TY - JOUR

T1 - Two’s Company, Three’s a Crowd: Consensus-Halving for a Constant Number of Agents

AU - Deligkas, Argyrios

AU - Filos-Ratsikas, Aris

AU - Hollender, Alexandros

N1 - Funding Information:
Alexandros Hollender was supported by an EPSRC doctoral studentship (Reference 1892947 ).
Publisher Copyright:
© 2022 The Author(s)

PY - 2022/12/1

Y1 - 2022/12/1

N2 - 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.

AB - 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.

KW - Consensus-halving

KW - Fair division

KW - Computational complexity

KW - Query complexity

KW - Robertson-Webb

U2 - 10.1016/j.artint.2022.103784

DO - 10.1016/j.artint.2022.103784

M3 - Article

VL - 313

JO - Artificial Intelligence

JF - Artificial Intelligence

SN - 0004-3702

M1 - 103784

ER -