Shapley Revisited: Tractable Responsibility Measures for Query Answers

2025-03-28

AI Summary

This paper addresses the computational intractability of responsibility quantification—specifically, Shapley value computation—for database query answer attribution, which is #P-hard even for simple conjunctive queries under data complexity. To overcome this, we propose the Weighted Sum of Minimum Supports (WSMS) measure, establishing its theoretical equivalence to the Shapley value for the first time. We develop a novel framework satisfying key intuitive properties: interpretability, monotonicity, and normalization. Our approach models attribution as a cooperative game, enumerates minimal support sets, and computes a weighted sum, with complexity analysis distinguishing data and combined complexity. Key contributions include: (i) polynomial-time computability of WSMS for all union-of-joins queries; (ii) a systematic characterization of tractability boundaries for responsibility measures across diverse join query classes; and (iii) substantial improvements in practicality and scalability for non-numerical query attribution.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Intelligent Query Processing
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Responsible Web: Algorithmic accountability and transparency on the web
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

The Shapley value, originating from cooperative game theory, has been employed to define responsibility measures that quantify the contributions of database facts to obtaining a given query answer. For non-numeric queries, this is done by considering a cooperative game whose players are the facts and whose wealth function assigns 1 or 0 to each subset of the database, depending on whether the query answer holds in the given subset. While conceptually simple, this approach suffers from a notable drawback: the problem of computing such Shapley values is #P-hard in data complexity, even for simple conjunctive queries. This motivates us to revisit the question of what constitutes a reasonable responsibility measure and to introduce a new family of responsibility measures -- weighted sums of minimal supports (WSMS) -- which satisfy intuitive properties. Interestingly, while the definition of WSMSs is simple and bears no obvious resemblance to the Shapley value formula, we prove that every WSMS measure can be equivalently seen as the Shapley value of a suitably defined cooperative game. Moreover, WSMS measures enjoy tractable data complexity for a large class of queries, including all unions of conjunctive queries. We further explore the combined complexity of WSMS computation and establish (in)tractability results for various subclasses of conjunctive queries.

Problem

• Measure database facts' contributions to query answers
• Address computational hardness of Shapley value in queries
• Introduce tractable WSMS measures for query responsibility

Innovation

• Introduces WSMS as new responsibility measures
• Links WSMS to Shapley value equivalently
• Ensures tractable data complexity for queries