Spines of Random Constraint Satisfaction Problems: Definition and Connection with Computational Complexity

TL;DR

This paper explores the relationship between phase transition phenomena and computational complexity in random constraint satisfaction problems, introducing the spine order parameter and linking discontinuities to exponential resolution complexity.

Contribution

It extends the definition of the spine to random CSPs and rigorously connects spine discontinuity with high decision complexity, providing new theoretical insights.

Findings

Discontinuity of the spine correlates with exponential resolution complexity.

Random CSPs with sharp thresholds and continuous spine resemble random 2-SAT.

The spine's behavior impacts the decision complexity more than the backbone.

Abstract

We study the connection between the order of phase transitions in combinatorial problems and the complexity of decision algorithms for such problems. We rigorously show that, for a class of random constraint satisfaction problems, a limited connection between the two phenomena indeed exists. Specifically, we extend the definition of the spine order parameter of Bollobas et al. to random constraint satisfaction problems, rigorously showing that for such problems a discontinuity of the spine is associated with a $2^{\Omega(n)}$ resolution complexity (and thus a $2^{\Omega(n)}$ complexity of DPLL algorithms) on random instances. The two phenomena have a common underlying cause: the emergence of ``large'' (linear size) minimally unsatisfiable subformulas of a random formula at the satisfiability phase transition. We present several further results that add weight to the intuition that random constraint satisfaction problems with a sharp threshold and a continuous spine are ``qualitatively similar to random 2-SAT''. Finally, we argue that it is the spine rather than the backbone parameter whose continuity has implications for the decision complexity of combinatorial problems, and we provide experimental evidence that the two parameters can behave in a different manner.