Completeness for first-order properties on sparse structures with algorithmic applications

Abstract

Properties definable in first-order logic are algorithmically interesting for both theoretical and pragmatic reasons. Many of the most studied algorithmic problems, such as Hitting Set and Orthogonal Vectors, are first-order, and the first-order properties naturally arise as relational database queries. A relatively straightforward algorithm for evaluating a property with k+1 quantifiers takes time O(mk) and, assuming the Strong Exponential Time Hy- pothesis (SETH), some such properties require O(mk-ϵ) time for any ϵ > 0. (Here, m represents the size of the input structure, i.e. the number of tuples in all relations.) We give algorithms for every first-order property that improves this upper bound to mk=2Θ( √p log n), i.e., an improvement by a factor more than any poly-log, but less than the polynomial required to refute SETH. Moreover, we show that further improvement is equivalent to improving algorithms for sparse instances of the well-studied Orthogonal Vectors problem. Surprisingly, both results are obtained by showing completeness of the Sparse Or- thogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of [3, 16]. While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class.