Fortnow-The Status of the P vs NP Problem

7/28/2019 Fortnow-The Status of the P vs NP Problem

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Doi:10.1145/1562164.1562186

Its one of the fundamental mathematicalproblems of our time, and its importancegrows with the rise of powerful computers.

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wHEn Editor-in-CHiEF MosHE Vardi asked me t writethis piece r Communications, my frst reacti wasthe article culd be writte i tw wrds:

Still pe.Whe I started graduate schl i the mid-1980s,

may believed that the quickly develpig area circuit cmplexity wuld s settle the PversusNP prblem, whether every algrithmic prblem

with efcietly verifable slutis have efcietlycmputable slutis. But circuit cmplexity adther appraches t the prblem have stalled ad

we have little reas t believe we will see a prseparatigP rm NP i the ear uture.

nevertheless, the cmputer sciece ladscapehas dramatically chaged i the early ur decadessice Steve Ck preseted his semial NP-cmpleteess paper The Cmplexity Therem-Prvig Prcedures10 i Shaker Heights, oH i early

May, 1971. Cmputatial pwer has dramatically

increased, the cost o computing hasdramatically decreased, not to men-tion the power o the Internet. Com-putation has become a standard toolin just about every academic eld.

Whole subelds o biology, chemis-try, physics, economics and others aredevoted to large-scale computationalmodeling, simulations, and problemsolving.

As we solve larger and more com-plex problems with greater computa-

tional power and cleverer algorithms,the problems we cannot tackle beginto stand out. The theory o NP-com-pleteness helps us understand theselimitations and the P versus NP prob-lem begins to loom large not just asan interesting theoretical question incomputer science, but as a basic prin-ciple that permeates all the sciences.

So while we dont expect the P ver-sus NP problem to be resolved in thenear uture, the question has drivenresearch in a number o topics to helpus understand, handle, and even take

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advantage o the hardness o variouscomputational problems.

In this article I look at how peoplehave tried to solve the P versus NPproblem as well as how this questionhas shaped so much o the research incomputer science and beyond. I willlook at how to handle NP-completeproblems and the theory that hasdeveloped rom those approaches.I show how a new type o interac-tive proo systems led to limitations

o approximation algorithms andconsider whether quantum comput-ing can solve NP-complete problems(short answer: not likely). And I closeby describing a new long-term projectthat will try to separate P rom NP us-ing algebraic-geometric techniques.

This article does not try to be totallyaccurate or complete either technical-ly or historically, but rather inormallydescribes the P versus NP problemand the major directions in computerscience inspired by this question overthe past several decades.

Wt s t P vrss NP Prbl?

Suppose we have a large group o stu-dents that we need to pair up to workon projects. We know which studentsare compatible with each other and we

want to put them in compatible groupso two. We could search all possible pair-ings but even or 40 students we wouldhave more than 300 billion trillion pos-sible pairings.

In 1965, Jack Edmonds12 gave an e-cient algorithm to solve this match-

ing problem and suggested a ormaldenition o ecient computation(runs in time a xed polynomial o theinput size). The class o problems withecient solutions would later becomeknown as P or Polynomial Time.

But many related problems do notseem to have such an ecient algo-rithm. What i we wanted to makegroups o three students with each pairo students in each group compatible(Partition into Triangles)? What i we

wanted to nd a large group o studentsall o whom are compatible with each

other (Clique)? What i we wanted tosit students around a large round table

with no incompatible students sittingnext to each other (Hamiltonian Cycle)?

What i we put the students into threegroups so that each student is in thesame group with only his or her com-patibles (3-Coloring)?

All these problems have a similaravor: Given a potential solution, orexample, a seating chart or the roundtable, we can validate that solution e-

ciently. The collection o problemsthat have eciently veriable solutionsis known as NP (or NondeterministicPolynomial-Time, i you have to ask).

So P = NP means that or every prob-lem that has an eciently veriablesolution, we can nd that solution e-ciently as well.

