Smoothed Complexity Theory

TL;DR

This paper introduces a formal framework for smoothed complexity theory, defining classes and establishing initial hardness and tractability results, bridging the gap between smoothed analysis and computational complexity.

Contribution

It proposes the first formal framework for smoothed complexity theory, including definitions, complexity classes, and initial hardness and tractability results.

Findings

Defined smoothed complexity classes.

Proved hardness results for bounded halting and tiling.

Established tractability for binary optimization, graph coloring, and satisfiability.

Abstract

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.