Unbalanced Triangle Detection and Enumeration Hardness for Unions of Conjunctive Queries

We study the enumeration of answers to Unions of Conjunctive Queries (UCQs) with optimal time guarantees. More precisely, we wish to identify the queries that can be solved with linear preprocessing time and constant delay. Despite the basic nature of this problem, it was shown only recently that UCQs can be solved within these time bounds if they admit free-connex union extensions, even if all individual CQs in the union are intractable with respect to the same complexity measure. Our goal is to understand whether there exist additional tractable UCQs, not covered by the currently known algorithms. As a first step, we show that some previously unclassified UCQs are hard using the classic 3SUM hypothesis, via a known reduction from 3SUM to triangle listing in graphs. As a second step, we identify a question about a variant of this graph task that is unavoidable if we want to classify all self-join-free UCQs: is it possible to decide the existence of a triangle in a vertex-unbalanced tripartite graph in linear time? We prove that this task is equivalent in hardness to some family of UCQs. Finally, we show a dichotomy for unions of two self-join-free CQs if we assume the answer to this question is negative. In conclusion, this paper pinpoints a computational barrier in the form of a single decision problem that is key to advancing our understanding of the enumeration complexity of many UCQs. Without a breakthrough for unbalanced triangle detection, we have no hope of finding an efficient algorithm for additional unions of two self-join-free CQs. On the other hand, a sufficiently efficient unbalanced triangle detection algorithm can be turned into an efficient algorithm for a family of UCQs currently not known to be tractable.