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The Status of the P versus NP Problem
Lance Fortnow
Northwestern University
1. INTRODUCTION
When Moshe Vardi asked me to write this piece for CACM,
my first reaction was the article could be written in two
words
Still open.
When I started graduate school in the mid-1980’s, many be-
lieved that the quickly developing area of circuit complexity
would soon settle the Pversus NP problem, whether every
algorithmic problem with efficiently verifiable solutions have
efficiently computable solutions. But circuit complexity and
other approaches to the problem have stalled and we have
little reason to believe we will see a proof separating Pfrom
NP in the near future.
Nevertheless the computer science landscape has dramati-
cally changed in the nearly four decades since Steve Cook
presented his seminal NP-completeness paper The Complex-
ity of Theorem-Proving Procedures [11] in Shaker Heights,
Ohio in early May, 1971. Computational power has dra-
matically increased, the cost of computing has dramatically
decreased, not to mention the power of the Internet. Com-
putation has become a standard tool in just about every aca-
demic field. Whole subfields of biology, chemistry, physics,
economics and others are devoted to large-scale computa-
tional modeling, simulations and problem solving.
As we solve larger and more complex problems with greater
computational power and cleverer algorithms, the problems
we cannot tackle begin to stand out. The theory of NP-
completeness helps us understand these limitations and the
Pversus NP problems begins to loom large not just as an
interesting theoretical question in computer science, but as
a basic principle that permeates all the sciences.
So while we don’t expect the Pversus NP problem to be
resolved in the near future, the question has driven research
in a number of topics to help us understand, handle and
even take advantage of the hardness of various computa-
tional problems.
In this survey we will look at how people have tried to solve
the Pversus NP problem but also how this question has
shaped so much of the research in computer science and be-
yond. We will look at how to handle NP-complete problems
and the theory that has developed from those approaches.
We show how a new type of “interactive proof systems” led
to limitations of approximation algorithms. We consider
whether quantum computer can solve NP-complete prob-
lems (short answer: not likely). We end describing a new
long-term project that will try to separate Pfrom NP using
algebraic-geometric techniques.
This survey won’t try to be 100% technically or historically
accurate or complete but rather informally describe the P
versus NP problem and the major directions in comput-
ers science inspired by this question over the past several
decades.
2. WHAT IS THE P VERSUS NP PROBLEM?
Suppose we have a large group of students that we need
to pair up to work on projects. We know which students
are compatible with each other and we want to put them
in compatible groups of two. We could search all possible
pairings but even for 40 students we would have more than
three hundred billion trillion possible pairings.
In 1965, Jack Edmonds [13] gave an efficient algorithm to
solve this matching problem and suggested a formal defi-
nition of “efficient computation” (runs in time a fixed poly-
nomial of the input size). The class of problems with efficient
solutions would later become known as Pfor “Polynomial
Time”.
But many related problems do not seem to have such an effi-
cient algorithm. What if we wanted to make groups of three
students with each pair of students in each group compat-
ible (Partition Into Triangles)? What if we wanted to find
a large group of students all of whom are compatible with
each other (Clique)? What if we wanted to sit the students
around a large round table with no incompatible students
sitting next to each other (Hamiltonian Cycle)? What if
we put the students into three groups so that each student
is in the same group with only his or her compatibles (3-
Coloring)?
All these problems have a similar flavor: Given a potential
solution, for example a seating chart for the round table,
we can validate that solution efficiently. The collection of
problems that have efficiently verifiable solutions is known
as NP (for “Nondeterministic Polynomial-Time” if you have
to ask).
So P=NP means that for every problem that has an ef-
ficiently verifiable solution, we can find that solution effi-
ciently as well.
We call the very hardest NP problems (which include Parti-
tion Into Triangles, Clique, Hamiltonian Cycle and 3-Coloring)
“NP-complete”, i.e. given an efficient algorithm for one of
them, we can find an efficient algorithm for all of them and
in fact any problem in NP. Steve Cook, Leonid Levin and
Richard Karp [11, 28, 25] developed the initial theory of NP-
completeness that generated multiple ACM Turing Awards.
In the 1970’s, theoretical computer scientists showed hun-
dreds more problems NP-complete (see [17]). An efficient
solution to any NP-complete problem would imply P=NP
and an efficient solution to every NP-complete problem.
Most computer scientists quickly came to believe P6=NP
and trying to prove it quickly became the single most impor-
tant question in all of theoretical computer science and one
of the most important in all of mathematics. Soon the P
versus NP problem became an important computationally
issue in nearly every scientific discipline.
