Using Weighted Matching to Solve 2-Approval/Veto Control and Bribery

TL;DR

This paper proves that certain election control and bribery problems for 2-Approval and 2-Veto are solvable in polynomial time by transforming them into weighted matching problems, settling their computational complexity.

Contribution

It establishes polynomial-time algorithms for 2-Approval and 2-Veto control and bribery problems, resolving open complexity questions in election manipulation.

Findings

2-Approval constructive control by replacing voters is in P

Priced bribery for 2-Veto elections is in P

Priced bribery for 3-Veto elections is NP-complete

Abstract

Determining the complexity of election attack problems is a major research direction in the computational study of voting problems. The paper "Towards completing the puzzle: complexity of control by replacing, adding, and deleting candidates or voters" by Erd\'elyi et al. (JAAMAS 2021) provides a comprehensive study of the complexity of control problems. The sole open problem is constructive control by replacing voters for 2-Approval. We show that this case is in P, strengthening the recent RP (randomized polynomial-time) upper bound due to Fitzsimmons and Hemaspaandra (IJCAI 2022). We show this by transforming 2-Approval CCRV to weighted matching. We also use this approach to show that priced bribery for 2-Veto elections is in P. With this result, and the accompanying (unsurprising) result that priced bribery for 3-Veto elections is NP-complete, this settles the complexity for $k$-Approval and $k$-Veto standard control and bribery cases.