[Paper Review] A Polynomial-time Algorithm to Achieve Extended Justified Representation

Haris Aziz, Shenwei Huang|arXiv (Cornell University)|Mar 30, 2017

Game Theory and Voting Systems8 references3 citations

TL;DR

This paper presents MaxSwapPAV, the first polynomial-time algorithm to compute a committee satisfying extended justified representation (EJR) in approval-based multi-winner voting. By iteratively swapping candidates in and out of the sitting committee to improve Proportional Approval Voting (PAV) scores, the algorithm guarantees EJR while running in time O(n²mk⁴ln k).

ABSTRACT

We consider a committee voting setting in which each voter approves of a subset of candidates and based on the approvals, a target number of candidates are to be selected. In particular we focus on the axiomatic property called extended justified representation (EJR). Although a committee satisfying EJR is guaranteed to exist, the computational complexity of finding such a committee has been an open problem and explicitly mentioned in multiple recent papers. We settle the complexity of finding a committee satisfying EJR by presenting a polynomial-time algorithm for the problem. Our algorithmic approach may be useful for constructing other voting rules in multi-winner voting.

Motivation & Objective

• To resolve the long-standing open problem of whether a committee satisfying extended justified representation (EJR) can be computed in polynomial time.
• To develop an efficient algorithm that guarantees EJR without relying on NP-hard optimization or enumeration.
• To provide a constructive method that leverages the connection between EJR and PAV-score improvement via candidate swaps.
• To demonstrate that EJR is computationally tractable, closing a key gap in computational social choice theory.
• To offer a practical and efficient alternative to existing PAV-based rules, which are NP-hard to compute.

Proposed method

• Proposes MaxSwapPAV, a polynomial-time algorithm that improves PAV scores by swapping one sitting candidate with an outside candidate.
• Uses a swap criterion based on a lower bound of 1/(2k³) on PAV-score improvement per swap to ensure progress.
• Employs a key lemma showing that if a committee is not EJR-satisfying, then a beneficial swap exists that increases the PAV score by at least 1/(2k³).
• Relies on the concept of PAV-swap-freeness: a committee is EJR-satisfying if no such beneficial swap exists.
• Implements a greedy selection of swaps that maximizes PAV-score improvement at each step.
• Bounds the total number of swaps by O(nk³ log k), ensuring polynomial runtime of O(n²mk⁴ log k).

Experimental results

Research questions

• RQ1 Can a committee satisfying extended justified representation (EJR) be computed in polynomial time?
• RQ2 Is there a constructive and efficient algorithm to achieve EJR without resorting to NP-hard optimization?
• RQ3 What is the computational relationship between EJR and the Proportional Approval Voting (PAV) rule?
• RQ4 Can a swap-based approach outperform sequential construction methods in achieving strong representation axioms?
• RQ5 Does the existence of a PAV-score improving swap imply the failure of EJR, and can this be used algorithmically?

Key findings

• MaxSwapPAV is the first polynomial-time algorithm that guarantees a committee satisfying extended justified representation (EJR).
• The algorithm runs in O(n²mk⁴ln k) time, with at most O(nk³ log k) swaps, each improving the PAV score by at least 1/(2k³).
• The algorithm proves that EJR is computationally tractable, resolving an open problem explicitly mentioned in multiple recent papers.
• A committee satisfying EJR exists and can be computed efficiently, unlike PAV which is NP-hard and W[1]-hard even for 2 approvals per voter.
• The approach of candidate swapping provides a viable alternative to sequential construction, enabling fairer representation from an initial committee.
• The result implies that EJR is as efficiently computable as the weaker PJR property, which was only recently shown to be polynomial-time computable.