As computers grew cheaper and more powerful, computa-
tion started playing a major role in nearly every academic
field, especially the sciences. The more scientists can do with
computers, the more they realize some problems seem com-
putationally difficult. Many of these fundamental problems
turn out to be NP-complete. A small sample:
•Finding a DNA sequence that best fits a collection of
fragments of the sequence (see [21]).
•Finding a ground state in the Ising model of phase
transitions (see [9]).
•Finding Nash Equilbriums with specific properties in
a number of environments (see [10]).
•Finding optimal protein threading procedures [27].
•Determining if a mathematical statement has a short
proof (follows from Cook [11]).
In 2000, the Clay Math Institute named the Pversus NP
problems as one of the seven most important open questions
in mathematics and has offered a million-dollar prize for a
proof that determines whether or not P=NP.
3. WHAT IF P = NP?
To understand the importance of the Pversus NP problem
let us imagine a world where P=NP. Technically we could
have P=NP but not have practical algorithms for most
NP-complete problems. But suppose in fact that we do have
very quick algorithms for all these problems.
Many focus on the negative, that if P=NP then public-
key cryptography becomes impossible. True but what we
will gain from P=NP will make the whole internet look
like a footnote in history.
Since all the NP-complete optimization problems become
easy, everything will be much more efficient. Transporta-
tion of all forms will be scheduled optimally to move people
and goods around quicker and cheaper. Manufacturers can
improve their production to increase speed and create less
waste. And I’m just scratching the surface.
Learning becomes easy by using the principle of Occam’s
razor–we simply find the smallest program consistent with
the data. Near perfect vision recognition, language compre-
hension and translation and all other learning tasks become
trivial. We will also have much better predictions of weather
and earthquakes and other natural phenomenon.
P=NP would also have big implications in mathematics.
One could find short fully logical proofs for theorems but
these fully logical proofs are usually extremely long. But we
can use the Occam razor principle to recognize and verify
mathematical proofs as typically written in journals. We
can then find proofs of theorems that have reasonably length
proofs say in under 100 pages. A person who proves P=
NP would walk home from the Clay Institute not with one
million-dollar check but with seven (actually six since the
Poincar´e Conjecture appears solved).
Don’t get your hopes up. Complexity theorists generally
believe P6=NP and such a beautiful world cannot exist.
4. APPROACHES TO SHOWING P 6=NP
In this section we present a number of ways we have tried
and failed to prove P6=NP. The survey of Fortnow and
Homer [15] gives a fuller historical overview of these tech-
niques.
4.1 Diagonalization
Can we just construct an NP language Lspecifically de-
signed so that every single polynomial-time algorithm fails to
compute Lproperly on some input? This approach, known
as diagonalization, goes back to the 19th Century.
In 1874 Cantor [8] showed that the real numbers are un-
countable using a technique known as diagonalization. Given
a countable list of reals, Cantor showed how to create a new
real number not on that list.
Turing, in his seminal paper on computation [39], used a
similar technique to show that the Halting problem is not
computable. In the 1960’s complexity theorists used diag-
onalization to show that given more time or memory one
can solve more problems. Why not use diagonalization to
separate NP from P?
Diagonalization requires simulation and we don’t know how
a fixed NP machine can simulate an arbitrary Pmachine.
Also a diagonalization proof would likely relativize, i.e., work
even if all machines involved have access to the same addi-
tional information. Baker, Gill and Solovay [7] showed no
relativizable proof can settle the Pversus NP problem in
either direction.
Complexity theorists have used diagonalization techniques
to show some NP-complete problems like Boolean formula
satisfiability cannot have algorithms that use both a small
amount of time and memory [40] but this is a long way from
P6=NP.
4.2 Circuit Complexity
To show P6=NP it is sufficient to show some NP-complete
problem cannot be solved by relatively small circuits of AND,
OR and NOT gates (the number of gates bounded by a fixed
polynomial in the input size).
In 1984, Furst, Saxe and Sipser [16] showed that small cir-
cuits cannot solve the parity function if the circuits have
a fixed number of layers of gates. In 1985, Razborov [32]
showed the NP-complete problem of finding a large clique
does not have small circuits if one only allows AND and OR
gates (no NOT gates). If one extends Razborov’s result to
general circuits one will have proved P6=NP.
