A Radius-Sensitive Approximation Algorithm for Connected Submodular Maximization

2026-05-28

AI Summary

This work addresses the problem of finding a connected subtree (or directed out-tree) with $k$ edges in a graph that maximizes a non-negative monotone submodular function, a formulation with broad applications in wireless network deployment and path planning. Focusing on undirected, directed, and rooted directed variants, the paper introduces the first polynomial-time approximation framework whose guarantees are sublinear in the optimal solution’s radius $r$. The key contribution is the GreedyRadius algorithm, which transforms the conventional bicriteria approximation—traditionally parameterized by $k$—into a single-criterion approximation based on $r$, while allowing controlled violation of the cardinality constraint. The approach achieves an approximation ratio of $\Omega(\varepsilon^3 / r^\varepsilon)$ for both CSM and DCSM, and $\Omega(\delta \varepsilon^3 / r^\varepsilon)$ for DRCSM, with at most a $(1+\delta)$-factor violation of the size constraint.

Abstract

Connected Submodular Maximization (CSM) is a graph problem with important applications to wireless network deployment, path planning, epidemic outbreaks, and cancer genome studies. In CSM, we are given a graph $G$, a non-negative monotone submodular function $f$ on subsets of the vertex set of $G$, and an integer $k$. The goal is to select a tree in $G$, with $k$ edges, whose vertex set maximizes $f$. We also study the more general Directed and Directed Rooted variants of CSM (DCSM and DRCSM respectively). In both variants, $G$ is directed and the solution must be an out-tree in $G$, with $k$ edges, whose vertex set maximizes $f$; DRCSM further specifies a vertex to be the root of the selected out-tree. For CSM, several previous works have proposed polynomial time approximation algorithms; the state-of-the-art polynomial time algorithm achieves a $Ω(\frac{1}{\sqrt{k}})$-approximation. We can also parameterize the approximation factor by the radius of the optimal solution, denoted by $r$; the state-of-the-art polynomial time algorithm achieves a $Ω(\frac{1}{r})$-approximation. In this paper, we improve on the state-of-the-art approximation factor for CSM with respect to $r$ as well as $k$, noting that $r \leq k$. We propose a polynomial time framework that, for (Directed) CSM, achieves a $Ω(\frac{\varepsilon^{3}}{{r}^{\varepsilon}})$-approximation for every constant $\varepsilon \in (0, 1]$. For DRCSM, our framework achieves a $Ω(\frac{δ\varepsilon^{3}}{{r}^{\varepsilon}})$-approximation that violates the size constraint by at most a factor of $1 + δ$ for every $δ\in [\frac{1}{k}, 1]$. A key component of our framework is GreedyRadius, which is an algorithm for DRCSM that takes another algorithm with a bicriteria approximation factor in terms of $k$ and outputs a solution with the same bicriteria approximation factor (up to constants) in terms of $r$.

Problem

• Connected Submodular Maximization
• approximation algorithm
• graph radius
• submodular function
• out-tree

Innovation

• submodular maximization
• approximation algorithm
• radius-sensitive
• connected submodular optimization
• bicriteria approximation