NDLI: Improving Exhaustive Search Implies Superpolynomial Lower Bounds

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Williams, Ryan
Copyright Year	2013
Abstract	The P vs. NP problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions. We do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all NP problems, and to date no major consequences have been derived from the assumption that an improvement exists. We show that there are natural NP and BPP problems for which minor algorithmic improvements over the trivial deterministic simulation already entail lower bounds such as ${\sf NEXP} \not\subseteq {\sf P}/{\rm poly}$ and ${\sf LOGSPACE} \neq {\sf NP}$. These results are especially interesting given that similar improvements have been found for many other hard problems. Optimistically, one might hope our results suggest a new path to lower bounds; pessimistically, they show that carrying out the seemingly modest program of finding slightly better algorithms for all search problems may be extremely difficult (if not impossible). We also prove unconditional superpolynomial time-space lower bounds for improving on exhaustive search: there is a problem verifiable with $k(n)$ length witnesses in $O(n^a)$ time (for some $a$ and some function $k(n) \leq n$) that cannot be solved in $k(n)^c n^{a+o(1)}$ time and $k(n)^c n^{o(1)}$ space, for every $c \geq 1$. While such problems can always be solved by exhaustive search in $O(2^{k(n)} n^a)$ time and $O(k(n)+ n^a)$ space, we can prove a superpolynomial lower bound in the parameter $k(n)$ when space usage is restricted.
Starting Page	1218
Ending Page	1244
Page Count	27
File Format	PDF
ISSN	00975397
DOI	10.1137/10080703X
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3 (Special Section on the Forty-Second Annual ACM Symposium on Theory of Computing (STOC 2010))
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-06-25
Access Restriction	Subscribed
Subject Keyword	circuit complexity Computational difficulty of problems time-space trade-offs exact algorithms improved exponential algorithms lower bounds Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Improving exhaustive search implies superpolynomial lower bounds

Time-Space Trade-offs in Resolution: Superpolynomial Lower Bounds for Superlinear Space

Improving exhaustive search implies superpolynomial lower bounds (2009).

Amplifying circuit lower bounds against polynomial time, with applications

Limits on Alternation Trading Proofs for Time–Space Lower Bounds

Circuit Lower Bounds for MerlinArthur Classes

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Space-Time Trade-Offs for Orthogonal Range Queries

On Proving Circuit Lower Bounds against the Polynomial-Time Hierarchy

Improving Exhaustive Search Implies Superpolynomial Lower Bounds

Similar Documents

Improving exhaustive search implies superpolynomial lower bounds

Time-Space Trade-offs in Resolution: Superpolynomial Lower Bounds for Superlinear Space

Improving exhaustive search implies superpolynomial lower bounds (2009).

Amplifying circuit lower bounds against polynomial time, with applications

Limits on Alternation Trading Proofs for Time–Space Lower Bounds

Circuit Lower Bounds for MerlinArthur Classes

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Space-Time Trade-Offs for Orthogonal Range Queries

On Proving Circuit Lower Bounds against the Polynomial-Time Hierarchy

Improving Exhaustive Search Implies Superpolynomial Lower Bounds