We call the very hardestNPproblems(which include Partition into Triangles,Clique, Hamiltonian Cycle and 3-Col-oring) NP-complete, that is, given anecient algorithm or one o them, wecan nd an ecient algorithm or allIL

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P = NP then public-key cryptographybecomes impossible. True, but what

we will gain rom P = NPwill make thewhole Internet look like a ootnote inhistory.

Since all the NP-complete optimiza-tion problems become easy, everything

will be much more ecient. Transpor-tation o all orms will be scheduledoptimally to move people and goodsaround quicker and cheaper. Manuac-turers can improve their production toincrease speed and create less waste.

And Im just scratching the surace.Learning becomes easy by using the

principle o Occams razorwe simplynd the smallest program consistent

with the data. Near perect vision rec-ognition, language comprehension andtranslation and all other learning tasksbecome trivial. We will also have muchbetter predictions o weather and earth-quakes and other natural phenom-enon.

P = NPwould also have big implica-tions in mathematics. One could ndshort, ully logical proos or theorems

o them and in act any problem in NP.Steve Cook, Leonid Levin, and RichardKarp10, 24, 27 developed the initial theoryo NP-completeness that generatedmultiple ACM Turing Awards.

In the 1970s, theoretical comput-er scientists showed hundreds moreproblems NP-complete (see Garey and

Johnson16). An ecient solution to anyNP-complete problem would implyP =NPand an ecient solution to everyNP-complete problem.

Most computer scientists quicklycame to believe P NP and trying toprove it quickly became the single mostimportant question in all o theoreticalcomputer science and one o the mostimportant in all o mathematics. SoonthePversusNPproblem became an im-portant computational issue in nearly

every scientic discipline.As computers grew cheaper andmore powerul, computation startedplaying a major role in nearly every aca-demic eld, especially the sciences. Themore scientists can do with computers,the more they realize some problemsseem computationally dicult. Many othese undamental problems turn outto be NP-complete. A small sample:

Finding a DNA sequence that best

ts a collection o ragments o the se-quence (see Guseld20).

Finding a ground state in the Isingmodel o phase transitions (see Cipra8).

Finding Nash Equilbriums with

specic properties in a number o envi-ronments (see Conitzer9).

Finding optimal protein threading

procedures.26

Determining i a mathematical

statement has a short proo (ollowsrom Cook10).

In 2000, the Clay Math Institutenamed the Pversus NP problem as oneo the seven most important open ques-

tions in mathematics and has oered amillion-dollar prize or a proo that de-termines whether or not P = NP.

Wt i P = NP?

To understand the importance o theP versus NP problem let us imaginea world where P = NP. Technically wecould have P = NP, but not have practi-cal algorithms or most NP-completeproblems. But suppose in actwe dohave very quick algorithms or all theseproblems.

Many ocus on the negative, that i

Wt w wldgn r P = NP

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but these proos are usually extremelylong. But we can use the Occam razorprinciple to recognize and veriy math-ematical proos as typically written in

journals. We can then nd proos otheorems that have reasonable lengthproos say in under 100 pages. A person

who proves P = NP would walk homerom the Clay Institute not with $1 mil-lion check but with seven (actually sixsince the Poincar Conjecture appearssolved).

Dont get your hopes up. Complexity

theorists generally believe P NP andsuch a beautiul world cannot exist.

apprs t Swng P NP

Here, I present a number o ways wehave tried and ailed to prove P NP.The survey o Fortnow and Homer14gives a uller historical overview o thesetechniques.

Diagonalization. Can we just con-struct an NP language L specicallydesigned so that every single polyno-mial-time algorithm ails to compute Lproperly on some input? This approach,

agonalization techniques to show someNP-complete problems like Booleanormula satisability cannot have algo-rithms that use both a small amount otime and memory,39 but this is a long

way romP NP.Circuit Complexity. To show P NP

it is sucient to show some -completeproblem cannot be solved by relativelysmall circuits o AND, OR, and NOTgates (the number o gates bounded bya xed polynomial in the input size).

