Fredman’s Trick Meets Dominance Product: Fine-Grained Complexity of Unweighted APSP, 3SUM Counting, and More

In this paper we carefully combine Fredman’s trick [SICOMP’76] and Matoušek’s approach for dominance product [IPL’91] to obtain powerful results in fine-grained complexity. Under the hypothesis that APSP for undirected graphs with edge weights in {1, 2, …, n} requires n3−o(1) time (when ω=2), we show a variety of conditional lower bounds, including an n7/3−o(1) lower bound for unweighted directed APSP and an n2.2−o(1) lower bound for computing the Minimum Witness Product between two n × n Boolean matrices, even if ω=2, improving upon their trivial n2 lower bounds. Our techniques can also be used to reduce the unweighted directed APSP problem to other problems. In particular, we show that (when ω = 2), if unweighted directed APSP requires n2.5−o(1) time, then Minimum Witness Product requires n7/3−o(1) time. We show that, surprisingly, many central problems in fine-grained complexity are equivalent to their natural counting versions. In particular, we show that Min-Plus Product and Exact Triangle are subcubically equivalent to their counting versions, and 3SUM is subquadratically equivalent to its counting version. We also obtain new algorithms using new variants of the Balog-Szemerédi-Gowers theorem from additive combinatorics. For example, we get an O(n3.83) time deterministic algorithm for exactly counting the number of shortest paths in an arbitrary weighted graph, improving the textbook O(n4) time algorithm. We also get faster algorithms for 3SUM in preprocessed universes, and deterministic algorithms for 3SUM on monotone sets in {1, 2, …, n}d.