Simulating Branching Programs with Edit Distance and Friends or: A Polylog Shaved is a Lower Bound Made

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(n2−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 circuits (and therefore also NC-Circuit-SAT) can be solved in (2−ε)n time. An interesting feature of our work is that we get major consequences even from mildly subquadratic algorithms for Edit Distance or LCS. For example, we show that if an arbitrarily large polylog factor is shaved from n2 for Edit Distance then NEXP does not have non-uniform NC1 circuits.