In 1984, Furst, Saxe, and Sipser15

showed that small circuits cannot solvethe parity unction i the circuits have axed number o layers o gates. In 1985,Razborov31 showed the NP-completeproblem o nding a large clique doesnot have small circuits i one only allows

AND and OR gates (no NOT gates). I oneextends Razborovs result to general cir-cuits one will have proved P NP.

Razborov later showed his techniqueswould ail miserably i one allows NOTgates.32 Razborov and Rudich33 developa notion o natural proos and giveevidence that our limited techniques

known as diagonalization, goes back tothe 19th century.

In 1874, Georg Cantor7 showed thereal numbers are uncountable using atechnique known as diagonalization.Given a countable list o reals, Cantorshowed how to create a new real num-ber not on that list.

Alan Turing, in his seminal paper oncomputation,38 used a similar techniqueto show that the Halting problem is notcomputable. In the 1960s complexitytheorists used diagonalization to show

that given more time or memory onecan solve more problems. Why not usediagonalization to separate NP romP?

Diagonalization requires simula-tion and we dont know how a xed NPmachine can simulate an arbitrary Pmachine. Also a diagonalization proo

would likely relativize, that is, work eveni all machines involved have access tothe same additional inormation. Bak-er, Gill and Solovay6 showed no relativ-izable proo can settle the P versus NPproblem in either direction.

Complexity theorists have used di-ILLU

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in circuit complexity cannot be pushedmuch urther. And, in act, we haventseen any signicantly new circuit lowerbounds in the past 20 years.

Proof Complexity. Consider the seto Tautologies, the Boolean ormulas o variables over ANDs, ORs, and NOTssuch that every setting o the variables

to True and False makes true, or ex-ample the ormula

(x ANDy) OR (NOT x) OR (NOTy).

A literal is a variable or its negation,such as x or NOT x. A ormula, like theone here, is in Disjunctive Normal Form(DNF) i it is the OR o ANDs o one ormore literals.

I a ormula is not a tautology, wecan give an easy proo o that act by ex-

hibiting an assignment o the variablesthat makes alse. But i were indeed atautology, we dont expect short proos.I one could prove there are no shortproos o tautology that would implyP NP.

Resolution is a standard approach toproving tautologies o DNFs by ndingtwo clauses o the orm (y1 AND x) and(y2 AND NOT x) and adding the clause(y1 AND y2). A ormula is a tautology ex-actly when one can produce an emptyclause in this manner.

In 1985, Wolgang Haken21

showedthat tautologies that encode the pigeon-hole principle (n + 1 pigeons in n holesmeans some hole has more than onepigeon) do not have short resolutionproos.

Since then complexity theorists haveshown similar weaknesses in a numbero other proo systems including cuttingplanes, algebraic proo systems basedon polynomials, and restricted versionso proos using the Frege axioms, thebasic axioms one learns in an introduc-

tory logic course.But to prove P NP we would need

to show that tautologies cannot haveshort proos in an arbitrary proo sys-tem. Even a breakthrough result show-ing tautologies dont have short generalFrege proos would not suce in sepa-ratingNP rom P.

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So you have an NP-complete problemyou just have to solve. I, as we believe,P NPyou wont nd a general algorithmthat will correctly and accurately solve

gives a ne-grained analysis o the com-plexity oNP-complete problems basedon their parameter size.

Approximation. We cannot hope tosolve NP-complete optimization prob-lems exactly but oten we can get a goodapproximate answer. Consider the trav-eling salesperson problem again with

distances between cities given as thecrow fies (Euclidean distance). Thisproblem remains NP-complete but Aro-ra4 gives an ecient algorithm that gets

very close to the best possible route.Consider the MAX-CUT problem

o dividing people into two groups tomaximize the number o incompatiblesbetween the groups. Goemans and Wil-liamson17 use semi-denite program-ming to give a division o people only a.878567 actor o the best possible.

