Before revision:  The constraint density where algorithms break down for each CSP is known as the "algorithmic barrier" and provably corresponds to a phase transition in the geometry of the space of solutions Achlioptas and Coja-Oghlan 2008 . In this paper we aim to shed light on the following question: Can algorithmic success up to the barrier for each CSP be ascribed to some simple structural property of the inputs?
After revision:  The constraint density where algorithms break down for each CSP is known as the "algorithmic barrier" and provably corresponds to a phase transition in the geometry of the space of solutions  . In this paper we aim to shed light on the following question: Can algorithmic success up to the barrier for each CSP be ascribed to some simple structural property of the inputs?