Posimodular Function Optimization

TL;DR

This paper investigates the complexity of optimizing posimodular functions, revealing exponential lower bounds for both minimization and maximization, and proposes algorithms with specific time complexities for restricted ranges.

Contribution

It establishes exponential lower bounds for posimodular function optimization and provides algorithms with bounds for functions with bounded range.

Findings

Minimization requires exponential oracle calls, $ ext{Omega}(2^{n/7.54})$.

Restricted range functions allow for algorithms with $O(n^d T_f + n^{2d+1})$ complexity.

Maximization also requires exponential oracle calls, $ ext{Omega}(2^{n-1})$, with a known time complexity for bounded ranges.

Abstract

Given a posimodular function $f: 2^V \to \mathbb{R}$ on a finite set $V$, we consider the problem of finding a nonempty subset $X$ of $V$ that minimizes $f(X)$. Posimodular functions often arise in combinatorial optimization such as undirected cut functions. In this paper, we show that any algorithm for the problem requires $\Omega(2^{\frac{n}{7.54}})$ oracle calls to $f$, where $n=|V|$. It contrasts to the fact that the submodular function minimization, which is another generalization of cut functions, is polynomially solvable. When the range of a given posimodular function is restricted to be $D=\{0,1,...,d\}$ for some nonnegative integer $d$, we show that $\Omega(2^{\frac{d}{15.08}})$ oracle calls are necessary, while we propose an $O(n^dT_f+n^{2d+1})$-time algorithm for the problem. Here, $T_f$ denotes the time needed to evaluate the function value $f(X)$ for a given $X \subseteq V$. We also consider the problem of maximizing a given posimodular function. We show that $\Omega(2^{n-1})$ oracle calls are necessary for solving the problem, and that the problem has time complexity $\Theta(n^{d-1}T_f) $ when $D=\{0,1,..., d\}$ is the range of $f$ for some constant $d$.