Heuristics and Average-Case Com-plexity. The study o NP-completenessocuses on how algorithms perorm onthe worst possible inputs. However thespecic problems that arise in practicemay be much easier to solve. Many com-puter scientists employ various heuris-tics to solve NP-complete problems thatarise rom the specic problems in theirelds.

While we create heuristics or manyo the NP-complete problems, Booleanormula Satisability (SAT) receives

more attention than any other. Booleanormulas, especially those in conjunc-tive normal orm (CNF), the AND o ORso variables and their negations, have a

very simple description and yet are gen-eral enough to apply to a large numbero practical scenarios particularly insotware verication and articial in-telligence. Most natural NP-completeproblems have simple ecient reduc-tions to the satisability o Boolean or-mulas. In competition these SAT solverscan oten settle satisability o ormulas

o one million variables.a

Computational complexity theo-rists study heuristics by consideringaverage-case complexityhow well canalgorithms perorm on average rom in-stances generated by some specic dis-tribution.

Leonid Levin28 developed a theory oecient algorithms over a specic dis-tribution and ormulated a distribution-al version o the PversusNPproblem.

Some problems, like versions o the

a http://www.satcompetition.org.

your problem all the time. But some-times you need to solve the problemanyway. All hope is not lost. Here, I de-scribe some o the tools one can use onNP-complete problems and how com-putational complexity theory studiesthese approaches. Typically one needsto combine several o these approaches

when tackling NP-complete problemsin the real world.

Brute Force. Computers have gottenaster, much aster since NP-complete-ness was rst developed. Brute orcesearch through all possibilities is nowpossible or some small problem in-stances. With some clever algorithms

we can even solve some moderate sizeproblems with ease.

The NP-complete traveling sales-person problem asks or the smallest

distance tour through a set o speciedcities. Using extensions o the cutting-plane method we can now solve, inpractice, traveling salespeople prob-lems with more than 10,000 cities (see

Applegate3).Consider the 3SAT problem, solving

Boolean ormula satisability whereormulas are in the orm o the AND oseveral clauses where each clause is theOR o three literal variables or nega-tions o variables). 3SAT remains NP-complete but the best algorithms can

in practice solve SAT problems on about100 variables. We have similar resultsor other variations o satisability andmany other NP-complete problems.

But or satisability on general or-mulae and on many other NP-completeproblems we do not know algorithmsbetter than essentially searching all thepossibilities. In addition, all these algo-rithms have exponential growth in theirrunning times, so even a small increasein the problem size can kill what was anecient algorithm. Brute orce alone

will not solve NP-complete problems nomatter how clever we are.

Parameterized Complexity. Considerthe Vertex Cover problem, nd a set ok central people such that or everycompatible pair o people, at least oneo them is central. For small k we candetermine whether a central set o peo-ple exists eciently no matter the totalnumber n o people we are considering.For the Clique problem even or small kthe problem can still be dicult.

Downey and Fellows11 developed atheory o parameterized complexity that


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shortest vector problem in a lattice orcomputing the permanent o a ma-trix, are hard on average exactly whenthey are hard on worst-case inputs, butneither o these problems is believedto be NP-complete. Whether similar

worst-to-average reductions hold orNP-complete sets is an important open

problem.Average-case complexity plays an im-

portant role in many areas o computerscience, particularly cryptography, asdiscussed later.

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Previously, we saw how sometimes onecan get good approximate solutions toNP-complete optimization problems.Many times though we seem to hit a

limit on our ability to even get good ap-proximations. We now know that wecannot achieve better approximationson many o these problems unless P =NP and we could solve these problemsexactly. The techniques to show thesenegative results came out o a new mod-el o proo system originally developedor cryptography and to classiy grouptheoretic algorithmic problems.

As mentioned earlier, we dont ex-pect to have short traditional proos otautologies. But consider an interac-

tive proo model where a prover Peggytries to convince a verier Victor that aormula is a tautology. Victor can askPeggy randomly generated questionsand need only be convinced with highcondence. Quite surprisingly, theseproo systems have been shown to ex-ist not only or tautologies but or anyproblem computable in a reasonableamount o memory.

