The Smoothed Complexity of Computing Kemeny and Slater Rankings

Abstract

The computational complexity of winner determination under common voting rules is a classical and fundamental topic in the field of computational social choice. Previous work has established the NP-hardness of winner determination under some commonly-studied voting rules, such as the Kemeny rule and the Slater rule. In a recent position paper, Baumeister, Hogrebe, and Rothe (2020) questioned the relevance of the worst-case nature of NP-hardness in social choice and proposed to conduct smoothed complexity analysis (Spielman and Teng 2009) under Blaser and Manthey’s (2015) framework. In this paper, we develop the first smoothed complexity results for winner determination in voting. We prove the smoothed hardness of Kemeny and Slater using the classical smoothed runtime analysis, and prove a parameterized typical-case smoothed easiness result for Kemeny. We also make an attempt of applying Blaser and Manthey’s (2015) smoothed complexity framework in social choice contexts by proving that the framework categorizes an always-exponential-time brute force search algorithm as being smoothed poly-time, under a natural noise model based on the well-studied Mallows model in social choice and statistics. Overall, our results show that smoothed complexity analysis in computational social choice is a challenging and fruitful topic.