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Conflict and Fairness in Resource Allocation

Susobhan Bandopadhyay, Aritra Banik, Sushmita Gupta, Pallavi Jain, Abhishek Sahu, Saket Saurabh, Prafullkumar Tale

TL;DR

The parameterized complexity of the fair allocation problem (and its variants) is studied by considering several natural and structural parameters by considering several natural and structural parameters.

Abstract

In the standard model of fair allocation of resources to agents, every agent has some utility for every resource, and the goal is to assign resources to agents so that the agents' welfare is maximized. Motivated by job scheduling, interest in this problem dates back to the work of Deuermeyer et al. [SIAM J. on Algebraic Discrete Methods'82]. Recent works consider the compatibility between resources and assign only mutually compatible resources to an agent. We study a fair allocation problem in which we are given a set of agents, a set of resources, a utility function for every agent over a set of resources, and a {\it conflict graph} on the set of resources (where an edge denotes incompatibility). The goal is to assign resources to the agents such that $(i)$ the set of resources allocated to an agent are compatible with each other, and $(ii)$ the minimum satisfaction of an agent is maximized, where the satisfaction of an agent is the sum of the utility of the assigned resources. Chiarelli et al. [Algorithmica'22] explore this problem from the classical complexity perspective to draw the boundary between the cases that are polynomial-time solvable and those that are \NP-hard. In this article, we study the parameterized complexity of the problem (and its variants) by considering several natural and structural parameters.

Conflict and Fairness in Resource Allocation

TL;DR

The parameterized complexity of the fair allocation problem (and its variants) is studied by considering several natural and structural parameters by considering several natural and structural parameters.

Abstract

In the standard model of fair allocation of resources to agents, every agent has some utility for every resource, and the goal is to assign resources to agents so that the agents' welfare is maximized. Motivated by job scheduling, interest in this problem dates back to the work of Deuermeyer et al. [SIAM J. on Algebraic Discrete Methods'82]. Recent works consider the compatibility between resources and assign only mutually compatible resources to an agent. We study a fair allocation problem in which we are given a set of agents, a set of resources, a utility function for every agent over a set of resources, and a {\it conflict graph} on the set of resources (where an edge denotes incompatibility). The goal is to assign resources to the agents such that the set of resources allocated to an agent are compatible with each other, and the minimum satisfaction of an agent is maximized, where the satisfaction of an agent is the sum of the utility of the assigned resources. Chiarelli et al. [Algorithmica'22] explore this problem from the classical complexity perspective to draw the boundary between the cases that are polynomial-time solvable and those that are \NP-hard. In this article, we study the parameterized complexity of the problem (and its variants) by considering several natural and structural parameters.
Paper Structure (11 sections, 23 theorems, 1 figure)

This paper contains 11 sections, 23 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:np-hardness-results']}
  3. Proof of Theorem \ref{['thm:fpt-nr-of-vertices']}
  4. Proof of Theorem \ref{['thm:equiavlence']} and Theorem \ref{['lem:mwis-degenerate']}
  5. Randomized Algorithm.
  6. Deterministic Algorithm.
  7. Proof of Theorem \ref{['thm:fpt-nr-of-non-edges']}
  8. Proof of \ref{['thm:twocliques']}
  9. Proof of \ref{['thm:kernel-partial-complete']}
  10. Proof of \ref{['thm:kernel-contrast']}
  11. Conclusion

Key Result

Theorem 0

We prove that

Figures (1)

  • Figure 1: Bird's-eye view of the results in the article. An arrow from parameter $\ell_1$ to $\ell_2$ implies that $\ell_2$ is upper bounded by some computable function of $\ell_1$. The red-colored rectangles denote that the problem is NP-hard even when the parameter is a constant. The green-colored rectangles denote that there is an algorithm. Note that some of these results are conditional or restrictive. Please refer to the theorems for precise statements.

Theorems & Definitions (26)

  • Theorem 0
  • Theorem 0
  • Theorem 0
  • Definition : Independence Friendly Graph Class
  • Theorem 0
  • Theorem 0
  • Theorem 0
  • Theorem 0
  • Theorem 0
  • Theorem 0
  • ...and 16 more