This project aims to establish the time complexity of algorithms for two classes of problems. The first class consists of problems related to Boolean matrix multiplication and matrix multiplication over various semirings. This class contains problems such as computing transitive closure of a graph and determining the minimum distance between all-pairs of nodes in a graph. Known combinatorial algorithms for these problems run in slightly sub-cubic time. By combinatorial algorithms we mean algorithms that do not rely on the fast matrix multiplication over rings. Our goal is to show that the known combinatorial algorithms for these problems are essentially optimal. This requires designing a model of combinatorial algorithms and proving almost cubic lower bounds in it.

The other class of problems that we will focus on contains dynamic data structure problems such as dynamic graph reachability and related problems. Known algorithms for these problems exhibit trade-off between the query time and the update time, where at least one of them is always polynomial. Our goal is to show that indeed any algorithm for these problems must have update time or query time at least polynomial.

The two classes of problems are closely associated with so called 3SUM problem which serves as a benchmark for uncomputability in sub-quadratic time. Our goal is to deepen and extend the known connections between 3SUM, the other two classes and problems like formula satisfiability (SAT).