Minimizing Maximum Dissatisfaction in the Allocation of Indivisible Items under a Common Preference Graph
CoRR(2023)
摘要
We consider the task of allocating indivisible items to agents, when the
agents' preferences over the items are identical. The preferences are captured
by means of a directed acyclic graph, with vertices representing items and an
edge (a,b), meaning that each of the agents prefers item a over item b.
The dissatisfaction of an agent is measured by the number of items that the
agent does not receive and for which it also does not receive any more
preferred item. The aim is to allocate the items to the agents in a fair way,
i.e., to minimize the maximum dissatisfaction among the agents. We study the
status of computational complexity of that problem and establish the following
dichotomy: the problem is NP-hard for the case of at least three agents, even
on fairly restricted graphs, but polynomially solvable for two agents. We also
provide several polynomial-time results with respect to different underlying
graph structures, such as graphs of width at most two and tree-like structures
such as stars and matchings. These findings are complemented with fixed
parameter tractability results related to path modules and independent set
modules. Techniques employed in the paper include bottleneck assignment
problem, greedy algorithm, dynamic programming, maximum network flow, and
integer linear programming.
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