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Budget-feasible Egalitarian Allocation of Conflicting Jobs

Sushmita Gupta, Pallavi Jain, A. Mohanapriya, Vikash Tripathi

TL;DR

This work studies Budgeted Conflict-Free Egalitarian Allocation (BCFEA), a fair-division problem that assigns conflict-free task bundles to $k$ agents under per-agent budgets, with the goal of maximizing the worst-off utility. The authors develop a comprehensive complexity and algorithmic framework: BCFEA is NP-hard even for $k=2$, yet becomes tractable under structured inputs (e.g., connected conflict graphs, bounded treewidth, interval/ chordal graphs) and fixed-parameter regimes; they provide exact exponential, FPT, pseudo-polynomial, and bicriteria approximation algorithms. Key contributions include a two-agent dichotomy by connectivity, a suite of parameterized algorithms (by components, treewidth, bundle size, and type count), an exact $ ilde{O}(2^{n})$ algorithm via subset convolution, and a bicriteria FPTAS for bounded-parameter cases. The results connect BCFEA to classical problems (Partition, 3-Coloring, Knapsack, Santa Claus) and establish practical approaches for structured inputs, laying groundwork for future work on additional fairness notions and broader utility models. Overall, the paper advances both the theoretical understanding and algorithmic toolkit for budget-aware, conflict-sensitive fair division.

Abstract

Allocating conflicting jobs among individuals while respecting a budget constraint for each individual is an optimization problem that arises in various real-world scenarios. In this paper, we consider the situation where each individual derives some satisfaction from each job. We focus on finding a feasible allocation of conflicting jobs that maximize egalitarian cost, i.e. the satisfaction of the \nc{individual who is worst-off}. To the best of our knowledge, this is the first paper to combine egalitarianism, budget-feasibility, and conflict-freeness in allocations. We provide a systematic study of the computational complexity of finding budget-feasible conflict-free egalitarian allocation and show that our problem generalizes a large number of classical optimization problems. Therefore, unsurprisingly, our problem is \NPH even for two individuals and when there is no conflict between any jobs. We show that the problem admits algorithms when studied in the realm of approximation algorithms and parameterized algorithms with a host of natural parameters that match and in some cases improve upon the running time of known algorithms.

Budget-feasible Egalitarian Allocation of Conflicting Jobs

TL;DR

This work studies Budgeted Conflict-Free Egalitarian Allocation (BCFEA), a fair-division problem that assigns conflict-free task bundles to agents under per-agent budgets, with the goal of maximizing the worst-off utility. The authors develop a comprehensive complexity and algorithmic framework: BCFEA is NP-hard even for , yet becomes tractable under structured inputs (e.g., connected conflict graphs, bounded treewidth, interval/ chordal graphs) and fixed-parameter regimes; they provide exact exponential, FPT, pseudo-polynomial, and bicriteria approximation algorithms. Key contributions include a two-agent dichotomy by connectivity, a suite of parameterized algorithms (by components, treewidth, bundle size, and type count), an exact algorithm via subset convolution, and a bicriteria FPTAS for bounded-parameter cases. The results connect BCFEA to classical problems (Partition, 3-Coloring, Knapsack, Santa Claus) and establish practical approaches for structured inputs, laying groundwork for future work on additional fairness notions and broader utility models. Overall, the paper advances both the theoretical understanding and algorithmic toolkit for budget-aware, conflict-sensitive fair division.

Abstract

Allocating conflicting jobs among individuals while respecting a budget constraint for each individual is an optimization problem that arises in various real-world scenarios. In this paper, we consider the situation where each individual derives some satisfaction from each job. We focus on finding a feasible allocation of conflicting jobs that maximize egalitarian cost, i.e. the satisfaction of the \nc{individual who is worst-off}. To the best of our knowledge, this is the first paper to combine egalitarianism, budget-feasibility, and conflict-freeness in allocations. We provide a systematic study of the computational complexity of finding budget-feasible conflict-free egalitarian allocation and show that our problem generalizes a large number of classical optimization problems. Therefore, unsurprisingly, our problem is \NPH even for two individuals and when there is no conflict between any jobs. We show that the problem admits algorithms when studied in the realm of approximation algorithms and parameterized algorithms with a host of natural parameters that match and in some cases improve upon the running time of known algorithms.
Paper Structure (12 sections, 17 theorems, 1 table)

This paper contains 12 sections, 17 theorems, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work.
  4. Preliminaries
  5. Our Results
  6. When the number of agents is two
  7. When the number of agents is arbitrary
  8. When the type is a constant.
  9. Exact Exponential Algorithm.
  10. Parameterized Algorithm
  11. Approximation Algorithm.
  12. About general utility and cost functions

Key Result

Proposition 1

pc_book Let $(T, \{\beta_t\}_{t \in V(T)})$ be a tree decomposition of a graph $G$ and $t_1t_2$ be an edge of $T$. Further, let $T_{t_1}$ and $T_{t_2}$ be two connected components of $T \setminus \{t_1t_2\}$, $A = \left( \bigcup_{t \in V(T_{t_1})} \beta_t \right) \setminus (\beta_{t_1} \cap \beta

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Definition 1
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • ...and 11 more