Technical Notes on Complexity of the Satisfiability Problem

TL;DR

This paper discusses the complexity of the propositional satisfiability problem, proving that under certain assumptions, the average case running time of a simple solution is linear, and explores factors influencing algorithm tractability.

Contribution

It provides a proof that the average running time is linear under some assumptions and highlights the importance of input distribution in assessing algorithm tractability.

Findings

Average running time is linear under certain conditions

Input distribution significantly affects algorithm tractability

Criteria for evaluating complex algorithm efficiency are proposed

Abstract

These notes contain, among others, a proof that the average running time of an easy solution to the satisfiability problem for propositional calculus is, under some reasonable assumptions, linear (with constant 2) in the size of the input. Moreover, some suggestions are made about criteria for tractability of complex algorithms. In particular, it is argued that the distribution of probability on the whole input space of an algorithm constitutes an non-negligible factor in estimating whether the algorithm is tractable or not.