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 Borsuk-Ulam 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 $\varepsilon$-close 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 Borsuk-Ulam search problem defined by a continuous function that is computed by an algebraic circuit. The Borsuk-Ulam 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 Borsuk-Ulam 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.
翻译:在将问题减半的共识中,我们被赋予了在间隔时间($[10,1]美元)上有估价的代理商。目标是将间隔分为最多为n+1美元的分块(在最多n分裁时放置最多为n+1美元),这可以将[$0,1]美元分成两组,所有代理商都同等地估价。Borsuk-Ulam 理论可以确定解决办法的存在。我们认为计算一个精确解决将问题减半的近似值的任务,在这种问题上,估值是通过高温电路计算的分配功能来计算的。这里的近似值指的是计算一个点,即美元+1美元接近于精确的解决方案,也称为强烈的近似值。我们表明,这项任务是多数值时间,相当于计算出一种接近于Borsuk-Ulam搜索问题的准确解决办法的近似近似值,而该办法是由一个连续的函数所定义的,该函数由代数变电路计算。博苏克-Ulam 搜索问题是复杂的BUBU。我们引入了一个新的复杂级BUBU,以捕捉到一种替代的Bsuk-Ulamm的配方-Ulam的配方的配方配方,我们从这些结构的复杂度的角度对若干结构进行了调查。