We say an algorithm on n × n matrices with entries in [−M, M ] (or n-node graphs with edge weights from [−M, M ]) is truly subcubic if it runs in O(n 3−δ · poly(log M )) time for some δ > 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n 3 ) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do:• The all-pairs shortest paths problem on weighted digraphs (APSP).• Detecting if a weighted graph has a triangle of negative total edge weight.• Listing up to n 2.99 negative triangles in an edge-weighted graph.• Finding a minimum weight cycle in a graph of non-negative edge weights.• The replacement paths problem on weighted digraphs.• Finding the second shortest simple path between two nodes in a weighted digraph.• Checking whether a given matrix defines a metric.• Verifying the correctness of a matrix product over the (min, +)-semiring.Therefore, if APSP cannot be solved in n 3−ε time for any ε > 0, then many other problems also need essentially cubic time. In fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure. These equivalences simplify prior work on subcubic algorithms for all-pairs path problems, since it now suffices to give appropriate subcubic triangle detection algorithms.Other consequences of our work are new combinatorial approaches to Boolean matrix multiplication over the (OR,AND)-semiring (abbreviated as BMM). We show that practical advances in triangle detection would imply practical BMM algorithms, among other results. Building on our techniques, we give two new BMM algorithms: a derandomization of the recent combinatorial BMM algorithm of Bansal and Williams (FOCS'09), and an improved quantum algorithm for BMM. * This work originated while the authors were members of the Institute for Advanced Study, Princeton, NJ and visiting the Com-

A recent, active line of work achieves tight lower bounds for fundamental problems under the Strong Exponential Time Hypothesis (SETH). A celebrated result of Backurs and Indyk (STOC'15) proves that computing the Edit Distance of two sequences of length n in truly subquadratic O(n 2−ε) time, for some ε > 0, is impossible under SETH. The result was extended by follow-up works to simpler looking problems like finding the Longest Common Subsequence (LCS). SETH is a very strong assumption, asserting that even linear size CNF formulas cannot be analyzed for satisfiability with an exponential speedup over exhaustive search. We consider much safer assumptions, e.g. that such a speedup is impossible for SAT on more expressive representations, like subexponential-size NC circuits. Intuitively, this assumption is much more plausible: NC circuits can implement linear algebra and complex cryptographic primitives, while CNFs cannot even approximately compute an XOR of bits. Our main result is a surprising reduction from SAT on Branching Programs to fundamental problems in P like Edit Distance, LCS, and many others. Truly subquadratic algorithms for these problems therefore have far more remarkable consequences than merely faster CNF-SAT algorithms. For example, SAT on arbitrary o(n)-depth bounded fan-in

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Fine-grained reductions have established equivalences between many core problems withÕ(n 3 )-time algorithms on n-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also haveÕ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn bounds optimal when m n 2 ?Starting from the hypothesis that the minimum weight (2 + 1)-Clique problem in edge weighted graphs requires n 2 +1−o(1) time, we prove that for all sparsities of the form m = Θ(n 1+1/ ), there is no O(n 2 + mn 1−ε ) time algorithm for ε > 0 for any of the below problems• Minimum Weight (2 + 1)-Cycle in a directed weighted graph, • Shortest Cycle in a directed weighted graph, • APSP in a directed or undirected weighted graph, • Radius (or Eccentricities) in a directed or undirected weighted graph, • Wiener index of a directed or undirected weighted graph, • Replacement Paths in a directed weighted graph, • Second Shortest Path in a directed weighted graph, • Betweenness Centrality of a given node in a directed weighted graph. That is, we prove hardness for a variety of sparse graph problems from the hardness of a dense graph problem. Our results also lead to new conditional lower bounds from several related hypothesis for unweighted sparse graph problems including k-cycle, shortest cycle, Radius, Wiener index and APSP. * andreali@mit.edu. Supported by the EECS Merrill Lynch Fellowship.†

Properties definable in first-order logic are algorithmically interesting for both theoretical and pragmatic reasons. Many of the most studied algorithmic problems, such as Hitting Set and Orthogonal Vectors, are first-order, and the first-order properties naturally arise as relational database queries. A relatively straightforward algorithm for evaluating a property with k + 1 quantifiers takes time O (m k) and, assuming the Strong Exponential Time Hypothesis (SETH), some such properties require O (m k−ϵ) time for any ϵ > 0. (Here, m represents the size of the input structure, i.e., the number of tuples in all relations.) We give algorithms for every first-order property that improves this upper bound to m k /2 Θ(√ log n) , i.e., an improvement by a factor more than any poly-log, but less than the polynomial required to refute SETH. Moreover, we show that further improvement is equivalent to improving algorithms for sparse instances of the well-studied Orthogonal Vectors problem. Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of References [3, 16]. While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class.