Higher Lower Bounds from the 3SUM Conjecture

TL;DR

This paper improves the connection between the 3SUM conjecture and lower bounds for various computational problems by providing more efficient reductions, leading to near-optimal bounds for algorithms in graph enumeration, set intersection, and dynamic graph problems.

Contribution

It introduces new, efficient reductions from 3SUM to offline SetDisjointness and SetIntersection, enabling tighter conditional lower bounds and proving the near-optimality of several algorithms.

Findings

Triangle enumeration in graphs is essentially optimal.

Listing t triangles is essentially optimal under certain assumptions.

Static SetDisjointness query answering requires near-quadratic preprocessing time.

Abstract

The 3SUM conjecture has proven to be a valuable tool for proving conditional lower bounds on dynamic data structures and graph problems. This line of work was initiated by P\v{a}tra\c{s}cu (STOC 2010) who reduced 3SUM to an offline SetDisjointness problem. However, the reduction introduced by P\v{a}tra\c{s}cu suffers from several inefficiencies, making it difficult to obtain tight conditional lower bounds from the 3SUM conjecture. In this paper we address many of the deficiencies of P\v{a}tra\c{s}cu's framework. We give new and efficient reductions from 3SUM to offline SetDisjointness and offline SetIntersection (the reporting version of SetDisjointness) which leads to polynomially higher lower bounds on several problems. Using our reductions, we are able to show the essential optimality of several algorithms, assuming the 3SUM conjecture. - Chiba and Nishizeki's $O(m\alpha)$-time algorithm (SICOMP 1985) for enumerating all triangles in a graph with arboricity/degeneracy $\alpha$ is essentially optimal, for any $\alpha$. - Bj{\o}rklund, Pagh, Williams, and Zwick's algorithm (ICALP 2014) for listing $t$ triangles is essentially optimal (assuming the matrix multiplication exponent is $\omega=2$). - Any static data structure for SetDisjointness that answers queries in constant time must spend $\Omega(N^{2-o(1)})$ time in preprocessing, where $N$ is the size of the set system. These statements were unattainable via P\v{a}tra\c{s}cu's reductions. We also introduce several new reductions from 3SUM to pattern matching problems and dynamic graph problems. Of particular interest are new conditional lower bounds for dynamic versions of Maximum Cardinality Matching, which introduce a new technique for obtaining amortized lower bounds.