November 6 – 11 , 2016, Dagstuhl Seminar 16451

Structure and Hardness in P


Moshe Lewenstein (Bar-Ilan University – Ramat Gan, IL)
Seth Pettie (University of Michigan – Ann Arbor, US)
Virginia Vassilevska Williams (Stanford University, US)

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Aims & Scope
List of Participants
Dagstuhl's Impact: Documents available


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 Omega(n1.5) time or Omega(n2) 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 3-SUM), 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 3-SAT).

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.
Summary text license
  Creative Commons BY 3.0 Unported license
  Moshe Lewenstein, Seth Pettie, and Virginia Vassilevska Williams


  • Data Structures / Algorithms / Complexity


  • Classifying P
  • Lower bounds
  • Hardness assumptions
  • Algorithmic equivalences


In the series Dagstuhl Reports each Dagstuhl Seminar and Dagstuhl Perspectives Workshop is documented. The seminar organizers, in cooperation with the collector, prepare a report that includes contributions from the participants' talks together with a summary of the seminar.


Download overview leaflet (PDF).

Dagstuhl's Impact

Please inform us when a publication was published as a result from your seminar. These publications are listed in the category Dagstuhl's Impact and are presented on a special shelf on the ground floor of the library.


Furthermore, a comprehensive peer-reviewed collection of research papers can be published in the series Dagstuhl Follow-Ups.