"Orthogonal Vectors is hard for first-order properties on sparse graphs"
Monday, April 18th, 2016, 2:00 pm
EBU3B, Room 4258
Fine-grained reductions, introduced by Vassilevska-Williams and Williams, preserve any improvement in the known algorithms. These have been used very successfully in relating the exact complexities of a wide range of problems, from NP-complete problems like SAT to important quadratic time solvable problems within P such as Edit Distance. However, until now, there have been few equivalences between problems and in particular, no problems that were complete for natural classes under fine-grained reductions. We give the first such completeness results. We consider the class of first-order graph property problems, viewing the input in adjacency list format (aka "sparse graph representation"). For this class, we show that the sparse Orthogonal Vectors problem is complete under randomized fine-grained reductions. In proving completeness for this problem, we also show that this sparse problem is equivalent to the standard Orthogonal Vectors problem when the number of dimensions is polynomially related to the number of vectors. Finally, we also establish a completeness and hardness result for k-Orthogonal Vectors.
Our results imply that the conjecture "not every first-order graph problem has an improved algorithm" is a useful intermediary between SETH and the conjectured hardness of problems such as Edit Distance. It follows that, if Edit Distance has a substantially subquadratic algorithm, then every first order graph problem has an improved algorithm. On the other hand, if first order graph property problems have improved algorithms, this falsifies SETH (and even some weaker versions of SETH) and gives new circuit lower bounds. We hope that this is the beginning of extending fine-grained complexity to include classes of problems as well as individual problems.
Joint work with Russell Impagliazzo.