Abstract. We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete [28,27], we show that the exact version is much harder. Specifically, finding a solution with n cuts is FIXPhard, and deciding whether there exists a solution with fewer than n cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP ⊆ BU ⊆ TFETR and that LinearBU = PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit.
Recently, REF studied the complexity of exact Consensus-Halving and showed that the problem is FIXP-hard.
Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem
ArXiv
ArXiv
Computer Science Mathematics