Minimizing the Cost of EFx Allocations

TL;DR

This work defines minCost-EFx Allocation, seeking allocations that are envy-free up to any item (EFx) while minimizing total costs from item transportation or processing. It establishes foundational complexity results for general costs, proving NP-hardness even with two agents, along with a polynomial kernel in the number of items and a dynamic-programming framework for bounded valuations that yields polynomial-time solutions in key regimes. The paper further identifies parameterized and restricted-cost settings where tractability improves (e.g., a fixed number of agents, a bounded number of item types), while proving strong inapproximability results for the general model. It also shows that even with restricted cost structures, EFx remains hard, but presents DP-based approaches that exploit item-types to achieve practical runtimes, highlighting a nuanced landscape of fairness and cost in indivisible-resource allocation with implications for disaster relief, scheduling, and budget-constrained allocations.

Abstract

Ensuring fairness while limiting costs, such as transportation or storage, is an important challenge in resource allocation, yet most work has focused on cost minimization without fairness or fairness without explicit cost considerations. We introduce and formally define the minCost-EFx Allocation problem, where the objective is to compute an allocation that is envy-free up to any item (EFx) and has minimum cost. We investigate the algorithmic complexity of this problem, proving that it is NP-hard already with two agents. On the positive side, we show that the problem admits a polynomial kernel with respect to the number of items, implying that a core source of intractability lies in the number of items. Building on this, we identify parameter-restricted settings that are tractable, including cases with bounded valuations and a constant number of agents, or a limited number of item types under restricted cost models. Finally, we turn to cost approximation, proving that for any $ρ>1$ the problem is not $ρ$-approximable in polynomial time (unless $P=NP$), while also identifying restricted cost models where costs are agent-specific and independent of the actual items received, which admit better approximation guarantees.

Figures (5)

• Figure 1: Representation of an EFx allocation using stacks
• Figure 2: Representation of the maximum and minimum stackable values
• Figure 3: An EFx allocation averaging the bundle values
• Figure 4: An EFx Allocation minimizing the cost function
• Figure 5: Example illustrating the minimum threshold each set must meet in a feasible EFx allocation

Theorems & Definitions (39)

• Theorem 1: $\star$
• proof : Proof (\ref{['thm:2np']})
• Theorem 2
• proof
• Corollary 1
• Theorem 3
• proof
• Lemma 4
• proof
• Theorem 5: FT87