Objective
Traditional complexity theory focuses on the dichotomy between P and NP-hard
problems. Lately, it has become increasingly clear that this misses a major part
of the picture. Results by the PI and others offer glimpses on a fascinating structure
hiding inside NP: new computational problems that seem to lie between polynomial
and NP-hard have been identified; new conditional lower bounds for problems with
large polynomial running times have been found; long-held beliefs on the difficulty
of problems in P have been overturned. Computational geometry plays a major role
in these developments, providing some of the main questions and concepts.
We propose to explore this fascinating landscape inside NP from the perspective
of computational geometry, guided by three complementary questions:
(A) What can we say about the complexity of search problems derived from
existence theorems in discrete geometry? These problems offer a new
perspective on complexity classes previously studied in algorithmic game
theory (PPAD, PLS, CLS). Preliminary work indicates that they have the
potential to answer long-standing open questions on these classes.
(B) Can we provide meaningful conditional lower bounds on geometric
problems for which we have only algorithms with large polynomial running
time? Prompted by a question raised by the PI and collaborators, such lower
bounds were developed for the Frechet distance. Are similar results possible
for problems not related to distance measures? If so, this could dramatically
extend the traditional theory based on 3SUM-hardness to a much more
diverse and nuanced picture.
(C) Can we find subquadratic decision trees and faster algorithms for
3SUM-hard problems? After recent results by Pettie and Gronlund on
3SUM and by the PI and collaborators on the Frechet distance, we
have the potential to gain new insights on this large class of well-studied
problems and to improve long-standing complexity bounds for them.
Fields of science
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Funding Scheme
ERC-STG - Starting GrantHost institution
14195 Berlin
Germany