beyond just the P versus NP problem
that I haven’t discussed here. In “
Further Reading,” a number of references
are presented for those interested in a
deeper understanding of the P versus
NP problem and computational complexity.
The P versus NP problem has gone
from an interesting problem related to
logic to perhaps the most fundamental
and important mathematical question
of our time, whose importance only
grows as computers become more powerful and widespread. The question has
even hit popular culture appearing in
television shows such as The Simpsons
and Numb3rs. Yet many only know of
the basic principles of P versus NP and
I hope this survey has given you a small
feeling of the depth of research inspired
by this mathematical problem.
Proving P ≠ NP would not be the end
of the story, it would just show that NP-
complete problem, don’t have efficient
algorithms for all inputs but many questions might remain. Cryptography, for
example, would require that a problem
like factoring (not believed to be NP-
complete) is hard for randomly drawn
Proving P ≠ NP might not be the start
of the story either. Weaker separations
remain perplexingly difficult, for example showing that Boolean-formula Satisfiability cannot be solved in near-linear
time or showing that some problem
using a certain amount of memory cannot be solved using roughly the same
amount of time.
None of us truly understands the P
versus NP problem, we have only begun
to peel the layers around this increasingly complex question. Perhaps we will
see a resolution of the P versus NP
problem in the near future but I almost hope
not. The P versus NP problem continues
to inspire and boggle the mind and continued exploration of this problem will
lead us to yet even new complexities in
that truly mysterious process we call
Recommendations for a more in-depth
look at the P versus NP problem and the
other topics discussed in this article:
Steve Homer and I have written a ˲
detailed historical view of computational complexity. 14
The 1979 book of Garey and John- ˲
son still gives the best overview of the P
versus NP problem with an incredibly
useful list of NP-complete problems. 16
Scott Aaronson looks at the unlikely ˲
possibility that the P versus NP problem
is formally independent. 1
Russell Impagliazzo gives a wonder- ˲
ful description of five possible worlds of
Sanjeev Arora and Boaz Barak have ˲
a new computational complexity textbook with an emphasis on recent research directions. 5
The ˲ Foundations and Trends in Theoretical Computer Science journal and the
Computational Complexity columns of
the Bulletin of the European Association
of Theoretical Computer Science and
SI-GACT News have many wonderful surveys on various topics in theory including those mentioned in this article.
Read the blog Computational Com- ˲
plexity and you will be among the first to
know about any updates of the status of
the P versus NP problem. 13
Thanks to Rahul Santhanam for many
useful discussions and comments. Josh
Grochow wrote an early draft. The anonymous referees provided critical advice.
Some of the material in this article has
appeared in my earlier surveys and my
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Bulletin of the European Association for Theoretical
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Lance Fortnow ( firstname.lastname@example.org) is a
professor of electrical engineering and computer science
at northwestern University’s McCormick school of
Engineering, Evanston, IL.