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 Goldberg [2018, 2019] 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 [Filos-Ratsikas and Goldberg, 2019], 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 [Filos-Ratsikas and Goldberg, 2019]. 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 [Simmons and Su, 2003]. In particular, we prove that ε-Consensus-1/3-Division is PPAD-hard.

Efficient Splitting of Measures and Necklaces

“…It is worth noting that this algorithm can be implemented online and does not require any assumption on the behavior of the measure functions µ i . In [17], Filos-Ratsikas et al provide an efficient algorithm for the case k = 2 that gives both agents at least 1 4 of each measure, under the assumption that each one of the measures is uniform in some subinterval of [0, 1] and is 0 on the rest of [0, 1]. The next result deals with online algorithms.…”