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 ? ? [0,1] denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which ? = 1/2. Stromquist and Woodall [Stromquist and Woodall, 1985] showed that for any ?, there exists a solution using 2n cuts of the segment. They also showed that if ? is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values ?. For ? = ?/k, we show a lower bound of (k-1)/k ? 2n - O(1) cuts; we also obtain almost matching upper bounds for a large subset of rational ?. 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 ?; 2) for a large subset of rational ? = ?/k, when (k-1)/k ? 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 ?; more generally, the problem belong to PPA-p for any prime p if 2(p-1)??p/2?/?p/2? ? n cuts are available

DPA Load Balancer: Load balancing for Data Parallel Actor-based systems

In this project we explore ways to dynamically load balance actors in a streaming framework. This is used to address input data skew that might lead to stragglers. We continuously monitor actors' input queue lengths for load, and redistribute inputs among reducers using consistent hashing if we detect stragglers. To ensure consistent processing post-redistribution, we adopt an approach that uses input forwarding combined with a state merge step at the end of the processing. We show that this approach can greatly alleviate stragglers for skewed data.Comment: 7 page

Conservation impediments and incentives – progressing the understanding of linkages between the adoption of conservation practices and the motivational orientation of graziers in the tropical savannas

The adoption of conservation practices is a complex matter – rural landholders consider a wide variety of factors and characteristics when deciding whether to adopt a conservation practice. To confound the issue, recent research has suggested that the goals of landholders affect the adoption of conservation practices by creating a subjective consideration of the relative importance of impediments and effectiveness of incentives in the adoption decision. In this research we describe an empirical link between graziers’ goals and their perceptions of the relative importance of impediments and the effectiveness of incentives in the adoption of conservation practices. The research was carried out in the tropical savannas region of Australia where pastoral production dominates the landscape and where it is of prime importance to ensure that grazing land is included in the conservation estate. The results suggest that to increase the adoption of conservation practices, schemes will have to be developed with reference to graziers subjective views on impediments and on the effectiveness of incentives.Graziers, goals, conservation, tropical savannas, impediments, incentives, Crop Production/Industries,

A Fixed-Parameter Algorithm for the Schrijver Problem

The Schrijver graph $S(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $\{1,2,\ldots,n\}$ that do not include two consecutive elements modulo $n$, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of $S(n,k)$ is $n-2k+2$ (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of $S(n,k)$ with $n-2k+1$ colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class $\mathsf{PPA}$. We prove that it can be solved by a randomized algorithm with running time $n^{O(1)} \cdot k^{O(k)}$, hence it is fixed-parameter tractable with respect to the parameter $k$.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2204.0676