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arXiv · 2007.15125

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

Argyrios Deligkas·Aris Filos-Ratsikas·Alexandros Hollender
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Abstract

We consider the $\varepsilon$-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 $\varepsilon$). 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.

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Argyrios Deligkas, Aris Filos-Ratsikas, Alexandros Hollender. 2022-09-08. Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents. https://doi.org/10.1016/j.artint.2022.103784

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