Committee Elections with Candidate Attribute Constraints

TL;DR

This paper studies a committee election problem with candidate attribute constraints, providing a complexity classification and identifying conditions for polynomial solvability and fixed-parameter tractability.

Contribution

It offers a dichotomy theorem for the problem's complexity and analyzes its parameterized complexity under various parameters.

Findings

Polynomial-time solvable when each candidate has one attribute and each attribute appears once in constraints.

NP-hard if these conditions are violated.

Fixed-parameter tractability when parameterized by attributes or candidates.

Abstract

In many real-world applications of committee elections, the candidates are associated with certain attributes and the chosen committee is required to satisfy some constraints posed on the candidate attributes. For instance, when dress collocation, it is generally acknowledged that when wearing a tie, you'd better wear a shirt, and wearing a suit, you'd better wear leather shoes. Here, dresses are categorized by upper garment, lower garment, shoes et.al, and upper garment is with the attribute tie and shirt, lower garment is with the attribute suit, and shoes is with the attribute leather. And two constraints "tie infers shirt" and "suit infers leather shoes" are proposed. We study this variant of committee elections from the computational complexity viewpoint. Given a set of candidates, each with some attributes and a profit, and a set of constraints, given as propositional logical expressions of the attributes, the task is to compute a set of k candidates, whose attributes satisfy all constraints and whose total profit achieves a given bound. We achieve a dichotomy concerning classical complexity with no length limit on constraints: the problem is polynomial-time solvable, if the following two conditions are fulfilled: 1) each candidate has only one attribute and 2) each attribute occurs at most once in the constraints. It becomes NP-hard if one of the two conditions is violated. Moreover, we examine its parameterized complexity. The parameterization with the number of constraints, the size of the committee, or the total profit bound as parameter leads to para-NP-hardness or W[1]-hardness, while with the number of attributes or the number of candidates as parameter, the problem turns out to be fixed-parameter tractable.