Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Deligkas, Argyrios, Filos-Ratsikas, Aris ORCID: 0000-0001-7868-8114 and Hollender, Alexandros (2021) Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents. In: EC '21: The 22nd ACM Conference on Economics and Computation, Budapest, Hungary.

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Official URL: http://dx.doi.org/10.1145/3465456.3467625

Abstract

We consider the ϵ-Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to ϵ). This problem was recently shown to be PPA-complete, for n agents and n cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem. For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents.

Item Type:	Conference or Workshop Item (Unspecified)
Divisions:	Faculty of Science and Engineering > School of Electrical Engineering, Electronics and Computer Science
Depositing User:	Symplectic Admin
Date Deposited:	24 Jun 2021 10:26
Last Modified:	18 Jan 2023 21:37
DOI:	10.1145/3465456.3467625
Related URLs:	Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3127547