Timo von Oertzen | Searcharxiv

arXiv2012

Essential Convexity and Complexity of Semi-Algebraic Constraints

Let Γbe 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 Γthat contain the relations \leq, {(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\inS, there are only finitely many points on the line segment between a and b that are not in S. If Γcontains a relation S that is not essentially convex and this is witnessed by rational points a,b, then we show that the CSP for Γis 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 Γis a first-order expansion of (R;*,+), then the CSP for Γcan be solved in polynomial time if and only if all relations in Γare essentially convex (unless P=NP).

cs.CC↗

arXiv2010

Horn versus full first-order: complexity dichotomies in algebraic constraint satisfaction

We study techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems. For certain fundamental algebraic structures Delta, we prove definability dichotomy theorems of the following form: for every first-order expansion Gamma of Delta, either Gamma has a quantifier-free Horn definition in Delta, or there is an element d of Gamma such that all non-empty relations in Gamma contain a tuple of the form (d,...,d), or all relations with a first-order definition in Delta have a primitive positive definition in Gamma. The results imply that several families of constraint satisfaction problems exhibit a complexity dichotomy: the problems are in P or NP-hard, depending on the choice of the allowed relations. As concrete examples, we investigate fundamental algebraic constraint satisfaction problems. The first class consists of all first-order expansions of (Q;+). The second class is the affine variant of the first class. In both cases, we obtain full dichotomies by utilising our general methods.

cs.LO↗