A variation known as a probabilisti-cally checkable proo system (PCPs),

where Peggy writes down an encoded

proo and Victor can make random-ized queries to the bits o the proo, hasapplications or approximations. ThePCP Theorem optimizes parameters,

which in its strong orm shows that ev-ery language in NPhas a PCP where Vic-tor uses a tiny number o random coinsand queries only three bits o the proo.

One can use this PCP theorem toshow the limitations o approximationor a large number o optimization ques-tions. For example, one cannot approxi-mate the largest clique in a group o npeople by more than a multiplicative ra-

tio o nearlyn unless P = NP. See Mad-hu Sudans recent article in Communi-cations or more details and reerenceson PCPs.36

One can do even better assuminga Unique Games Conjecture thatthere exists PCPs or NP problems withsome stronger properties. Consider the

MAX-CUT problem o dividing peoplediscussed earlier. I the unique gamesconjecture holds one cannot do bet-ter than the .878567 actor given by theGoemans-Williamson approximationalgorithm.26 Recent work shows how toget a provably best approximation oressentially any constrained problem as-suming this conjecture.30

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In What I P = NP? we saw the nice

world that arises when we assumeP

=NP. But we expect P NP to hold in verystrong ways. We can use strong hard-ness assumptions as a positive tool,particularly to create cryptographic pro-tocols and to reduce or even eliminatethe need o random bits in probabilisticalgorithms.

Cryptography.We take it or grantedthese days, the little key or lock on our

Web page that tells us that someonelistening to the network wont get thecredit card number I just sent to an on-

line store or the password to the bankthat controls my money. But public-keycryptography, the ability to send securemessages between two parties that havenever privately exchanged keys, is a rela-tively new development based on hard-ness assumptions o computationalproblems.

IP = NP then public-key cryptogra-phy is impossible. AssumingP NP isnot enough to get public-key protocols,instead we need strong average-case as-sumptions about the diculty o actor-

ing or related problems.We can do much more than just pub-

lic-key cryptography using hard prob-lems. Suppose Alices husband Bob is

working on a Sudoku puzzle and Aliceclaims she has a solution to the puzzle(solving a n n Sudoku puzzle is NP-complete). Can Alice convince Bob thatshe knows a solution without revealingany piece o it?

Alice can use a zero-knowledgeproo, an interactive proo with the ad-ditional eature that the verier learnsnothing other than some property

W xpt P NPt ld n vry

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search problems so any algorithm wouldhave to use some special structure oNP-complete problems that we dont knowabout. We have used some algebraicstructure oNP-complete problems orinteractive and zero-knowledge proosbut quantum algorithms would seem torequire much more.

Lov Grover19 did nd a quantum al-gorithm that works on general NPprob-lems but that algorithm only achievesa quadratic speed-up and we have evi-dence that those techniques will not go

urther.Meanwhile quantum cryptography,

using quantum mechanics to achievesome cryptographic protocols withouthardness assumptions, has had somesuccess both in theory and in practice.

a Nw hp?

Ketan Mulmuley and Milind Sohonihave presented an approach to the P

versus NP problem through algebraicgeometry, dubbed Geometric Complex-ity Theory, or GCT.29

This approach seems to avoid the di-

holds, like a Sudoku puzzle having a so-lution. EveryNP search problem has azero-knowledge proo under the appro-priate hardness assumptions.

Online poker is generally playedthrough some trusted Web site, usu-ally somewhere in the Caribbean. Can

we play poker over the Internet without

a trusted server? Using the right cryp-tographic assumptions, not only pokerbut any protocol that uses a trusted par-ty can be replaced by one that uses notrusted party and the players cant cheator learn anything new beyond what theycould do with the trusted party.b

Eliminating Randomness. In the 1970swe saw a new type o algorithm, one thatused random bits to aid in nding a so-lution to a problem. Most notably wehad probabilistic algorithms35 or deter-

mining whether a number is prime, animportant routine needed or moderncryptography. In 2004, we discovered

we dont need randomness at all to e-ciently determine i a number is prime.2Does randomness help us at all in nd-ing solutions to NPproblems?

