TY - JOUR

T1 - Consensus-Halving: Does It Ever Get Easier?

AU - Filos-Ratsikas, Aris

AU - Hollender, Alexandros

AU - Sotiraki, Katerina

AU - Zampetakis, Manolis

N1 - Alexandros Hollender was supported by an EPSRC doctoral studentship (Reference 1892947).

PY - 2022/10/14

Y1 - 2022/10/14

N2 - 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 [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

AB - 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 [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

M3 - Article

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

ER -