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A Polynomial Algorithm for Minimizing $k$-Distant Submodular Functions

Ryuhei Mizutani

TL;DR

The paper introduces $k$-distant submodular functions as a natural relaxation of submodularity and proves a polynomial-time algorithm for minimizing integer-valued instances when $k$ is fixed, via a linear-programming reduction and a dual-uncrossing analysis that bounds the dual-support to a polynomial-sized family. This LP-centric approach reproduces and extends the methodology used for $2/3$-submodular minimization and yields two key applications: a complexity dichotomy for $p/q$-submodular function minimization and a polynomial-time weighted matroid-intersection result under the minimum rank oracle for a broad class of matroids. The paper also establishes W[1]-hardness, showing that no FPT algorithm with respect to $k$ is likely to exist unless standard complexity-theoretic collapses occur. Collectively, these results place $k$-distant submodular minimization at a tractable boundary for fixed $k$ while clarifying its limits and connections to matroid theory and complexity dichotomies.

Abstract

This paper considers the minimization problem of relaxed submodular functions. For a positive integer $k$, a set function is called $k$-distant submodular if the submodular inequality holds for every pair whose symmetric difference is at least $k$. This paper provides a polynomial time algorithm to minimize $k$-distant submodular functions for a fixed positive integer $k$. This result generalizes the tractable result of minimizing 2/3-submodular functions, which satisfy the submodular inequality for at least two pairs formed from every distinct three sets.

A Polynomial Algorithm for Minimizing $k$-Distant Submodular Functions

TL;DR

The paper introduces -distant submodular functions as a natural relaxation of submodularity and proves a polynomial-time algorithm for minimizing integer-valued instances when is fixed, via a linear-programming reduction and a dual-uncrossing analysis that bounds the dual-support to a polynomial-sized family. This LP-centric approach reproduces and extends the methodology used for -submodular minimization and yields two key applications: a complexity dichotomy for -submodular function minimization and a polynomial-time weighted matroid-intersection result under the minimum rank oracle for a broad class of matroids. The paper also establishes W[1]-hardness, showing that no FPT algorithm with respect to is likely to exist unless standard complexity-theoretic collapses occur. Collectively, these results place -distant submodular minimization at a tractable boundary for fixed while clarifying its limits and connections to matroid theory and complexity dichotomies.

Abstract

This paper considers the minimization problem of relaxed submodular functions. For a positive integer , a set function is called -distant submodular if the submodular inequality holds for every pair whose symmetric difference is at least . This paper provides a polynomial time algorithm to minimize -distant submodular functions for a fixed positive integer . This result generalizes the tractable result of minimizing 2/3-submodular functions, which satisfy the submodular inequality for at least two pairs formed from every distinct three sets.
Paper Structure (9 sections, 10 theorems, 46 equations)

This paper contains 9 sections, 10 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. Organization.
  4. Preliminaries
  5. A polynomial algorithm
  6. Applications
  7. A complexity dichotomy for $p/q$-submodular function minimization
  8. Weighted matroid intersection under the minimum rank oracle
  9. W[1]-hardness

Key Result

Theorem 1.5

Let $f$ be an integer-valued $k$-distant submodular function on $S$. A minimizer of $f$ can be found in a polynomial number of arithmetic steps and function evaluations in $|S|^k$ and $\log B$, where $B$ is an upper bound on the absolute values of $f$.

Theorems & Definitions (22)

  • Example 1.1: Cut function with negative edge weight
  • Example 1.2: Perturbation of strictly submodular function
  • Example 1.3: Minimum rank function of two sparse paving matroids
  • Example 1.4: 2/3-Submodular function
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Claim 3.2
  • ...and 12 more