
Computing Exact Solutions of Consensus Halving and the BorsukUlam Theorem
We study the problem of finding an exact solution to the consensus halvi...
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Consensus Halving for Sets of Items
Consensus halving refers to the problem of dividing a resource into two ...
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ConsensusHalving: Does it Ever Get Easier?
In the εConsensusHalving problem, a fundamental problem in fair divisi...
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A Universality Theorem for Nested Polytopes
In a nutshell, we show that polynomials and nested polytopes are topolog...
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On the Complexity of Moduloq Arguments and the ChevalleyWarning Theorem
We study the search problem class PPA_q defined as a moduloq analog of ...
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Approximating the Existential Theory of the Reals
The existential theory of the reals (ETR) consists of existentially quan...
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AND Testing and Robust Judgement Aggregation
A function f{0,1}^n→{0,1} is called an approximate ANDhomomorphism if c...
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Strong Approximate Consensus Halving and the BorsukUlam Theorem
In the consensus halving problem we are given n agents with valuations over the interval [0,1]. The goal is to divide the interval into at most n+1 pieces (by placing at most n cuts), which may be combined to give a partition of [0,1] into two sets valued equally by all agents. The existence of a solution may be established by the BorsukUlam theorem. We consider the task of computing an approximation of an exact solution of the consensus halving problem, where the valuations are given by distribution functions computed by algebraic circuits. Here approximation refers to computing a point that εclose to an exact solution, also called strong approximation. We show that this task is polynomial time equivalent to computing an approximation to an exact solution of the BorsukUlam search problem defined by a continuous function that is computed by an algebraic circuit. The BorsukUlam search problem is the defining problem of the complexity class BU. We introduce a new complexity class BBU to also capture an alternative formulation of the BorsukUlam theorem from a computational point of view. We investigate their relationship and prove several structural results for these classes as well as for the complexity class FIXP.
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