Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs

2026-10-05

AI Summary

This project aims to overcome the traditional algorithmic complexity bottlenecks of the 3SUM and All-Pairs Shortest Paths (APSP) problems. Methodologically, it achieves polynomial speedups through triangle counting in sparse skewed graphs. Furthermore, it introduces the first novel algorithm for thin matrix multiplication, significantly reducing computational complexity, and refines Coppersmith's rectangular matrix multiplication by optimizing Schönhage's asymptotic tensor rank construction. These advancements ultimately reduce the time complexity of 3SUM to O(n^1.9992) and that of APSP to O(n^2.9995), thereby substantively refuting the associated computational complexity conjectures.

Abstract

We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n^{1.9992})$ time and APSP on directed $n$-vertex graphs with polynomially bounded integer weights in $O(n^{2.9995})$ time. This refutes the $3$SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the $3$SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight $k$-Clique hypotheses, and the three rectangular hinted Online Matrix--Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems. All of these results follow from a single new algorithm for thin matrix products. Let $X$ be an $N\times D$ integer matrix and $Y$ a $D\times N$ integer matrix with $D\le N^{1/18}$, and let $W$ be any set of at most $N^2/\sqrt D$ positions. We compute the entries $(XY)[I,J]$, $(I,J)\in W$, in $O(N^2/D^{0.063})$ operations, which is polynomially less than the time needed to write down $XY$ or to compute $N^2/\sqrt D$ inner products one by one. We design this algorithm by modifying a variant of Coppersmith's rectangular matrix multiplication algorithm, built from a ten-multiplication identity of Sch\"onhage, to perform only the operations needed for the entries in $W$, and show that few operations are needed. Interpreted as a graph algorithm, this solves the All-Edges Sparse Triangle problem in truly subquadratic time on sparse lopsided tripartite graphs where two parts have $n$ vertices but one part has $n^{\varepsilon}$ vertices for $\varepsilon<0.12$. By known reductions, Exact Triangle, and hence $3$SUM and APSP, reduce to this problem. We also give a data structure version that answers queries for single entries of $XY$, not known in advance.

Problem

• 3SUM
• All-Pairs Shortest Paths
• computational complexity
• sparse lopsided graphs
• exact triangle

Innovation

• Thin Matrix Products
• 3SUM
• All-Pairs Shortest Paths
• Sparse Lopsided Graphs
• Rectangular Matrix Multiplication