Razborov later showed his techniques would fail miserably
if one allows NOT gates [33]. Razborov and Rudich [34] de-
velop a notion of“natural”proofs and give evidence that our
limited techniques in circuit complexity cannot be pushed
much further. And in fact we haven’t seen any significantly
new circuit lower bounds in the past twenty years.
4.3 Proof Complexity
Consider the set of Tautologies, the Boolean formulas φof
variables over ANDs, ORs an NOTs such that every setting
of the variables to True and False makes φtrue, for example
the formula
(xAND y) OR (NOT x) OR (NOT y).
A literal is a variable or its negation, i.e. xor NOT x. A
formula, like the one above, is in Disjunctive Normal Form
(DNF) if it is the OR of ANDs of one or more literals.
If a formula φis not a tautology, we can give an easy proof
of that fact by exhibiting an assignment of the variables that
makes φfalse. But if φwere indeed a tautology we don’t
expect short proofs of that. If one could prove there are no
short proofs of tautology that would imply P6=NP.
Resolution is a standard approach to proving tautologies of
DNFs by finding two clauses of the form (ψ1AND x) and
(ψ2AND NOT x) and adding the clause (ψ1AND ψ2). A
formula is a tautology exactly when one can produce an
empty clause in this manner.
In 1985, Haken [22] showed that tautologies that encode the
pigeonhole principle (n+ 1 pigeons in nholes means some
hole has more than one pigeon) do not have short resolution
proofs.
Since then complexity theorists have shown similar weak-
nesses in a number of other proof systems including cutting
planes, algebraic proof systems based on polynomials and re-
stricted versions of proofs using the Frege axioms, the basic
axioms one learns in an introductory logic course.
But to prove P6=NP we would need to show that tautolo-
gies cannot have short proofs in an arbitrary proof system.
Even a breakthrough result showing tautologies don’t have
short general Frege proofs would not suffice in separating
NP from P.
5. DEALING WITH HARDNESS
So you have an NP-complete problem you just have to solve.
If as we believe P6=NP you won’t find a general algorithm
that will correctly and accurately solve your problem all of
the time. But sometimes you need to solve the problem any-
way. All hope is not lost. In this section we describe some
of the tools one can use on NP-complete problems and how
computational complexity theory studies these approaches.
Typically one needs to combine several of these approaches
when tackling NP-complete problems in the real world.
5.1 Brute Force
Computers have gotten faster, much much faster since NP-
completeness was first developed. Brute force search through
all possibilities is now possible for some small problem in-
stances. With some clever algorithms we can even solve
some moderate size problems with ease.
The NP-complete traveling salesperson problem asks for the
smallest distance tour through a set of specified cities. Using
extensions of the cutting-plane method we can now solve, in
practice, traveling salespeople problems with more than ten
thousand cities (see [4]).
Consider the 3SAT problem, solving Boolean formula satisfi-
ability where formulas are in the form of the AND of several
clauses where each clause is the OR of three literal (variables
or negations of variables). 3SAT remains NP-complete but
the best algorithms can in practice solve SAT problems on
about 100 variables. We have similar results for other vari-
ations of satisfiability and many other NP-complete prob-
lems.
But for satisfiability on general formulae and on many other
NP-complete problems we do not know algorithms better
than essentially searching all the possibilities. In addition all
these algorithms have exponential growth in their running
times, so even a small increase in the problem size can kill
what was an efficient algorithm. Brute force alone will not
solve NP-complete problems no matter how clever we are.
5.2 Parameterized Complexity
Consider the Vertex Cover problem, find a set of k“central
people” such that for every compatible pair of people, at
least one of them is central. For small kwe can determine
whether a central set of people exists efficiently no matter
the total number nof people we are considering. For the
Clique problem even for small kthe problem can still be
difficult.
Downey and Fellows [12] developed a theory of parameter-
ized complexity that gives a fine-grained analysis of the com-
plexity of NP-complete problems based on their parameter
size.
5.3 Approximation
We cannot hope to solve NP-complete optimization prob-
lems exactly but often we can get a good approximate an-
swer. Consider the traveling salesperson problem again with
distances between cities given as the crow flies (Euclidean
distance). This problem remains NP-complete but Arora [5]
gives an efficient algorithm that gets very close to the best
possible route.
Consider the MAX-CUT problem of dividing people into two
groups to maximize the number of incompatibles between
the groups. Goemans and Williamson [18] uses semi-definite
programming to give a division of people only a .878567
factor of the best possible.
In Section 6 we will see how a theory of probabilistic proof
systems leads to provable limitations of approximation algo-
rithms.
