Consensus Division in an Arbitrary Ratio

We consider the problem of partitioning a line segment into two subsets, so that $n$ finite measures all have 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.