Abstract. We study the consensus-halving problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. The ǫ-approximate consensus-halving problem allows each agent to have an ǫ discrepancy on the values of the portions. We prove that computing ǫ-approximate consensus-halving solution using n cuts is in PPA, and is PPAD-hard, where ǫ is some positive constant; the problem remains PPAD-hard when we allow a constant number of additional cuts. It is NP-hard to decide whether a solution with n − 1 cuts exists for the problem. As a corollary of our results, we obtain that the approximate computational version of the Continuous Necklace Splitting Problem [2, 18] is PPAD-hard when the number of portions t is two.
A precursor of this paper REF established that the Consensushalving problem is PPAD-hard, even when we allow constant-size approximation errors for the agents.
Hardness Results for Consensus-Halving
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Computer Science Mathematics