Essential Convexity and Complexity of Semi-Algebraic Constraints

Let Γ be a structure with a finite relational signature and a first-order definition in (R; ∗,+) with parameters from R, that is, a relational structure over the real numbers where all relations are semi-algebraic sets. In this article, we study the computational complexity of constraint satisfaction problem (CSP) for Γ: the problem to decide whether a given primitive positive sentence is true in Γ. We focus on those structures Γ that contain the relations ≤, {(x, y, z) | x+y = z} and {1}. Hence, all CSPs studied in this article are at least as expressive as the feasibility problem for linear programs. The central concept in our investigation is essential convexity: a relation S is essentially convex if for all a, b ∈ S, there are only finitely many points on the line segment between a and b that are not in S. If Γ contains a relation S that is not essentially convex and this is witnessed by rational points a, b, then we show that the CSP for Γ is NP-hard. Furthermore, we characterize essentially convex relations in logical terms. This different view may open up new ways for identifying tractable classes of semi-algebraic CSPs. For instance, we show that if Γ is a first-order expansion of (R; +, 1,≤), then the CSP for Γ can be solved in polynomial time if and only if all relations in Γ are essentially convex (unless P=NP). 1998 ACM Subject Classification: F.2.2, F.4.1, G.1.6. 2010 Mathematics Subject Classification: 68Q17.