Search versus Decision for Election Manipulation Problems

TL;DR

This paper investigates the complexity difference between recognizing manipulable election instances and actually finding successful manipulations, showing they can differ significantly under certain computational assumptions.

Contribution

It demonstrates that for some election systems, the decision problem is polynomial-time solvable while the search problem is computationally hard, assuming integer factoring is hard.

Findings

Recognition of manipulable instances is in P for certain systems.

Finding successful manipulations can be computationally hard.

The complexity gap depends on the hardness of integer factoring.

Abstract

Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If integer factoring is hard, then for election manipulation, election bribery, and some types of election control, there are election systems for which recognizing which instances can be successfully manipulated is in polynomial time but producing the successful manipulations cannot be done in polynomial time.