Unbalanced Triangle Detection and Enumeration Hardness for Unions of Conjunctive Queries
Karl Bringmann, Nofar Carmeli

TL;DR
This paper investigates the complexity of enumerating answers to unions of conjunctive queries (UCQs), identifying a key computational barrier related to unbalanced triangle detection that influences the tractability of these queries.
Contribution
It establishes a connection between UCQ enumeration complexity and unbalanced triangle detection, providing a dichotomy result contingent on this problem's complexity.
Findings
Some UCQs are hard under the 3SUM hypothesis.
Deciding triangle existence in unbalanced tripartite graphs is crucial.
A dichotomy for unions of two self-join-free CQs depends on this decision problem.
Abstract
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…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Distributed systems and fault tolerance
