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

2026-03-11

AI Summary

This work addresses the efficient representation of all cuts whose values equal a given small integer \(k\) in integer-valued symmetric submodular functions. By generalizing the low-rank structure theorem for graph cut rank functions to arbitrary connectivity functions and introducing a compact encoding based on inclusion–exclusion–partition triples, the authors devise a constructive algorithm that leverages submodularity, combinatorial decomposition, and an oracle query model. The method exactly characterizes all \(k\)-valued cuts using \(O(n^{4k})\) space and constructs them in \(O(n^{2k+7}\gamma + n^{2k+8} + n^{4k+2})\) time, where \(\gamma\) denotes the cost of an oracle call. This is the first polynomial-time approach capable of handling cardinality-constrained \(k\)-valued cut enumeration.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $A$.

Problem

• connectivity functions
• symmetric submodular functions
• polynomial-size encoding
• small-value cuts
• cut representation

Innovation

• symmetric submodular functions
• polynomial-size encoding
• connectivity functions
• low-rank structure
• cut enumeration