Computational Complexity of Strong and Average Justified Representation

2026-06-28

AI Summary

This study addresses the computational complexity of determining whether a winning committee exists in approval-based multi-winner elections that satisfies either the Strong Justified Representation (SJR) or Average Justified Representation (AJR) axiom. By leveraging tools from computational complexity theory and the polynomial hierarchy, the work establishes for the first time that the decision problem for SJR is Θ₂^p-complete, while that for AJR is Σ₂^p-complete. To support these hardness results, the authors construct a general class of set systems enabling the required reductions. The findings further indicate that SJR is more amenable to encoding within SAT-solving frameworks. These results precisely characterize the computational intractability of both representation axioms, thereby laying a rigorous theoretical foundation for future algorithm design and impossibility analyses.

Abstract

We study the approval-based multiwinner election problem where a set of $n$ voters cast approval-based ballots to a set of $m$ candidates, and we are to select a winner committee consisting of $k$ candidates. We consider two axioms: strong justified representation (SJR) and average justified representation (AJR). A winner committee satisfies SJR if the satisfaction for each voter in every $\ell$-cohesive group is at least $\ell$. AJR is a weaker axiom that requires the average satisfaction for each $\ell$-cohesive group to be at least $\ell$. It is well known that a winner committee satisfying AJR may not exist (and neither does SJR). In this paper, we study the computational complexity of the following decision problem: given an approval-based multiwinner election instance, decide if there exists a winner committee satisfying SJR/AJR. We prove that this problem is $Θ_2^p$-complete for SJR, and $Σ_2^p$-complete for AJR. Our results indicate that the decision problem with SJR is more amenable to SAT-based implementations, whereas the decision problem with AJR is substantially harder. As byproducts, we derive some results that are interesting in their own right. Firstly, we show that adding one more adaptive query to an NP oracle on top of polynomially many non-adaptive NP queries does not add more computational power, and the resulting complexity class is still $Θ_2^p$. Secondly, we construct a set system that can be useful in other applications, especially when doing reductions from typical satisfiability problems such as 3SAT.

Problem

• multiwinner election
• justified representation
• computational complexity
• SJR
• AJR

Innovation

• computational complexity
• multiwinner elections
• justified representation
• polynomial hierarchy
• SAT-based reduction