The Complexity of Computing Minimal Unidirectional Covering Sets

TL;DR

This paper investigates the computational complexity of finding minimal unidirectional covering sets in binary dominance relations, proving that these problems are generally hard to solve efficiently, unlike bidirectional covering sets.

Contribution

It establishes the precise complexity classes of minimal unidirectional covering set problems, showing they are Theta_{2}^{p}-hard and not solvable in polynomial time unless P=NP.

Findings

Minimal upward and downward covering sets are computationally hard to find.

The complexity of these problems is Theta_{2}^{p}-complete.

Contrasts with polynomial-time computability of bidirectional covering sets.

Abstract

Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal upward or downward covering set. For both problems, we raise this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size covering sets are hard or complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer's result that minimal bidirectional covering sets (i.e., sets that are both minimal upward and minimal downward covering sets) are polynomial-time computable.