On the complexity of join predicates

We consider the complexity of join problems, focusing on equijoins, spatial-overlap joins, and set-containment joins. We use a graph pebbling model to characterize these joins combinatorially, by the length of their optimal pebbling strategies and computationally, by the complexity of discovering these strategies. Our results show that equijoins are the easiest of all joins, with optimal pebbling strategies that meet the lower bound over all join problems and that can be found in linear time. By contrast, spatial-overlap and set-containment joins are the hardest joins, with instances where optimal pebbling strategies reach the upper bound over all join problems and with the problem of discovering optimal pebbling strategies being NP-complete. For set-containment joins, we show that discovering the optimal pebbling is also MAX-SNP-Complete. As a consequence, we show that unless NP = P, there is a constant ∈o, such that this problem cannot be approximated within a factor of 1 + ∈&Ogr; in polynomial time. Our results shed some light on the difficulty the applied community has had in finding “good” algorithms for spatial-overlap and set-containment joins.