### 06.11.16 - 11.11.16, Seminar 16451

# Structure and Hardness in P

### The following text appeared on our web pages prior to the seminar, and was included as part of the invitation.

## Motivation

The complexity class *P* (polynomial time) contains a vast variety of problems of practical interest and yet relatively little is known about the *structure* of P, or of the complexity of many *individual* problems in *P*. It is known that there exist contrived problems requiring Ω(n^{1.5}) time or Ω(n^{2}) time, and yet to date no unconditional nonlinear lower bounds have been proved for *any* problem of practical interest. However, the last few years have seen a new resurgence in *conditional* lower bounds, whose validity rests on the conjectured hardness of some archetypal computational problem. This work has imbued the class *P* with new structure and has valuable explanatory power.

To cite a small fraction of recent discoveries, it is now known that classic dynamic programming problems such as Edit Distance, LCS, and Fréchet distance require quadratic time (based on the conjectured hardness of *k*-CNF-SAT), that the best known triangle enumeration algorithms are optimal (based on the hardness of 3SUM), that Valiant's context-free grammar parser is optimal (based on the hardness of *k*-CLIQUE), and that the best known approximate Nash equilibrium algorithm is optimal (based on the hardness of 3SAT).

This Dagstuhl Seminar will bring together top researchers in diverse areas of theoretical computer science and include a mixture of both experts and non-experts in conditional lower bounds. Some specific goals of this seminar are listed below.

- Numerous important problems (such as Linear Programming) seem insoluble in linear time, and yet no conditional lower bounds are known to explain this fact. A goal is to discover conditional lower bounds for key problems for which little is currently known.
- Recent work has been based on both traditional hardness assumptions (such as the ETH, SETH, 3SUM, and APSP conjectures) and a variety of newly considered hardness assumptions (such as the OMv conjecture, the
*k*-CLIQUE conjecture, and the Hitting Set conjecture). Almost nothing is known about the relative plausibility of these conjectures, or if multiple conjectures are, in fact, equivalent. A goal is to discover formal relationships between the traditional and newer hardness assumptions. - A key goal of the seminar is to disseminate the
*techniques*used to prove conditional lower bounds, particularly to researchers from areas of theoretical computer science that have yet to benefit from this theory. To this end the seminar will include a number of tutorials from top experts in the field.