Supermodular Maximization with Cardinality Constraints
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915569709613056 |
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| author | Chen, Xujin Hu, Xiaodong Wang, Changjun Ye, Qingjie |
| author_facet | Chen, Xujin Hu, Xiaodong Wang, Changjun Ye, Qingjie |
| contents | Let $V$ be a finite set of $n$ elements, $f: 2^V \rightarrow \mathbb{R}_+$ be a nonnegative monotone supermodular function, and $k$ be a positive integer no greater than $n$. This paper addresses the problem of maximizing $f(S)$ over all subsets $S \subseteq V$ subject to the cardinality constraint $|S| = k$ or $|S|\le k$.
Let $r$ be a constant integer. The function $f$ is assumed to be {\em $r$-decomposable}, meaning there exist $m\,(\ge1)$ subsets $V_1, \dots, V_m$ of $V$, each with a cardinality at most $r$, and a corresponding set of nonnegative supermodular functions $f_i : 2^{V_i} \rightarrow \mathbb{R}_+$, $i=1,\ldots,m$ such that $f(S) =\sum_{i=1}^m f_i(S \cap V_i)$ holds for each $S \subseteq V$. Given $r$ as an input, we present a polynomial-time $O(n^{(r-1)/2})$-approximation algorithm for this maximization problem, which does not require prior knowledge of the specific decomposition.
When the decomposition $(V_i,f_i)_{i=1}^m$ is known, an additional connectivity requirement is introduced to the problem. Let $G$ be the graph with vertex set $V$ and edge set $\cup_{i=1}^m \{uv:u,v\in V_i,u\neq v\}$. The cardinality constrained solution set $S$ is required to induce a connected subgraph in $G$. This model generalizes the well-known problem of finding the densest connected $k$-subgraph. We propose a polynomial time $O(n^{(r-1)/2})$-approximation algorithm for this generalization. Notably, this algorithm gives an $O(n^{1/2})$-approximation for the densest connected $k$-subgraph problem, improving upon the previous best-known approximation ratio of $O(n^{2/3})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Supermodular Maximization with Cardinality Constraints Chen, Xujin Hu, Xiaodong Wang, Changjun Ye, Qingjie Optimization and Control Discrete Mathematics Data Structures and Algorithms Combinatorics 05C85, 68W25, 90C27 Let $V$ be a finite set of $n$ elements, $f: 2^V \rightarrow \mathbb{R}_+$ be a nonnegative monotone supermodular function, and $k$ be a positive integer no greater than $n$. This paper addresses the problem of maximizing $f(S)$ over all subsets $S \subseteq V$ subject to the cardinality constraint $|S| = k$ or $|S|\le k$. Let $r$ be a constant integer. The function $f$ is assumed to be {\em $r$-decomposable}, meaning there exist $m\,(\ge1)$ subsets $V_1, \dots, V_m$ of $V$, each with a cardinality at most $r$, and a corresponding set of nonnegative supermodular functions $f_i : 2^{V_i} \rightarrow \mathbb{R}_+$, $i=1,\ldots,m$ such that $f(S) =\sum_{i=1}^m f_i(S \cap V_i)$ holds for each $S \subseteq V$. Given $r$ as an input, we present a polynomial-time $O(n^{(r-1)/2})$-approximation algorithm for this maximization problem, which does not require prior knowledge of the specific decomposition. When the decomposition $(V_i,f_i)_{i=1}^m$ is known, an additional connectivity requirement is introduced to the problem. Let $G$ be the graph with vertex set $V$ and edge set $\cup_{i=1}^m \{uv:u,v\in V_i,u\neq v\}$. The cardinality constrained solution set $S$ is required to induce a connected subgraph in $G$. This model generalizes the well-known problem of finding the densest connected $k$-subgraph. We propose a polynomial time $O(n^{(r-1)/2})$-approximation algorithm for this generalization. Notably, this algorithm gives an $O(n^{1/2})$-approximation for the densest connected $k$-subgraph problem, improving upon the previous best-known approximation ratio of $O(n^{2/3})$. |
| title | Supermodular Maximization with Cardinality Constraints |
| topic | Optimization and Control Discrete Mathematics Data Structures and Algorithms Combinatorics 05C85, 68W25, 90C27 |
| url | https://arxiv.org/abs/2510.19191 |