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

Abstract

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.