Information inequality problem over set functions

TL;DR

This paper investigates the computational complexity of the information inequality problem in database contexts, showing it is coNP-complete for normal polymatroids but solvable in polynomial time for monotone functions, with implications for query size bounds.

Contribution

It characterizes the complexity of the information inequality problem under various syntactic and semantic restrictions relevant to database applications.

Findings

coNP-complete for normal polymatroids

polynomial-time solvable for monotone functions

alternative proof for entropic bounds over degree constraints

Abstract

Information inequalities appear in many database applications such as query output size bounds, query containment, and implication between data dependencies. Recently Khamis et al. proposed to study the algorithmic aspects of information inequalities, including the information inequality problem: decide whether a linear inequality over entropies of random variables is valid. While the decidability of this problem is a major open question, applications often involve only inequalities that adhere to specific syntactic forms linked to useful semantic invariance properties. This paper studies the information inequality problem in different syntactic and semantic scenarios that arise from database applications. Focusing on the boundary between tractability and intractability, we show that the information inequality problem is coNP-complete if restricted to normal polymatroids, and in polynomial time if relaxed to monotone functions. We also examine syntactic restrictions related to query output size bounds, and provide an alternative proof, through monotone functions, for the polynomial-time computability of the entropic bound over simple sets of degree constraints.