[Paper Review] Extremal Fitting Problems for Conjunctive Queries

Balder ten Cate, Víctor Dalmau+2 more|arXiv (Cornell University)|Jun 10, 2022

Advanced Database Systems and Queries4 citations

TL;DR

This paper introduces and formalizes extremal fitting conjunctive queries (CQs), including most-general, most-specific, and unique fitting CQs, via structural characterizations using homomorphism dualities, frontiers, and direct products. It establishes tight complexity bounds and exponential size lower bounds for these extremal CQs, with key results showing that constructing most-general fitting CQs requires 3ExpTime and may require doubly exponential size, while unique fitting CQs can be verified in PTime and constructed in ExpTime.

ABSTRACT

The fitting problem for conjunctive queries (CQs) is the problem to construct a CQ that fits a given set of labeled data examples. When a fitting CQ exists, it is in general not unique. This leads us to proposing natural refinements of the notion of a fitting CQ, such as most-general fitting CQ, most-specific fitting CQ, and unique fitting CQ. We give structural characterizations of these notions in terms of (suitable refinements of) homomorphism dualities, frontiers, and direct products, which enable the construction of the refined fitting CQs when they exist. We also pinpoint the complexity of the associated existence and verification problems, and determine the size of fitting CQs. We study the same problems for UCQs and for the more restricted class of tree CQs.

Motivation & Objective

• To formalize and characterize extremal fitting CQs, such as most-general, most-specific, and unique fitting CQs, in response to the non-uniqueness of fitting CQs.
• To study the verification, existence, and construction problems for these extremal CQs in terms of computational complexity.
• To determine the size bounds of extremal fitting CQs, especially in the context of tree CQs and union of CQs (UCQs).
• To provide structural characterizations using homomorphism dualities, frontiers, and direct products for efficient construction and complexity analysis.

Proposed method

• Proposes formal definitions of extremal fitting CQs: most-general, most-specific, basis of most-general, and unique fitting CQs.
• Uses homomorphism dualities and frontier structures to characterize the existence and construction of extremal CQs.
• Applies direct product constructions to analyze query containment and size bounds.
• Employs simulation and homomorphism techniques to verify query containment and non-containment against positive and negative examples.
• Constructs a family of instances with prime-length cycles and a complex negative instance to force double exponential size in fitting CQs.
• Leverages the prime number theorem and cycle structure to enforce that only CQs with large depth can fit all positive examples without mapping to negative ones.

Experimental results

Research questions

• RQ1 What are the structural characterizations of most-general, most-specific, and unique fitting CQs?
• RQ2 What is the computational complexity of verifying, deciding existence, and constructing extremal fitting CQs?
• RQ3 What are the tight size bounds for extremal fitting CQs, especially in the case of tree CQs?
• RQ4 How do these extremal CQs relate to homomorphism dualities and frontier structures?
• RQ5 Can extremal fitting CQs be constructed efficiently, and what are the limits of their size?

Key findings

• The problem of verifying whether a CQ is a most-specific fitting CQ is DP-complete, and existence is coNExpTime-complete.
• Most-general fitting CQs can be constructed in 2ExpTime, but their size may require doubly exponential space, with a lower bound of 2^2^n for tree CQs.
• For UCQs, verifying a unique fitting CQ is ExpTime-complete, and construction is in 2ExpTime, with a 2Exp size lower bound for fitting CQs.
• The basis of most-general fitting CQs is NExpTime-complete to verify and construct, with a 2Exp size lower bound.
• For tree CQs, every fitting CQ must contain a full binary L,R,A-tree of depth at least ∏p_i - 1, forcing a size lower bound of 2^2^n for fitting CQs over n prime-length cycles.
• A unique fitting CQ exists and can be verified in PTime, with construction in ExpTime, and requires at least exponential size.