Monotone Submodular Multiway Partition
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912494156513280 |
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| author | Bi, Richard Chandrasekaran, Karthekeyan Joshi, Soham |
| author_facet | Bi, Richard Chandrasekaran, Karthekeyan Joshi, Soham |
| contents | In submodular multiway partition (SUB-MP), the input is a non-negative submodular function $f:2^V \rightarrow \mathbb{R}_{\ge 0}$ given by an evaluation oracle along with $k$ terminals $t_1, t_2, \ldots, t_k\in V$. The goal is to find a partition $V_1, V_2, \ldots, V_k$ of $V$ with $t_i\in V_i$ for every $i\in [k]$ in order to minimize $\sum_{i=1}^k f(V_i)$. In this work, we focus on SUB-MP when the input function is monotone (termed MONO-SUB-MP). MONO-SUB-MP formulates partitioning problems over several interesting structures -- e.g., matrices, matroids, graphs, and hypergraphs. MONO-SUB-MP is NP-hard since the graph multiway cut problem can be cast as a special case. We investigate the approximability of MONO-SUB-MP: we show that it admits a $4/3$-approximation and does not admit a $(10/9-ε)$-approximation for every constant $ε>0$. Next, we study a special case of MONO-SUB-MP where the monotone submodular function of interest is the coverage function of an input graph, termed GRAPH-COVERAGE-MP. GRAPH-COVERAGE-MP is equivalent to the classic multiway cut problem for the purposes of exact optimization. We show that GRAPH-COVERAGE-MP admits a $1.125$-approximation and does not admit a $(1.00074-ε)$-approximation for every constant $ε>0$ assuming the Unique Games Conjecture. These results separate GRAPH-COVERAGE-MP from graph multiway cut in terms of approximability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_05255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monotone Submodular Multiway Partition Bi, Richard Chandrasekaran, Karthekeyan Joshi, Soham Data Structures and Algorithms In submodular multiway partition (SUB-MP), the input is a non-negative submodular function $f:2^V \rightarrow \mathbb{R}_{\ge 0}$ given by an evaluation oracle along with $k$ terminals $t_1, t_2, \ldots, t_k\in V$. The goal is to find a partition $V_1, V_2, \ldots, V_k$ of $V$ with $t_i\in V_i$ for every $i\in [k]$ in order to minimize $\sum_{i=1}^k f(V_i)$. In this work, we focus on SUB-MP when the input function is monotone (termed MONO-SUB-MP). MONO-SUB-MP formulates partitioning problems over several interesting structures -- e.g., matrices, matroids, graphs, and hypergraphs. MONO-SUB-MP is NP-hard since the graph multiway cut problem can be cast as a special case. We investigate the approximability of MONO-SUB-MP: we show that it admits a $4/3$-approximation and does not admit a $(10/9-ε)$-approximation for every constant $ε>0$. Next, we study a special case of MONO-SUB-MP where the monotone submodular function of interest is the coverage function of an input graph, termed GRAPH-COVERAGE-MP. GRAPH-COVERAGE-MP is equivalent to the classic multiway cut problem for the purposes of exact optimization. We show that GRAPH-COVERAGE-MP admits a $1.125$-approximation and does not admit a $(1.00074-ε)$-approximation for every constant $ε>0$ assuming the Unique Games Conjecture. These results separate GRAPH-COVERAGE-MP from graph multiway cut in terms of approximability. |
| title | Monotone Submodular Multiway Partition |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2411.05255 |