Consensus-Halving: Does It Ever Get Easier?

Filos-Ratsikas, Aris ORCID: 0000-0001-7868-8114, Hollender, Alexandros, Sotiraki, Katerina and Zampetakis, Manolis (2020) Consensus-Halving: Does It Ever Get Easier? In: EC '20: The 21st ACM Conference on Economics and Computation, 2020-7-13 - 2020-7-16, Virtual.

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Official URL: http://dx.doi.org/10.1137/20M1387493

Abstract

In the $\varepsilon$-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 [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 $\varepsilon$-Consensus-Halving for any $\varepsilon$, as well as a polynomial-time algorithm for $\varepsilon=1/2$. 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 $\varepsilon$-Consensus-$1/3$-Division is PPAD-hard.

Item Type:	Conference or Workshop Item (Unspecified)
Additional Information:	Journal version. Preliminary version appeared at EC '20
Uncontrolled Keywords:	cs.CC, cs.CC, cs.GT
Depositing User:	Symplectic Admin
Date Deposited:	15 Jul 2020 09:23
Last Modified:	03 May 2023 14:25
DOI:	10.1145/3391403.3399527
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3093991