ConsensusHalving: Does It Ever Get Easier?
Abstract
In the $\varepsilon$ConsensusHalving 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 FilosRatsikas and Goldberg [2019] to be the first "natural" complete problem for the computational class PPA, answering a decadeold 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 PPAhardness result of [FilosRatsikas 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 [FilosRatsikas and Goldberg, 2019]. Then, we consider the case of singleblock (uniform) valuations and provide a parameterized polynomial time algorithm for solving $\varepsilon$ConsensusHalving for any $\varepsilon$, as well as a polynomialtime algorithm for $\varepsilon=1/2$. Finally, an important application of our new techniques is the first hardness result for a generalization of ConsensusHalving, the Consensus$1/k$Division problem [Simmons and Su, 2003]. In particular, we prove that $\varepsilon$Consensus$1/3$Division is PPADhard.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2002.11437
 arXiv:
 arXiv:2002.11437
 Bibcode:
 2020arXiv200211437F
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Computer Science and Game Theory
 EPrint:
 Journal version. Preliminary version appeared at EC '20