🤖 AI Summary
This paper studies fair allocation of indivisible goods among agents with additive, heterogeneous preferences, where classical fairness notions—such as proportionality and envy-freeness—are often unattainable. To address this, we introduce *bounded sharing*: at most $k$ goods may be fractionally allocated to multiple agents. We provide the first systematic characterization of the fairness feasibility boundary under shared-item constraints, uncovering a fundamental complexity dichotomy between binary and non-degenerate valuations. Through combinatorial optimization and computational complexity analysis, we establish precise tractability thresholds: polynomial-time algorithms for feasibility checking and fair allocation construction under both valuation classes, alongside tight NP-hardness results beyond these boundaries. Our work significantly expands the scope of achievable fairness in indivisible goods allocation by quantifying the minimal sharing required to restore fairness guarantees.
📝 Abstract
A set of objects is to be divided fairly among agents with different tastes, modeled by additive utility-functions. An agent is allowed to share a bounded number of objects between two or more agents in order to attain fairness. The paper studies various notions of fairness, such as proportionality, envy-freeness, equitability, and consensus. We analyze the run-time complexity of finding a fair allocation with a given number of sharings under several restrictions on the agents' valuations, such as: binary generalized-binary and non-degenerate.
