Computing exact solutions of consensus halving and the Borsuk-Ulam theorem

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 [29] , [30] , we show that the exact version is much harder. Specifically, finding a solution with n agents and n cuts is FIXP -hard, and deciding whether there exists a solution with fewer than n cuts is ETR -complete. Along the way, we define a new complexity class, called 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.