Hardness results for consensus-halving

Filos-Ratsikas, Aris, Søren Kristoffer, Stiil Frederiksen, Goldberg, Paul W. and Zhang, Jie (2018) Hardness results for consensus-halving. In 43rd International Symposium on Mathematical Foundations of Computer Science. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.. (In Press)

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 ϵ-approximate consensus-halving problem allows each agent to have an ϵ discrepancy on the values of the portions. We prove that computing ϵ-approximate consensus-halving solution using n cuts is in PPA, and is PPAD-hard, where ϵ 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 is PPAD-hard when the number of portions t is two.

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