We consider the problem of partitioning a line segment into two subsets, so that $n$ finite measures all has the same ratio of values for the subsets. Letting $\alpha\in[0,1]$ denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which $\alpha=\frac{1}{2}$. It is known that for any $\alpha$, there exists a solution using $2n$ cuts of the segment. Here we show that if $\alpha$ is irrational, that upper bound is almost optimal. We also obtain bounds that are nearly exact for a large subset of rational values $\alpha$. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1. when using the minimal number of cuts for each instance is required, the problem is NP-hard for any $\alpha$; 2. for a large subset of rational $\alpha = \frac{\ell}{k}$, when $\frac{k-1}{k} \cdot 2n$ cuts are available, the problem is in the Turing closure of PPA-$k$; 3. when $2n$ cuts are allowed, the problem belongs to PPA for any $\alpha$; furthermore, the problem belong to PPA-$p$ for any prime $p$ if $2(p-1)\cdot \frac{\lceil p/2 \rceil}{\lfloor p/2 \rfloor} \cdot n$ cuts are available.
翻译:我们考虑的是将线段分割成两个子集的问题, 也就是说, 美元限量措施对于子集的值比都是一样的。 使用 $\ alpha\ in[ 0, 1]$ 表示理想比率, 这概括了 PPA 完整的协商一致- 通化问题, 其中$\ pha{ { 1\\\\\\2} 美元。 已知对于任何美元来说, 都有一个使用 $2 削减 的解决方案。 我们在这里显示, 如果 $\ alfa 是不合理的, 美元上限几乎是最佳的。 我们还得到了 相当合理值 $ $\ 的界限。 在计算方面, 我们探索它对于可用的削减数量的依赖性。 更具体地说, 当需要使用 $1 的最低削减数量时, 任何 $\ 的 问题是 NP- ; 对于 $\ palfa =\ plec\\ t r\\ pr\ k} 问题, 当 $ 2cdo\ c don listal 问题时, 问题是 rPA 。