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

TL;DR

This paper introduces a polynomial-time algorithm for minimizing a class of relaxed submodular functions called $k$-distant submodular functions, generalizing previous results for specific cases.

Contribution

It presents the first polynomial algorithm for minimizing $k$-distant submodular functions for any fixed positive integer $k$, extending prior work on 2/3-submodular functions.

Findings

Polynomial-time algorithm for fixed $k$-distant submodular functions

Generalizes previous results on 2/3-submodular functions

Provides theoretical foundation for broader class of submodular functions

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.