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

TL;DR

This paper investigates the computational and query complexity of the $ ext{ε}$-Consensus-Halving problem with a constant number of agents, revealing a complexity dichotomy for monotone valuations and establishing PPA-completeness and lower bounds.

Contribution

It establishes a complexity dichotomy for two versus three or more agents with monotone valuations and connects the problem to a monotone Borsuk-Ulam problem, advancing understanding of its computational hardness.

Findings

Two agents with monotone valuations: polynomial-time solvable.

Three or more agents with monotone valuations: PPA-complete.

For general valuations, the problem is PPA-complete even with two agents.

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.