Graph Homomorphisms and Universal Algebra

TL;DR

This course introduces the universal-algebraic approach to study the computational complexity of finite-domain CSPs, and covers in particular the cyclic terms and bounded width theorems.

Abstract

Constraint satisfaction problems are computational problems that naturally appear in many areas of theoretical computer science. One of the central themes is their computational complexity, and in particular the border between polynomial-time tractability and NP-hardness. In this course we introduce the universal-algebraic approach to study the computational complexity of finite-domain CSPs. The course covers in particular the cyclic terms and bounded width theorems. To keep the presentation accessible, we start the course in the tangible setting of directed graphs and graph homomorphism problems.

Figures (21)

• Figure 1: Illustration of the uniqueness proof for cores
• Figure 2: The arc-consistency procedure for $\mathop{\mathrm{CSP}}\nolimits(H)$.
• Figure 3: The AC-3 implementation of the arc-consistency procedure for $\mathop{\mathrm{CSP}}\nolimits(H)$.
• Figure 4: The graph from Exercise \ref{['exe:T3T2']}.
• Figure 5: One of the smallest orientations of a tree $H$ such that CSP$(H)$ is NP-complete (assuming P $\neq$ NP; all orientations of trees with less vertices can be solved by path consistency otrees).

Theorems & Definitions (471)

• Definition 2.1: direct product
• Proposition 2.2
• proof
• Conjecture 1: Hedetniemi
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5: Bulatov BulatovFVConjecture, Zhuk ZhukFVConjecture
• Theorem 2.6: of HellNesetril