Submodular Minimization Under Congruency Constraints

TL;DR

This paper extends the class of constraints under which submodular function minimization remains efficiently solvable, introducing a new approach that handles sets with cardinality constraints modulo prime powers, with implications for combinatorial optimization and integer programming.

Contribution

The paper introduces a novel method for efficient submodular minimization under cardinality constraints modulo prime powers, expanding known solvable constraint classes.

Findings

Efficient SFM is possible over sets of any given lattice of cardinality r mod m for prime power m.

Established a connection between algorithm correctness and the non-existence of certain set systems.

Extended results to settle open questions on girth and cogirth of binary matroids.

Abstract

Submodular function minimization (SFM) is a fundamental and efficiently solvable problem class in combinatorial optimization with a multitude of applications in various fields. Surprisingly, there is only very little known about constraint types under which SFM remains efficiently solvable. The arguably most relevant non-trivial constraint class for which polynomial SFM algorithms are known are parity constraints, i.e., optimizing only over sets of odd (or even) cardinality. Parity constraints capture classical combinatorial optimization problems like the odd-cut problem, and they are a key tool in a recent technique to efficiently solve integer programs with a constraint matrix whose subdeterminants are bounded by two in absolute value. We show that efficient SFM is possible even for a significantly larger class than parity constraints, by introducing a new approach that combines techniques from Combinatorial Optimization, Combinatorics, and Number Theory. In particular, we can show that efficient SFM is possible over all sets (of any given lattice) of cardinality r mod m, as long as m is a constant prime power. This covers generalizations of the odd-cut problem with open complexity status, and with relevance in the context of integer programming with higher subdeterminants. To obtain our results, we establish a connection between the correctness of a natural algorithm, and the inexistence of set systems with specific combinatorial properties. We introduce a general technique to disprove the existence of such set systems, which allows for obtaining extensions of our results beyond the above-mentioned setting. These extensions settle two open questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of computing the girth and cogirth of certain types of binary matroids.