Conditional Hardness for Sensitivity Problems

Wednesday, May 31, 2017 - 4:00pm to 5:00pm
Andrea Lincoln

In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity.

In this paper we prove conditional lower bounds for these and additional problems in a sensitivity setting. For example, we show that under the Boolean Matrix Multiplication (BMM) conjecture combinatorial algorithms cannot compute the diameter of an undirected unweighted dense graph with truly subcubic preprocessing time and truly subquadratic update/query time. This result is surprising since in the static setting it is not clear whether a reduction from BMM to diameter is possible.

We further show under the BMM conjecture that many problems, such as approximate shortest paths, cannot be solved faster than by recomputation from scratch even after only one or two edge insertions. We further give a reduction from All Pairs Shortest Paths to Diameter under 1 deletion in weighted graphs. This is intriguing, as in the static setting it is a big open problem whether Diameter is as hard as APSP. We further get a nearly tight lower bound for shortest paths after two edge deletions based on the APSP conjecture. We give more lower bounds under the Strong Exponential Time Hypothesis. Many of our lower bounds also hold for static oracle data structures where no sensitivity is required.
Joint work with: Monika Henzinger, Stefan Neumann and Virginia Vassilevska Williams.