), Minimum Connected Set Cover, and Node-Weighted Group Steiner Tree. Methodologically, the authors integrate hierarchical decomposition, probabilistic rounding, tree embedding, and dynamic programming, introducing a (1+ε)-budget relaxation to trade budget feasibility for improved approximation guarantees. Their contributions are threefold: (i) For CBC and DCBC, they achieve the first approximations depending only on log²|X| (where X is the ground set) and √|V| (where V is the vertex set), eliminating linear dependence on budget size or graph diameter—yielding O(log²|X|/ε²) for CBC and O(√|V| log²|X|/ε²) for DCBC; (ii) They improve the approximation ratios for Minimum Connected Set Cover and Node-Weighted Group Steiner Tree to O(log³|X|); and (iii) They unify and tighten approximation bounds across several classical network design and coverage problems.
Technology Category
Application Category
📝 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 $Oleft(frac{log^2|X|}{epsilon^2}
ight)$ and $Oleft(frac{sqrt{|V|}log^2|X|}{epsilon^2}
ight)$, resp., with a budget violation of a factor $1+epsilon$, where $epsilonin (0,1]$. 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 $Oleft(frac{1}{epsilon^2}|V|^{2/3}log|V|
ight)$ to $Oleft(frac{1}{epsilon^2}|V|^{1/2}log^2|V|
ight)$. 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.
Problem
Research questions and friction points this paper is trying to address.
Develop algorithms for Connected Budgeted maximum Coverage problem (CBC).
Improve approximation ratios for Directed rooted CBC (DCBC).
Enhance solutions for Node-Weighted Group Steiner Tree problems.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Approximation algorithms for connected coverage problems
Budget violation factor of 1+ε
Improved approximation factors for related problems
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