The k-Allocation Problem and Its Variants

Abstract

AbstractIn the process of reviewing and ranking projects by a group of reviewers, each reviewer is assumed to review a partial list of projects, up to k projects. Each individual reviewer then ranks and compares all pairs of k projects. The k-allocation problem is to determine the allocation of up to k projects to each reviewer within the expertise set of the reviewer so that the resulting union of reviewed projects has certain desirable properties. One property of the k-allocation is to have all pairs of projects compared by at least one reviewer. This we call the k-complete problem.In cases when the property of k-complete cannot be achieved, one might settle for other properties. One such basic requirement is that each pair of projects is comparable via a ranking path which is a sequence of pairwise rankings of projects implying a comparison of all pairs on the path. A k-allocation with a ranking path between each pair is the connectivity-k-aloc. Since the robustness of relative comparisons deteriorates with the length of the ranking path, another property is that between each pair of projects there will be at least one ranking path that has at most two hops or q hops for fixed values of q. Another property that increases robustness of the ranking is to find a k-allocation so there are at least p disjoint ranking paths between each pair.We model all these problems as graph problems and show that the connectivity-k-aloc problem is polynomially solvable using matroid intersection, the k-complete problem is NP-hard unless k = 2, and all other considered variants of the k-allocation properties problem are NP-complete for all values of k ≥2. We provide approximation algorithms for an optimization problem related to the k-complete problem.KeywordsApproximation algorithmsallocation problemmaximum coverage problem