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.

翻译：我们研究了在最优时间保证下对联合合取查询（UCQs）答案的枚举问题。更准确地说，我们希望识别那些能够以线性预处理时间和常数延迟求解的查询。尽管该问题具有基础性，但直到最近才证明：如果UCQs允许自由连接联合扩展，则可以在上述时间界限内求解，即使并集中的所有单个CQ在相同复杂度度量下都是难解的。我们的目标是理解是否存在当前已知算法未覆盖的其他易处理UCQ。作为第一步，我们基于经典的3SUM假设，通过从3SUM到图中三角形列举的已知归约，证明了一些先前未分类的UCQ是难解的。作为第二步，我们识别出一个关于该图任务变体的问题，该问题在分类所有无自连接UCQ时不可避免：是否能在线性时间内判定顶点非平衡三部图中三角形的存在性？我们证明该任务在难度上等价于某个UCQ族。最后，若假设该问题的答案为否定，我们展示了两个无自连接CQ的并集的二分性。总之，本文以单个判定问题的形式指出了一个计算障碍，该问题是推进我们对众多UCQ枚举复杂度理解的关键。若无非平衡三角形检测的突破，我们无法期望为额外的两个无自连接CQ的并集找到高效算法。另一方面，一个足够高效的非平衡三角形检测算法可转化为当前未知是否易处理的某个UCQ族的高效算法。