Consensus-Halving: Does It Ever Get Easier?

Abstract

In the ε-Consensus-Halving problem, a fundamental problem in fair division, there are n agents with valuations over the interval [0, 1], and the goal is to divide the interval into pieces and assign a label "+" or "−" to each piece, such that every agent values the total amount of "+" and the total amount of "−" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [18,19] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question.In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [19], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [19]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving ε-Consensus-Halving for any ε, as well as a polynomial-time algorithm for ε = 1/2; these are the first algorithmic results for the problem. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-1/k-Division problem [33]. In particular, we prove that ε-Consensus-1/3-Division is PPAD-hard.

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

“…Concretely for our case, all of our reductions will be efficient black-box reductions, thus allowing us to obtain both PPA-completeness results and query complexity bounds matching those of the problems that we reduce from/to. We remark that the reductions constructed for proving the PPA-hardness of the problem in previous works (for a non-constant number of agents) [28][29][30] are not black-box reductions, and therefore have no implications on the query complexity of the problem.…”

“…In a follow-up paper, [29] used the PPA-completeness of Consensus-Halving to prove that the Necklace Splitting problem with two thieves is also PPA-complete. Very recently, Filos-Ratsikas et al [30] strengthened the PPA-hardness result to the case of very simple valuation functions, namely piecewise constant valuations with at most two blocks of value. Deligkas et al [22] studied the computational complexity of the exact version of the problem, and obtained among other results its membership in a newly introduced class BU (for "Borsuk-Ulam" [12]) and its computational hardness for the well-known class FIXP of Etessami and Yannakakis [25].…”

“…Additionally, Filos-Ratsikas and Goldberg [29], reduced from this problem to establish the PPA-completeness of Necklace Splitting with two thieves; these PPA-completeness results provided the first definitive evidence of intractability for these two classic problems, establishing for instance that solving them is at least as hard as finding a Nash equilibrium of a strategic game [18,19]. Filos-Ratsikas et al [30] improved on the results for the -Consensus-Halving problem, by showing that the problem remains PPA-complete, even if one restricts the attention to very small classes of agents' valuations, namely piecewise uniform valuations with only two valuation blocks.…”

“…This latter result falls under the general umbrella of imposing restrictions on the structure of the problem, to explore if the computational hardness persists or whether we can obtain polynomialtime algorithms. Filos-Ratsikas et al [30] applied this approach along the axis of the valuation functions, while considering a general number of agents, similarly to [28,29]. In this paper, we take a different approach, and we restrict the number of agents to be constant.…”

Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem

“…In the same way, while FIXP ⊆ BU, FIXP-hardness of computing an exact consensus halving is not implied by our reduction, since Theorem 2 establishes BU a -hardness rather than BU-hardness. Recently a considerably simpler proof of PPA-hardness for computing an ε-consensus halving was given by Filos-Ratsikas, Hollender, Sotiraki and Zampetakis [FHSZ20], and our reduction is inspired by this work.…”