Truly independent and uniormrandom bits are either very dicult orimpossible to produce (depending on

your belies about quantum mechan-ics). Computer algorithms instead usepseudorandom generators to gener-

ate a sequence o bits rom some givenseed. The generators typically ound onour computers usually work well but oc-casionally give incorrect results both intheory and in practice.

We can create theoretically betterpseudorandom generators in two di-erent ways, one based on the stronghardness assumptions o cryptographyand the other based on worst-case com-plexity assumptions. I will ocus on thissecond approach.

We need to assume a bit more than

P NP, roughly that NP-complete prob-lems cannot be solved by smaller thanexpected AND-OR-NOT circuits. A longseries o papers showed that, under thisassumption, any problem with an e-cient probabilistic algorithm also hasan ecient algorithm that uses a pseu-dorandom generator with a very shortseed, a surprising connection betweenhard languages and pseudo-random-ness (see Impagliazzo23). The seed is soshort we can try all possible seeds e-

b See the survey o Goldreich18 or details.

ciently and avoid the need or random-ness altogether.

Thus complexity theorists generallybelieve having randomness does nothelp in solvingNP search problems andthat NP-complete problems do not haveecient solutions, either with or with-out using truly random bits.

While randomness doesnt seemnecessary or solving search problems,the unpredictability o random bitsplays a critical role in cryptography andinteractive proo systems and likely can-not be avoided in these scenarios.

cld Qnt cptrs

Slv NP-cplt Prbls?

While we have randomized and non-randomized ecient algorithms or de-termining whether a number is prime,

these algorithms usually dont give usthe actors o a composite number.Much o modern cryptography relies onthe act that actoring or similar prob-lems do not have ecient algorithms.

In the mid-1990s, Peter Shor34showed how to actor numbers usinga hypothetical quantum computer. Healso developed a similar quantum al-gorithm to solve the discrete logarithmproblem. The hardness o discrete log-arithm on classical computers is alsoused as a basis or many cryptographic

protocols. Nevertheless, we dont ex-pect that actoring or nding discretelogarithms are NP-complete. While wedont think we have ecient algorithmsto solve actoring or discrete logarithm,

we also dont believe we can reduceNP-complete problems like Clique to theactoring or discrete logarithm prob-lems.

So could quantum computers oneday solve NP-complete problems? Un-likely.

Im not a physicist so I wont address

the problem as to whether these ma-chines can actually be built at a largeenough scale to solve actoring prob-lems larger than we can with currenttechnology (about 200 digits). Ater bil-lions o dollars o unding o quantumcomputing research we still have a long

way to go.Even i we could build these ma-

chines, Shors algorithm relies heavily onthe algebraic structures o numbers that

we dont see in the known NP-completeproblems. We know that his algorithmcannot be applied to generic black-box

NP n b sn s grp wr vrylnt s nntd t vry trlnt. ovr ts pgs dnstrtn t grp s swn.


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culties mentioned earlier, but requiresdeep mathematics that could requiremany years or decades to carry through.

In essence, they dene a amily ohigh-dimension polygons Pn based ongroup representations on certain alge-braic varieties. Roughly speaking, oreach n, iPn contains an integral point,then any circuit amily or the Hamil-tonian path problem must have size atleast nlogn on inputs o size n, which im-plies P NP. Thus, to show that P NPit suces to show that Pn contains an

integral point or all n.Although all that is necessary is to

show that Pn contains an integral pointor all n, Mulmuley and Sohoni arguethat this direct approach would be di-cult and instead suggest rst showingthat the integer programming problemor the amilyPn is, in act, in P. Underthis approach, there are three signi-cant steps remaining:

1. Prove that the LP relaxation solvesthe integer programming problem or

Pn in polynomial time;2. Find an ecient, simple combi-

provide some insight to the PversusNPproblem.

