FPT Approximation Schemes for Maximizing Submodular Functions

TL;DR

This paper develops fixed parameter tractable approximation algorithms for maximizing submodular functions, applicable to various real-world problems including social choice and election systems.

Contribution

It introduces FPT approximation schemes for submodular maximization based on $p$-separability properties, broadening applicability to practical problems.

Findings

Algorithms are effective for low $p$ and $K$ values.

Applicable to social choice and multiwinner election problems.

Provides theoretical guarantees for approximation quality.

Abstract

We investigate the existence of approximation algorithms for maximization of submodular functions, that run in fixed parameter tractable (FPT) time. Given a non-decreasing submodular set function $v: 2^X \to \mathbb{R}$ the goal is to select a subset $S$ of $K$ elements from $X$ such that $v(S)$ is maximized. We identify three properties of set functions, referred to as $p$-separability properties, and we argue that many real-life problems can be expressed as maximization of submodular, $p$-separable functions, with low values of the parameter $p$. We present FPT approximation schemes for the minimization and maximization variants of the problem, for several parameters that depend on characteristics of the optimized set function, such as $p$ and $K$. We confirm that our algorithms are applicable to a broad class of problems, in particular to problems from computational social choice, such as item selection or winner determination under several multiwinner election systems.