Minimization of Tree Patterns

Many of today’s graph query languages are based on graph pattern matching. We investigate optimization of tree-shaped patterns that have transitive closure operators. Such patterns not only appear in the context of graph databases but also were originally studied for querying tree-structured data, where they can perform child, descendant, node label, and wildcard tests. The minimization problem aims at reducing the number of nodes in patterns and goes back to the early 2000s. We provide an example showing that, in contrast to earlier claims, tree patterns cannot be minimized by deleting nodes only. The example resolves the M =? NR problem, which asks if a tree pattern is minimal if and only if it is nonredundant. The example can be adapted to prove that minimization is ΣP2-complete, which resolves another question that was open since the early research on the problem. The latter result shows that, unless NP = ΠP2, more general approaches for minimizing tree patterns are also bound to fail in general.