Consensus-Halving: Does It Ever Get Easier?

Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, Manolis Zampetakis

Abstract: In the -Consensus-Halving problem, a fundamental problem in fair division, there are 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 [Proceedings of the 50th Annual ACM Symposium on Theory of Computing, 2018, pp. 51–64; Proceedings of the 51st Annual ACM Symposium on Theory of Computing, 2019, pp. 638–649] 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 [Proceedings of the 51st Annual ACM Symposium on Theory of Computing, 2019, pp. 638–649] 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 [Proceedings of the 51st Annual ACM Symposium on Theory of Computing, 2019, pp. 638–649]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial-time algorithm for solving -Consensus-Halving for any , as well as a polynomial-time algorithm for . Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus- -Division problem [F. W. Simmons and F. E. Su, Math. Social Sci., 45 (2003), pp. 15–25]. In particular, we prove that -Consensus- -Division is PPAD-hard.