Submodular function minimization (SFM) is a fundamental and efficiently solvable problem 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 effi- cient 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 has interesting links to integer programming with bounded subdeterminants. To obtain our results, we establish a connection between the correctness of a natural algorithm, and the nonexistence 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  in the context of computing the girth and cogirth of certain types of binary matroids.
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We thank Karthekeyan Chandrasekaran for interesting discussions on related topics. Moreover, we are grateful to the anonymous referees for various comments and suggestions that helped to improve the quality of the presentation.
A preliminary version of this paper appeared at the ACM-SIAM Symposium on Discrete Algorithms (SODA18).
Research supported in part by the Swiss National Science Foundation grant 200021_175573.
Research supported in part by the Swiss National Science Foundation grant 200021_165866.
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Nägele, M., Sudakov, B. & Zenklusen, R. Submodular Minimization Under Congruency Constraints. Combinatorica 39, 1351–1386 (2019). https://doi.org/10.1007/s00493-019-3900-1
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