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Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions

Sang-il Oum, Marek Sokołowski

TL;DR

The low rank structure theorem of Boja\'nczyk, Pilipczuk, Przybyszewski, Przybyszewski, Soko{\l}owski, and Stamoulis on cut-rank functions on graphs is generalized to general connectivity functions.

Abstract

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 $A$.

Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions

TL;DR

The low rank structure theorem of Boja\'nczyk, Pilipczuk, Przybyszewski, Przybyszewski, Soko{\l}owski, and Stamoulis on cut-rank functions on graphs is generalized to general connectivity functions.

Abstract

We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining on the empty set. For a connectivity function on an -element set and an integer , we show that the family of all sets with admits a polynomial-size representation: it can be described by a list of at most 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 with . We also give an algorithm that constructs this representation in time , where is the oracle time to evaluate . 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 , we obtain a polynomial-time algorithm for finding a set with and a prescribed cardinality constraint on .
Paper Structure (8 sections, 12 theorems, 14 equations, 1 figure)

This paper contains 8 sections, 12 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Blocking digraphs
  4. Forbidden skew matching
  5. Covering all closed sets
  6. Final proof
  7. Application to submodular minimum bisection
  8. Acknowledgements.

Key Result

theorem 1

Let $n$ be a positive integer. Let $f:2^V\to\mathbb Z$ be a submodular function on an $n$-element set $V$. Let us assume that $\gamma$ is the time to compute $f(X)$ for any subset $X$ of $V$. Then we can find $X\subseteq V$ minimizing $f(X)$ in time $O(\gamma n^5 + n^6)$.

Figures (1)

  • Figure 1: A skew matching of size $5$. The solid arcs form the matching $(a_i,b_i)$, while dotted arcs indicate other allowable arcs. No arcs $a_i\to b_j$ with $i<j$ and no arcs $b_i\to a_j$ are present. Arcs between $a_i$ and $a_j$ or between $b_i$ and $b_j$ are allowed.

Theorems & Definitions (23)

  • theorem 1: note=Orlin Orlin2009,label=thm:orlin
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • lemma 5
  • ...and 13 more