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Minimizing Submodular Functions over Hierarchical Families

Ryuhei Mizutani

TL;DR

This work studies submodular function minimization (SFM) when restricted to complements of structured set families. It introduces $k$-hierarchical lattices and proves that for such complements, any minimizer lies in a lattice of the form $\\mathcal{F}_{ST}$ with disjoint $S,T$ and $|S|,|T|\\le k$, enabling a polynomial-time reduction to standard SFM. Consequently, SFM over complements of intersecting or crossing families, and more generally unions of lattices, becomes polynomial-time; it also partially resolves the open question on intersections of parity families and provides a method to obtain the $k$-th smallest value of $f$ in polynomial time. The approach relies on a key structural lemma and a reduction via the auxiliary objective $g(X)=(n+1)f(X)+|X|$, achieving a running time of $O(n^{2k}\\tau(n))$ with $\\tau(n)$ the time for unconstrained SFM. This broadens the tractable landscape of SFM and unifies parity, lattice, and congruency constraints within a single framework.

Abstract

This paper considers submodular function minimization (SFM) restricted to a family of subsets. We show that SFM over complements of families with certain hierarchical structures can be solved in polynomial-time. This yields a polynomial-time algorithm for SFM over complements of various families, such as intersecting families, crossing families, and the unions of lattices. Moreover, this tractability result partially settles the open question posed by Nägele, Sudakov, and Zenklusen on polynomial-solvability of SFM over the intersection of parity families. Furthermore, our tractability result implies that for a constant positive integer k, the k-th smallest value of a submodular function can be obtained in polynomial-time.

Minimizing Submodular Functions over Hierarchical Families

TL;DR

This work studies submodular function minimization (SFM) when restricted to complements of structured set families. It introduces -hierarchical lattices and proves that for such complements, any minimizer lies in a lattice of the form with disjoint and , enabling a polynomial-time reduction to standard SFM. Consequently, SFM over complements of intersecting or crossing families, and more generally unions of lattices, becomes polynomial-time; it also partially resolves the open question on intersections of parity families and provides a method to obtain the -th smallest value of in polynomial time. The approach relies on a key structural lemma and a reduction via the auxiliary objective , achieving a running time of with the time for unconstrained SFM. This broadens the tractable landscape of SFM and unifies parity, lattice, and congruency constraints within a single framework.

Abstract

This paper considers submodular function minimization (SFM) restricted to a family of subsets. We show that SFM over complements of families with certain hierarchical structures can be solved in polynomial-time. This yields a polynomial-time algorithm for SFM over complements of various families, such as intersecting families, crossing families, and the unions of lattices. Moreover, this tractability result partially settles the open question posed by Nägele, Sudakov, and Zenklusen on polynomial-solvability of SFM over the intersection of parity families. Furthermore, our tractability result implies that for a constant positive integer k, the k-th smallest value of a submodular function can be obtained in polynomial-time.
Paper Structure (2 sections, 8 theorems, 6 equations)

This paper contains 2 sections, 8 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem 1.5

For a positive integer $k$, let $\mathcal{F}\subseteq 2^V$ be the complement of a $k$-hierarchical lattice. Let $f:2^V\to \mathbb{Z}$ be a submodular function. Let $X^*\in \mathcal{F}$ be a minimizer of $f$ over $\mathcal{F}$. Then, there exist $S,T\subseteq V$ with $S\cap T=\emptyset$ and $\max\{|S

Theorems & Definitions (18)

  • Example 1.1: Intersecting family
  • Example 1.2: Crossing family
  • Example 1.3: Union of $k$ lattices
  • Example 1.4: $k$-th smallest value of a submodular function
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 8 more