The Complexity of Fair Division of Indivisible Items with Externalities
Argyrios Deligkas, Eduard Eiben, Viktoriia Korchemna, Šimon Schierreich
TL;DR
The paper tackles the computational complexity of fairly dividing indivisible items under externalities, focusing on EF, EF1, and EFX. Through reductions from classic partition problems and parameterized analyses, it shows NP-hardness for EFX existence even with as few as 3 agents or 6 value types, and identifies fixed-parameter tractable regimes by combining parameters such as the number of agents and item-types or distinct values. It also proves strong equivalences under restricted valuations (two-valued/binary) and demonstrates that agent-item-correlated valuations reduce to standard fair division without externalities, allowing the reuse of known EF1/EFX guarantees. The work thus delineates the boundary between hard and tractable cases, and provides insights and tools for algorithm design in fair division problems with externalities.
Abstract
We study the computational complexity of fairly allocating a set of indivisible items under externalities. In this recently-proposed setting, in addition to the utility the agent gets from their bundle, they also receive utility from items allocated to other agents. We focus on the extended definitions of envy-freeness up to one item (EF1) and of envy-freeness up to any item (EFX), and we provide the landscape of their complexity for several different scenarios. We prove that it is NP-complete to decide whether there exists an EFX allocation, even when there are only three agents, or even when there are only six different values for the items. We complement these negative results by showing that when both the number of agents and the number of different values for items are bounded by a parameter the problem becomes fixed-parameter tractable. Furthermore, we prove that two-valued and binary-valued instances are equivalent and that EFX and EF1 allocations coincide for this class of instances. Finally, motivated from real-life scenarios, we focus on a class of structured valuation functions, which we term agent/item-correlated. We prove their equivalence to the ``standard'' setting without externalities. Therefore, all previous results for EF1 and EFX apply immediately for these valuations.