How to assign volunteers to tasks compatibly ? A graph theoretic and parameterized approach

TL;DR

This paper investigates a complex resource allocation problem involving conflict graphs, aiming to maximize utility for agents, and explores its computational complexity, parameterized tractability, and efficient algorithms for specific cases.

Contribution

It characterizes parameter regimes where the problem is polynomial-time solvable and introduces a faster algorithm using FFT-based polynomial multiplication.

Findings

Problem is NP-hard even with few agents and restrictions

Certain parameters allow polynomial-time algorithms

Developed a faster $2^{m}|I|^{O(1)}$ algorithm using FFT

Abstract

In this paper we study a resource allocation problem that encodes correlation between items in terms of \conflict and maximizes the minimum utility of the agents under a conflict free allocation. Admittedly, the problem is computationally hard even under stringent restrictions because it encodes a variant of the {\sc Maximum Weight Independent Set} problem which is one of the canonical hard problems in both classical and parameterized complexity. Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from the classical complexity perspective to draw the boundary between {\sf NP}-hardness and tractability for a constant number of agents. The problem was shown to be hard even for small constant number of agents and various other restrictions on the underlying graph. Notwithstanding this computational barrier, we notice that there are several parameters that are worth studying: number of agents, number of items, combinatorial structure that defines the conflict among the items, all of which could well be small under specific circumstancs. Our search rules out several parameters (even when taken together) and takes us towards a characterization of families of input instances that are amenable to polynomial time algorithms when the parameters are constant. In addition to this we give a superior $2^{m}|I|^{\Co{O}(1)}$ algorithm for our problem where $m$ denotes the number of items that significantly beats the exhaustive $\Oh(m^{m})$ algorithm by cleverly using ideas from FFT based fast polynomial multiplication; and we identify simple graph classes relevant to our problem's motivation that admit efficient algorithms.