Mulmuley and Sohoni have reduceda question about the nonexistence opolynomial-time algorithms or all NP-complete problems to a question aboutthe existence o a polynomial-time al-gorithm (with certain properties) or aspecic problem. This should give ussome hope, even in the ace o problems(1)(3).

Nevertheless, Mulmuley believes itwill take about 100 years to carry out this

program, i it works at all.

cnlsn

This survey ocused on the Pversus NPproblem, its importance, our attemptsto prove P NP and the approaches weuse to deal with the NP-complete prob-lems that nature and society throwsat us. Much o the work mentionedrequired a long series o mathemati-cally dicult research papers that Icould not hope to adequately coverin this short article. Also the eld ocomputational complexity goes well

natorial algorithm or the integer pro-gramming problem orPn, and;

3. Prove that this simple algorithmalways answers yes.

Since the polygonsPn are algebro-geo-metric in nature, solving (1) is thought torequire algebraic geometry, representa-tion theory, and the theory o quantumgroups. Mulmuley and Sohoni have giv-en reasonable algebro-geometric condi-tions that imply (1). These conditionshave classical analogues that are knownto hold, based on the Riemann Hypoth-

esis over nite elds (a theorem provedby Andr Weil in the 1960s). Mulmuleyand Sohoni suggest that an analogousRiemann Hypothesis-like statement isrequired here (though not the classicalRiemann Hypothesis).

Although step (1) is dicult, Mulmu-ley and Sohoni have provided deniteconjectures based on reasonable math-ematical analogies that would solve(1). In contrast, the path to completingsteps (2) and (3) is less clear. Despitethese remaining hurdles, even solvingthe conjectures involved in (1) couldIL

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(jue 2003).15. Furt, M., saxe, j., ad siper, M. Parit, circuit ad

the plmial-time hierarch. Mathematical SystemsTheory 17(1984), 1327.

16. Gare, M. ad jh, D.Computers and Intractability.A Guide to the Theory of NP-Completeness. W. H.Freema ad Cmpa, ny, 1979.

17. Gema, M. ad William, D. Imprvedapprximati algrithm r maximum cut adatifabilit prblem uig emidefite prgrammig.Journal of the ACM 42, 6 (1995), 11151145.

18. Gldreich, o. Fudati crptgraph|a primer.

Foundations and Trends in Theoretical ComputerScience 1, 1 (2005) 1116.

19. Grver, L. A at quatum mechaical algrithm rdatabae earch. I Proceedings of the 28th ACMSymposium on the Theory of Computing. ACM, ny,1996, 212219.

20. Gufeld, D. Algorithms on Strings, Trees andSequences: Computer Science and ComputationalBiology. Cambridge Uiverit Pre, 1997.

21. Hake, A. The itractabilit reluti. TheoreticalComputer Science, 39 (1985) 297305.

22. Impagliazz, R. A peral view average-caecmplexit ther. I Proceedings of the 10th AnnualConference on Structure in Complexity Theory. IEEECmputer sciet Pre, 1995, 134147.

23. Impagliazz, R. ad Wigder, A. P = BPP i E requireexpetial circuit: Deradmizig the XoR lemma. IProceedings of the 29th ACM Symposium on the Theoryof Computing. ACM, ny, 1997, 220229.

24. Karp, R. Reducibilit amg cmbiatrial prblem.Complexity of Computer Computations. R. Miller ad j.Thatcher, Ed. Pleum Pre, 1972, 85103.

25. Kht, s., Kidler, G., Mel, E., ad oDell, R.optimal iapprximabilit reult r MAX-CUT adther 2-variable CsP? SIAM Journal on Computing37, 1 (2007), 319357.

26. Lathrp, R. The prtei threadig prblem withequece ami acid iteracti preerece i nP-cmplete. Protein Engineering 7, 9 (1994), 10591068.

27. Levi, L. Uiveralie perebrie zadachi (Uiveralearch prblem: i Ruia). Problemy PeredachiInformatsii 9, 3 (1973), 265266. Crrected Eglihtralati.37

28. Levi, L. Average cae cmplete prblem. SIAMJournal on Computing 15, (1986), 285286.

29. Mulmule, K. ad shi, M. Gemetric cmplexitther I: A apprach t the P v. nP ad relatedprblem. SIAM Journal on Computing 31, 2, (2001)496526.