5.4 Heuristics and Average-Case Complexity
The study of NP-completeness focuses on how algorithms
perform on the worst possible inputs. However the specific
problems that arise in practice may be much easier to solve.
Many computer scientists employ various heuristics to solve
NP-complete problems that arise from the specific problems
in their fields.
While we create heuristics for many of the NP-complete
problems, Boolean formula Satisfiability (SAT) receives more
attention than any other. Boolean formulas, especially those
in conjunctive normal form (CNF), the AND of ORs of vari-
ables and their negations, have a very simple description and
yet are general enough to apply to a large number of practi-
cal scenarios particularly in software verification and artifi-
cial intelligence. Most natural NP-complete problems have
simple efficient reductions to the satisfiability of Boolean
formula. In competition these SAT solvers can often settle
satisfiability of formulas of a million variables [1].
Computational complexity theorists study heuristics by con-
sidering average-case complexity—how well can algorithms
perform on average from instances generated by some spe-
cific distribution.
Levin [29] developed a theory of efficient algorithms over a
specific distribution and formulated a distributional version
of the Pversus NP problem.
Some problems like versions of the shortest vector problem
in a lattice or computing the permanent of a matrix are
hard on average exactly when they are hard on worst-case
inputs, but neither of these problems is believed to be NP-
complete. Whether similar worst-to-average reductions hold
for NP-complete sets is an important open problem.
Average-case complexity plays an important role in many
areas of computer science, particularly cryptography which
we discuss in Section 7.1.
6. INTERACTIVE PROOFS AND
LIMITS OF APPROXIMATION
In Section 5.3 we saw how sometimes one can get good ap-
proximate solutions to NP-complete optimization problems.
Many times though we seem to hit a limit on our ability to
even get good approximations. We now know that we cannot
achieve better approximations on many of these problems
unless P=NP and we could solve these problems exactly.
The techniques to show these negative results came out of a
new model of proof system originally developed for cryptog-
raphy and to classify group theoretic algorithmic problems.
As mentioned in Section 4.3 we don’t expect to have short
traditional proofs of tautologies. But consider an “interac-
tive proof” model where a prover Peggy tries to convince a
verifier Victor that a formula φis a tautology. Victor can ask
Peggy randomly generated questions and need only be con-
vinced with high confidence. Quite surprisingly these proof
systems have been shown to exist not only for tautologies
but for any problem computable in a reasonable amount of
memory.
A variation known as a “probabilistically checkable proof
system” (PCPs), where Peggy writes down an encoded proof
and Victor can make randomized queries to the bits of the
proof, has applications for approximations. The “PCP The-
orem” which optimizes parameters which in its strong form
shows that every language in NP has a PCP where Victor
uses a tiny number of random coins and queries only three
bits of the proof.
One can use this PCP theorem to show the limitations of
approximation for a large number of optimization questions.
For example one cannot approximate the largest clique in a
group of npeople by more than a multiplicative ratio of
nearly √nunless P=NP. See the recent CACM survey by
Madhu Sudan for more details and references on PCPs [37].
One can do even better assuming a “Unique Games Conjec-
ture,” that there exists PCPs for NP problems with some
stronger properties. Consider the MAX-CUT problem of di-
viding people discussed in Section 5.3. If the unique games
conjecture holds one cannot do better than the .878567 fac-
tor given by the Goemans-Williamson approximation algo-
rithm [26]. Recent work shows how to get a provably best
approximation for essentially any constrained problem as-
suming this conjecture [31].
7. USING HARDNESS
In Section 3 we saw the nice world that arises when we
assume P=NP. But we expect P6=NP to hold in very
strong ways. We can use strong hardness assumptions as a
positive tool, particularly to create cryptographic protocols
and to reduce or even eliminate the need of random bits in
probabilistic algorithms.
7.1 Cryptography
We take it for granted these days, the little key or lock on
our web page that tells us that someone listening to the
network won’t get the credit card number I just sent to an
on-line store or the password to the bank that controls my
money. But public-key cryptography, the ability to send se-
cure messages between two parties that have never privately
exchanged keys, is a relatively new development based on
hardness assumptions of computational problems.
If P=NP then public-key cryptography is impossible. As-
suming P6=NP is not enough to get public-key protocols,
instead we need strong average-case assumptions about the
difficulty of factoring or related problems.
We can do much more than just public-key cryptography us-
ing hard problems. Suppose Alice’s husband Bob is working
on a Sudoku puzzle and Alice claims she has a solution to the
puzzle (solving a n2×n2-Sudoku puzzle is NP-complete).
