Ranking with Partitioning

TL;DR

This paper explores combinatorial optimization problems for ranking items based on similarity and additive measures, analyzing their complexity, solutions, and applications in real-world scenarios like climate change.

Contribution

It classifies the computational complexity of ranking problems involving item similarity and proposes solutions for special cases, with applications in environmental assessment.

Findings

Most problems are computationally hard in the worst case.

Exact and approximate solutions are found for specific problem variants.

Applications include assessing greenhouse gas emission sources.

Abstract

Given an undirected graph representing similarities between a set of items and an additive measure evaluating the items, we treat the position of a special subset of items in an ordinal ranking through a collection of combinatorial optimization problems in which items may be combined if they are similar. The objective for these problems is to either maximize or minimize the absolute or relative rank of the special subset, with a meta-goal of assessing the robustness of the rank, even in the presence of a well-defined criterion. We classify the computational complexity of all four problems, mostly finding worst-case hardness, then find exact and approximate solutions to special cases and variants of the problems. These structured cases are inspired by several real-world examples and may be used to assess commonly cited facts across disparate domains, as we demonstrate for sources of greenhouse gas emissions that contribute to climate change.