How hard is control in single-crossing elections?

Abstract

Election control problems model situations where some entity (traditionally called the election chair) wants to ensure some candidate’s victory by either adding or deleting candidates or voters. The complexity of deciding if such control actions can be successful is well-studied for many typical voting rules and, usually, such control problems are mathrm {NP}-complete. However, Faliszewski et al. (Inf Comput 209(2):89–107, 2011) have shown that many control problems become polynomial-time solvable when we consider single-peaked elections. In this paper we show that a similar phenomenon applies to the case of single-crossing elections. Specifically, we consider the complexity of control by adding/deleting candidates/voters under plurality, Condorcet, and approval voting. For each of these control types and each of the rules, we show that if the control type is mathrm {NP}-complete in general, it becomes polynomial-time solvable for single-crossing elections.