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.
Highlights
Dividing resources among agents in a fair manner is among the most fundamental problems in multi-agent systems [16]
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
138:2 Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem valuations over A, this can always be achieved using at most n cuts, and this fact can be proved via the Borsuk-Ulam theorem from algebraic topology [44]
Summary
Dividing resources among agents in a fair manner is among the most fundamental problems in multi-agent systems [16]. The problem of deciding whether there exists an approximate solution with k-cuts when k < n is NP-complete [27] These results are notable, because they identify consensus halving as one of the first natural PPA-complete problems. For problems in the complexity class PPAD, which is a subclass of both TFNP and PPA, prior work has found that there is a sharp contrast between exact and approximate solutions. This deceptively simple us to decide for problem is not integers a1, known to lie a2, in NP, and can be reduced to the problem of finding an exact Brouwer fixed point [26], which provides evidence that FIXP may be significantly harder than FNP. A large number of problems are known to be ETR-complete: geometric intersection problems [34, 39], graph-drawing problems [1, 9, 18, 40], matrix factorization problems [42, 43], the Art Gallery problem [2], and deciding the existence of constrained (symmetric) Nash equilibria in (symmetric) normal form games with at least three players [10, 11, 12, 13, 31]
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