Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics

SIAM Journal on Computing, Ahead of Print.
Abstract. The “short cycle removal” technique was recently introduced by Abboud et al. [Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2022, pp. 1487–1500] to prove the fine-grained hardness of approximation. Its main technical result is that listing all triangles in an [math]-regular graph is [math]-hard even when the number of short cycles is small, namely, when the number of [math]-cycles is [math] for [math]. Its corollaries are based on the 3-SUM conjecture and their strength depends on [math], i.e., on how effectively the short cycles are removed. Abboud et al. achieve [math] by applying structure versus randomness arguments on graphs. In this paper, we take a step back and apply conceptually similar arguments on the numbers of the 3-SUM problem, from which the hardness of triangle listing is derived. Consequently, we achieve [math] and the following lower bound corollaries under the 3-SUM conjecture: Approximate distance oracles: The seminal Thorup–Zwick distance oracles achieve stretch [math] after preprocessing a graph in [math] time. For the same stretch, and assuming the query time is [math], Abboud et al. proved an [math] lower bound on the preprocessing time; we improve it to [math], which is only a factor 2 away from the upper bound. Additionally, we obtain tight bounds for stretch [math] and [math] and higher lower bounds for dynamic shortest paths. Listing 4-cycles: Abboud et al. proved the first superlinear lower bound for listing all 4-cycles in a graph, ruling out [math] time algorithms where [math] is the number of 4-cycles. We settle the complexity of this basic problem by showing that the [math] upper bound is tight up to [math] factors. Our results exploit a rich tool set from additive combinatorics, most notably the Balog–Szemerédi–Gowers theorem and Rusza’s covering lemma. A key ingredient that may be of independent interest is a truly subquadratic algorithm for 3-SUM if one of the sets has small doubling.