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

Abstract

We consider the -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 ). This problem was recently shown to be PPA-complete, for agents and 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.

Pizza Sharing Is PPA-Hard

“…We begin by showing PPA-hardness. By combining the PPA-hardness result of (Filos-Ratsikas et al 2020) with Lemma 7 we get the following.…”

“…Since mass partitions lie in the intersection of topology, discrete geometry, and computer science there are several surveys on the topic; (Blagojević et al 2018;De Loera et al 2019;Matoušek 2008;Živaljević 2017) focus on the topological point of view, while (Agarwal, Erickson et al 1999;Edelsbrunner 2012;Kaneko and Kano 2003;Kano and Urrutia 2020;Matousek 2013) focus on computational aspects. Consensus halving (Simmons and Su 2003) is the mass partition problem that received the majority of attention in Economics and Computation so far (Deligkas, Filos-Ratsikas, and Hollender 2020;Filos-Ratsikas and Goldberg 2019;Filos-Ratsikas et al 2020, 2021.…”

Constant Inapproximability for PPA

“…But what is the complexity of the problem if we are not given the whole valuations upfront, but instead can only efficiently evaluate them? In [Deligkas et al, 2021b] it was proven that in this setting the problem is PPA-complete for 3 agents with non-additive valuations. It seems that proving hardness for the more standard additive valuation model would require radically new ideas.…”

“…In a recent work by Deligkas et al [2021b] the main question was "How does the complexity of consensus halving depend on the number of agents?". This paper's main result is a dichotomy between 2 and 3 agents when the valuations are monotone (but possibly non-additive).…”

Almost Envy-Freeness for Groups: Improved Bounds via Discrepancy Theory

“…Like fair division in general, consensus 1/k-division and consensus halving have been studied by mathematicians and economists for several decades [Hobby and Rice, 1965;Alon, 1987;Simmons and Su, 2003], and attracted recent interest from computer scientists in light of new computational complexity results , 2020Deligkas et al, 2020Deligkas et al, , 2021Goldberg et al, 2020].…”