
ConsenusHalving: Does it Ever Get Easier?
In the εConsensusHalving problem, a fundamental problem in fair divisi...
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Consensus Halving for Sets of Items
Consensus halving refers to the problem of dividing a resource into two ...
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Consensus Halving is PPAComplete
We show that the computational problem CONSENSUSHALVING is PPAcomplete...
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Strong Approximate Consensus Halving and the BorsukUlam Theorem
In the consensus halving problem we are given n agents with valuations o...
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Dividing Bads is Harder than Dividing Goods: On the Complexity of Fair and Efficient Division of Chores
We study the chore division problem where a set of agents needs to divid...
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The Bike Sharing Problem
Assume that m ≥ 1 autonomous mobile agents and 0 ≤ b ≤ m singleagent tr...
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A rigorous formulation of and partial results on Lorenz's "consensus strikes back" phenomenon for the HegselmannKrause model
In a 2006 paper, Jan Lorenz observed a curious behaviour in numerical si...
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ConsensusHalving: Does it Ever Get Easier?
In the εConsensusHalving 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 εConsensusHalving for any ε, as well as a polynomialtime 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 ConsensusHalving, the Consensus1/kDivision problem. In particular, we prove that εConsensus1/3Division is PPADhard.
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