A Radius-Sensitive Approximation Algorithm for Connected Submodular Maximization
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| Format: | Preprint |
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2026
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| _version_ | 1866910270261035008 |
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| author | Cervenjak, Philip Gan, Junhao Kakimura, Naonori Umboh, Seeun William Wirth, Anthony |
| author_facet | Cervenjak, Philip Gan, Junhao Kakimura, Naonori Umboh, Seeun William Wirth, Anthony |
| contents | 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 $r$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30053 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Radius-Sensitive Approximation Algorithm for Connected Submodular Maximization Cervenjak, Philip Gan, Junhao Kakimura, Naonori Umboh, Seeun William Wirth, Anthony Data Structures and Algorithms 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 $r$. |
| title | A Radius-Sensitive Approximation Algorithm for Connected Submodular Maximization |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2605.30053 |