Banzhaf Values for Facts in Query Answering

TL;DR

This paper introduces efficient algorithms for computing Banzhaf values, a measure of fact contribution in query answering, enabling better explanations of database query results with significant performance improvements.

Contribution

The paper presents three novel algorithms for exact, approximate, and ranking computations of Banzhaf values in database queries, along with a complexity dichotomy for fact ranking.

Findings

Algorithms outperform prior methods by up to two orders of magnitude.

The methods handle complex query instances previously infeasible.

A dichotomy identifies tractable and intractable cases for fact ranking.

Abstract

Quantifying the contribution of database facts to query answers has been studied as means of explanation. The Banzhaf value, originally developed in Game Theory, is a natural measure of fact contribution, yet its efficient computation for select-project-join-union queries is challenging. In this paper, we introduce three algorithms to compute the Banzhaf value of database facts: an exact algorithm, an anytime deterministic approximation algorithm with relative error guarantees, and an algorithm for ranking and top-$k$. They have three key building blocks: compilation of query lineage into an equivalent function that allows efficient Banzhaf value computation; dynamic programming computation of the Banzhaf values of variables in a Boolean function using the Banzhaf values for constituent functions; and a mechanism to compute efficiently lower and upper bounds on Banzhaf values for any positive DNF function. We complement the algorithms with a dichotomy for the Banzhaf-based ranking problem: given two facts, deciding whether the Banzhaf value of one is greater than of the other is tractable for hierarchical queries and intractable for non-hierarchical queries. We show experimentally that our algorithms significantly outperform exact and approximate algorithms from prior work, most times up to two orders of magnitude. Our algorithms can also cover challenging problem instances that are beyond reach for prior work.