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 $\alpha=\frac{1}{2}$. Stromquist and Woodall showed that for any $\alpha$, there exists a solution using $2n$ cuts of the segment. They also showed that if $\alpha$ is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values $\alpha$. For $\alpha = \frac{\ell}{k}$, we show a lower bound of $\frac{k-1}{k} \cdot 2n - O(1)$ cuts; we also obtain almost matching upper bounds for a large subset of rational $\alpha$. 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 PPA-$k$ under Turing reduction; 3. when $2n$ cuts are allowed, the problem belongs to PPA for any $\alpha$; more generally, 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均值相同。 在这项工作中, 我们为合理值设定了范围 $\ alpha\ in[ 0, 1美元] 表示理想比率, 这泛泛了 PPPA 完全协商一致的问题, 其中$\ alpha\\\frac{ 1\\\\\\\\\2美元。 Stromquist 和 Woodall 显示, 对于任何$( $) 来说, 都存在一个解决方案, 使用 $$( $) 的宽度削减 。 在计算方面, 如果 $( $) 是不合理的, 则使用 $( $) 最低的平价( 美元) ; 当 平价( 美元) 平价( 美元) 时, 平价( 美元) 平价( 平价) 直值( 美元) 平价( 美元) 平价( 2美元) 平价) 平价( ) 通常情况下, 平价( 平价) 平价( 平价) 平价) 。 当使用 任何问题时, 平价( 平价) 平价( 平价) 平价) 平价)