The Smallest Hard Trees

TL;DR

This paper identifies the smallest trees with specific computational complexities for their associated CSPs, demonstrating that tree orientation critically influences problem difficulty and providing computational tools for their classification.

Contribution

It determines the smallest trees with NP-complete, NL-hard, and other complexity properties, and introduces methods for efficiently analyzing tree orientations in CSP complexity classification.

Findings

A 20-vertex tree with NP-complete CSP orientation.

Smallest trees that are NL-hard, not solvable by arc consistency, or Datalog.

Experimental support for the 'easy trees lack counting' conjecture.

Abstract

We find an orientation of a tree with 20 vertices such that the corresponding fixed-template constraint satisfaction problem (CSP) is NP-complete, and prove that for every orientation of a tree with fewer vertices the corresponding CSP can be solved in polynomial time. We also compute the smallest tree that is NL-hard (assuming L is not NL), the smallest tree that cannot be solved by arc consistency, and the smallest tree that cannot be solved by Datalog. Our experimental results also support a conjecture of Bulin concerning a question of Hell, Nesetril and Zhu, namely that "easy trees lack the ability to count". Most proofs are computer-based and make use of the most recent universal-algebraic theory about the complexity of finite-domain CSPs. However, further ideas are required because of the huge number of orientations of trees. In particular, we use the well-known fact that it suffices to study orientations of trees that are cores and show how to efficiently decide whether a given orientation of a tree is a core using the arc-consistency procedure. Moreover, we present a method to generate orientations of trees that are cores that works well in practice. In this way we found interesting examples for the open research problem to classify finite-domain CSPs in NL.