Introduction

Abstract

The first systematic complexity-theoretic study of constraints was carried out by T.J. Schaefer in 1978 with a paper on Boolean constraint satisfaction problems (CSPs). This volume opens with a survey of Boolean constraints by N. Creignou and H. Vollmer. Schaefer proved a Dichotomy Theorem about the satisfiability problem for Boolean CSPs, which asserts that each Boolean CSP is either NP-complete or in P (hence, assuming P ≠ NP, all infinitely many intermediate complexity degrees are avoided). Creignou and Vollmer present a modern algebraic proof of Schaefer’s Dichotomy Theorem, a proof that makes use of Galois theory and of the structure of the lattice of Boolean clones, known as Post’s lattice. In addition to the satisfiability problem, several other algorithmic problems, including counting, optimization, and circumscription, can be completely classified from a complexity point of view using Post’s lattice. The first contribution to this volume surveys these results and poses the question: what is so special about certain algorithmic tasks that makes Post’s lattice applicable to their classification.