Can Alice convince Bob that she knows a solution without
revealing any piece of it?
Turns out Alice can using a “zero-knowledge proof,” an in-
teractive proof with the additional feature that the verifier
learns nothing other than some property holds, like a Su-
doku puzzle having a solution. Every NP search problem
has a zero-knowledge proof under the appropriate hardness
assumptions.
On-line poker is generally played through some “trusted”
website, usually somewhere in the Caribbean. Can we play
poker over the Internet without a trusted server? Using the
right cryptographic assumptions, not only poker but any
protocol that uses a trusted party can be replaced by one
that uses no trusted party and the players can’t cheat or
learn anything new beyond what they could do with the
trusted party.
See the survey of Goldreich [19] for details on the above.
7.2 Eliminating Randomness
In the 70’s we saw a new type of algorithm, one that used
random bits to aid in finding a solution to a problem. Most
notably we had probabilistic algorithms [36] for determining
whether a number is prime, an important routine needed
for modern cryptography. In 2004 we discovered we don’t
need randomness at all to efficiently determine if a number is
prime [3]. Does randomness help us at all in finding solutions
to NP problems?
Truly independent and uniform random bits are either very
difficult or impossible to produce (depending on your beliefs
about quantum mechanics). Computer algorithms instead
use pseudorandom generators to generate a sequence of bits
from some given seed. The generators typically found on our
computers usually work well but occasionally give incorrect
results both in theory and in practice.
We can create theoretically better pseudorandom genera-
tors in two different ways, one based on the strong hardness
assumptions of cryptography and the other based on worst-
case complexity assumptions. I will focus on this second
approach.
We need to assume a bit more than P6=NP, roughly that
NP-complete problems cannot be solved by smaller than
expected AND-OR-NOT circuits. A long series of papers
showed that, under this assumption, any problem with an ef-
ficient probabilistic algorithm also has an efficient algorithm
that uses a pseudorandom generator with a very short seed,
a surprising connection between hard languages and pseu-
dorandomness (see [24]). The seed is so short we can try all
possible seeds efficiently and avoid the need for randomness
altogether.
Thus complexity theorists generally believe having random-
ness does not help in solving NP search problems and that
NP-complete problems do not have efficient solutions, either
with or without using truly random bits.
While randomness doesn’t seem necessary for solving search
problems, the unpredictability of random bits plays a critical
role in cryptography and interactive proof systems and likely
cannot be avoided in these scenarios.
8. COULD QUANTUM COMPUTERS
SOLVE NP-COMPLETE PROBLEMS?
While we have randomized and non-randomized efficient al-
gorithms for determining whether a number is prime, these
algorithms usually don’t give us the factors of a composite
number. Much of modern cryptography relies on the fact
that factoring or similar problems do not have efficient al-
gorithms.
In the mid-90’s, Peter Shor [35] showed how to factor num-
bers using a hypothetical quantum computer. He also gave
a similar quantum algorithm to solve the discrete logarithm
problem. The hardness of discrete logarithm on classical
computers is also used as a basis for many cryptographic pro-
tocols. Nevertheless we don’t expect that factoring or find-
ing discrete logarithms are NP-complete. While we don’t
think we have efficient algorithms to solve factoring or dis-
crete logarithm, we also don’t believe we can reduce NP-
complete problems like Clique to the factoring or discrete
logarithm problems.
So could quantum computers one day solve NP-complete
problems? Unlikely.
I’m not a physicist so I won’t address the problem as to
whether these machines can actually be built at a large
enough scale to solve factoring problems larger than we can
with current technology (about 200 digits). After billions of
dollars of funding of quantum computing research we still
have a long way to go.
Even if we could build these machines, Shor’s algorithm re-
lies heavily on the algebraic structures of numbers that we
don’t see in the known NP-complete problems. We know
that his algorithm cannot be applied to generic “black-box”
search problems so any algorithm would have to use some
special structure of NP-complete problems that we don’t
know about. We have used some algebraic structure of NP-
complete problems for interactive and zero-knowledge proofs
but quantum algorithms would seem to require much more.
Lov Grover [20] did find a quantum algorithm that works
on general NP problems but that algorithm only achieves
a quadratic speed-up and we have evidence that those tech-
niques will not go further.
Meanwhile quantum cryptography, using quantum mechan-
ics to achieve some cryptographic protocols without hard-
ness assumptions, has had some success both in theory and
in practice.
