Insight into Voting Problem Complexity Using Randomized Classes

TL;DR

This paper explores the complexity of a voting control problem, revealing its equivalence to a well-known matching problem and demonstrating its polynomial-time solvability, which impacts broader complexity theory questions.

Contribution

It establishes the equivalence of CCRV for First-Last to Exact Perfect Bipartite Matching and shows CCRV for 2-Approval is in P, resolving key open problems.

Findings

CCRV for First-Last is equivalent to Exact Perfect Bipartite Matching.

CCRV for 2-Approval is in P, settling an open problem.

Implications for P vs NP based on CCRV complexity.

Abstract

The first step in classifying the complexity of an NP problem is typically showing the problem in P or NP-complete. This has been a successful first step for many problems, including voting problems. However, in this paper we show that this may not always be the best first step. We consider the problem of constructive control by replacing voters (CCRV) introduced by Loreggia et al. (2015) for the scoring rule First-Last, which is defined by $\langle 1, 0, \dots, 0, -1\rangle$. We show that this problem is equivalent to Exact Perfect Bipartite Matching, and so CCRV for First-Last can be determined in random polynomial time. So on the one hand, if CCRV for First-Last is NP-complete then RP = NP, which is extremely unlikely. On the other hand, showing that CCRV for First-Last is in P would also show that Exact Perfect Bipartite Matching is in P, which would solve a well-studied 40-year-old open problem. By considering RP as an option we also gain insight into the complexity of CCRV for 2-Approval, ultimately showing it in P, which settles the complexity of the sole open problem in the comprehensive table from Erd\'{e}lyi et al. (2021).