Some hardness escalation results in computational complexity theory

In this thesis, we prove new hardness escalation results in computational complexity theory; a phenomenon where hardness results against seemingly weak models of computation for any problem can be lifted, in a black box manner, to much stronger models of computation by considering a simple gadget composed version of the original problem. For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we show that a gadget-composed version of F is hard to refute in any proof system whose lines are computed by efficient communication protocols. This allows us to prove new lower bounds for: -- Monotone Circuit Size : we get an exponential lower bound for an explicit monotone function computable by linear sized monotone span programs and also in (non-monotone) NC². -- Real Monotone Circuit Size : Our proof technique extends to real communication protocols, which yields similar lower bounds against real monotone circuits. -- Cutting Planes Length : we get exponential lower bound for an explicit CNF contradiction that is refutable with logarithmic Nullstellensatz degree. Finally, we describe an intimate connection between computational models and communication complexity analogs of the sub-classes of TFNP, the class of all total search problems in NP. We show that the communication analog of PPA[subscript p] captures span programs over F[subscript p] for any prime p. This complements previously known results that communication FP captures formulas (Karchmer- Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995).

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2020

Author dc:creator

• Kamath, Pritish.

Advisor dc:contributor.advisor

• Ronitt Rubinfeld and Madhu Sudan.

• Electrical Engineering and Computer Science.

Statement dc:rights

• MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/128290

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/128290