The Impact of Negation on the Complexity of the Shapley Value in Conjunctive Queries

TL;DR

This paper explores how negation in conjunctive queries affects the computational complexity of calculating the Shapley value, revealing fundamental changes in tractability and approximation difficulty.

Contribution

It generalizes existing complexity dichotomies to include negation in conjunctive queries and analyzes its impact on Shapley value computation.

Findings

Negation alters the complexity landscape of Shapley value computation.

A generalized dichotomy for queries with negation is established.

Negation significantly impacts the difficulty of approximate Shapley value calculations.

Abstract

The Shapley value is a conventional and well-studied function for determining the contribution of a player to the coalition in a cooperative game. Among its applications in a plethora of domains, it has recently been proposed to use the Shapley value for quantifying the contribution of a tuple to the result of a database query. In particular, we have a thorough understanding of the tractability frontier for the class of Conjunctive Queries (CQs) and aggregate functions over CQs. It has also been established that a tractable (randomized) multiplicative approximation exists for every union of CQs. Nevertheless, all of these results are based on the monotonicity of CQs. In this work, we investigate the implication of negation on the complexity of Shapley computation, in both the exact and approximate senses. We generalize a known dichotomy to account for negated atoms. We also show that negation fundamentally changes the complexity of approximation. We do so by drawing a connection to the problem of deciding whether a tuple is "relevant" to a query, and by analyzing its complexity.