9. A NEW HOPE?
Ketan Mulmuley and Milind Sohoni have presented an ap-
proach to the Pvs. NP problem through algebraic geom-
etry, dubbed Geometric Complexity Theory, or GCT [30].
This approach seems to avoid the difficulties mentioned in
Section 4 but requires deep mathematics that could require
many years or decades to carry through.
In essence, they defines a family of high-dimension poly-
gons Pnbased on group representations on certain algebraic
varieties. Roughly speaking, for each n, if Pncontains an
integral point, then any circuit family for the Hamiltonian
path problem must have size at least nlog non inputs of size
n, which implies P6=NP. Thus, to show that P6=NP it
suffices to show that Pncontains an integral point for all n.
Although all that is necessary is to show that Pncontains
an integral point for all n, Mulmuley and Sohoni argue that
this direct approach would be difficult and instead suggests
first showing that the integer programming problem for the
family Pnis in fact in P. Under this approach, there are
three significant steps remaining:
1. Prove that the LP relaxation solves the integer pro-
gramming problem for Pnin polynomial time,
2. find an efficient, simple combinatorial algorithm for the
integer programming problem for Pn, and
3. prove that this simple algorithm always answers“yes.”
Since the polygons Pnare algebro-geometric in nature, solv-
ing (1) is thought to require algebraic geometry, representa-
tion theory, and the theory of quantum groups. Mulmuley
and Sohoni have given reasonable algebro-geometric condi-
tions that imply (1). These conditions have classical ana-
logues that are known to hold, based on the Riemann Hy-
pothesis over finite fields (a theorem proved by Andr´e Weil
in the 1960s). Mulmuley and Sohoni suggest that an anal-
ogous Riemann Hypothesis-like statement is required here
(though not the classical Riemann Hypothesis).
Although step (1) is difficult, Mulmuley and Sohoni have
provided definite conjectures based on reasonable mathe-
matical analogies that would solve (1). In contrast, the path
to completing steps (2) and (3) is less clear. Despite these
remaining hurdles, even solving the conjectures involved in
(1) could provide some insight to the Pversus NP problem.
Mulmuley and Sohoni have reduced a question about the
non-existence of polynomial-time algorithms for all NP-
complete problems to a question about the existence of a
polynomial-time algorithm (with certain properties) for a
specific problem. This should give us some hope, even in
the face of problems (1)-(3).
Nevertheless, Mulmuley himself believes it will take about
100 years to carry out this program, if it works at all.
10. CONCLUSION
This survey focused on the Pversus NP problem, its impor-
tance, our attempts to prove P6=NP and the approaches
we use to deal with the NP-complete problems that nature
and society throws at us. Much of the work mentioned re-
quired a long series of mathematically difficult research pa-
pers that we could not hope to adequately cover in this short
article. Also the field of computational complexity goes well
beyond just the Pversus NP problem that we haven’t dis-
cussed here. In Section 11 we give a number of references for
those interested in a deeper understanding of the Pversus
NP problem and computational complexity.
The Pversus 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
Pversus NP and I hope this survey has given you a small
feeling of the depth of research inspired by this mathematical
problem.
Proving P6=NP would not be the end of the story, it
would just show that NP-complete problem don’t have ef-
ficient 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 composite numbers.
Proving P6=NP might not be the start of the story ei-
ther. 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 understand the Pversus 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 Pversus 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 computation.
11. FURTHER READING
Recommendations for a more in-depth look at the Pversus
NP problem and the other topics discussed in this article:
•Steve Homer and myself have written a detailed his-
torical view of computational complexity [15].
•The 1979 book of Garey and Johnson still gives the
best overview of the Pversus NP problem with an
incredibly useful list of NP-complete problems [17].
•Scott Aaronson looks at the unlikely possibility that
the Pversus NP problem is formally independent [2].
•Russell Impagliazzo gives a wonderful description of
five possible worlds of complexity [23].
•Sanjeev Arora and Boaz Barak have a new computa-
tional complexity textbook with an emphasis on recent
research directions [6].
•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 SIGACT News
have many wonderful surveys on various topics in the-
ory including those mentioned in this article.
•Read the weblog Computational Complexity and you
will be among the first to know on any updates of the
status of the Pversus NP problem [14].
Acknowledgments
Thanks to Rahul Santhanam for many useful discussions and
comments. Josh Grochow wrote an early draft of Section 9.
The anonymous referees provided critical advice. Some of
the material in this article has appeared in my earlier surveys
and my blog [14].
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