Size Bounds for Conjunctive Queries with General Functional Dependencies

TL;DR

This paper develops a theoretical framework using information theory to establish size bounds for conjunctive query results under general functional dependencies, extending previous work and providing new insights into worst-case scenarios.

Contribution

It introduces a linear programming approach based on entropy to bound query result sizes with arbitrary functional dependencies, generalizing prior bounds and connecting to open problems in information theory.

Findings

Characterizes entropy structure of worst-case instances for simple dependencies

Provides upper and lower bounds for general functional dependencies

Polynomial-time method to determine if query result size can exceed input

Abstract

This paper extends the work of Gottlob, Lee, and Valiant (PODS 2009)[GLV], and considers worst-case bounds for the size of the result Q(D) of a conjunctive query Q to a database D given an arbitrary set of functional dependencies. The bounds in [GLV] are based on a "coloring" of the query variables. In order to extend the previous bounds to the setting of arbitrary functional dependencies, we leverage tools from information theory to formalize the original intuition that each color used represents some possible entropy of that variable, and bound the maximum possible size increase via a linear program that seeks to maximize how much more entropy is in the result of the query than the input. This new view allows us to precisely characterize the entropy structure of worst-case instances for conjunctive queries with simple functional dependencies (keys), providing new insights into the results of [GLV]. We extend these results to the case of general functional dependencies, providing upper and lower bounds on the worst-case size increase. We identify the fundamental connection between the gap in these bounds and a central open question in information theory. Finally, we show that, while both the upper and lower bounds are given by exponentially large linear programs, one can distinguish in polynomial time whether the result of a query with an arbitrary set of functional dependencies can be any larger than the input database.