Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions
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arXiv
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| Natura: | Preprint |
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2026
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| author | Oum, Sang-il Sokołowski, Marek |
| author_facet | Oum, Sang-il Sokołowski, Marek |
| contents | We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10710 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions Oum, Sang-il Sokołowski, Marek Combinatorics Data Structures and Algorithms 05C85, 68R05 We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$. |
| title | Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions |
| topic | Combinatorics Data Structures and Algorithms 05C85, 68R05 |
| url | https://arxiv.org/abs/2603.10710 |