30. Raghavedra, P. optimal algrithm adiapprximabilit reult r ever cp? I Proceedingsof the 40th ACM Symposium on the Theory ofComputing. ACM, ny, 2008, 245254.

31. Razbrv, A. Lwer bud the mtecmplexit me Blea ucti. SovietMathematics-Doklady 31, (1985) 485493.

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33. Razbrv, A., ad Rudich, s. natural pr. Journalof Computer and System Sciences 55, 1 (Aug. 1997),2435.

34. shr. P. Plmial-time algrithm r primeactrizati ad dicrete lgarithm a quatumcmputer. SIAM Journal on Computing 26, 5 (1997)14841509.

35. slva, R. ad strae, V. A at Mte-Carl tet rprimalit. SIAM Journal on Computing 6(1977), 8485.

see al erratum 7:118, 1978.36. suda, M. Prbabiliticall checkable pr. Commun.ACM 52, 3 (Mar. 2009) 7684.

37. Trakhtebrt, R. A urve Ruia apprache tPerebr (brute-rce earch) algrithm. Annals of theHistory of Computing 6, 4 (1984), 384400.

38. Turig, A. o cmputable umber, with a applicatit the Etcheidug prblem. Proceedings of theLondon Mathematical Society 42 (1936), 230265.

39. va Melkebeek, D. A urve lwer bud ratifabilit ad related prblem. Foundations andTrends in Theoretical Computer Science 2, (2007),197303.

Lance Fortnow ([email protected]) i aprer electrical egieerig ad cmputer cieceat nrthweter Uiverit McCrmick schl Egieerig, Evat, IL.

2009 ACM 0001-0782/09/0900 $10.00

beyond just the P versus NP problemthat I havent discussed here. In Fur-ther Reading, a number o reerencesare presented or those interested in adeeper understanding o the P versusNP problem and computational com-plexity.

The P versus NP problem has gone

rom an interesting problem related tologic to perhaps the most undamentaland important mathematical questiono our time, whose importance onlygrows as computers become more pow-erul and widespread. The question haseven hit popular culture appearing intelevision shows such as The Simpsonsand Numb3rs. Yet many only know othe basic principles oPversus NP andI hope this survey has given you a smalleeling o the depth o research inspired

by this mathematical problem.ProvingP NPwould not be the endo the story, it would just show that NP-complete problem, dont have ecientalgorithms or all inputs but many ques-tions might remain. Cryptography, orexample, would require that a problemlike actoring (not believed to be NP-complete) is hard or randomly drawncomposite numbers.

ProvingP NPmight not be the starto the story either. Weaker separationsremain perplexingly dicult, or exam-

ple showing that Boolean-ormula Satis-ability cannot be solved in near-lineartime or showing that some problemusing a certain amount o memory can-not be solved using roughly the sameamount o time.

None o us truly understands the PversusNP problem, we have only begunto peel the layers around this increas-ingly complex question. Perhaps we willsee a resolution o the PversusNPprob-lem in the near uture but I almost hopenot. ThePversusNPproblem continues

to inspire and boggle the mind and con-tinued exploration o this problem willlead us to yet even new complexities inthat truly mysterious process we callcomputation.

frtr Rdng

Recommendations or a more in-depthlook at the PversusNPproblem and theother topics discussed in this article:

Steve Homer and I have written a

detailed historical view o computation-al complexity.14

The 1979 book o Garey and John-

son still gives the best overview o the Pversus NP problem with an incrediblyuseul list oNP-complete problems.16

Scott Aaronson looks at the unlikely

possibility that the PversusNPproblemis ormally independent.1

Russell Impagliazzo gives a wonder-

ul description o ve possible worlds o

complexity.22

Sanjeev Arora and Boaz Barak have

a new computational complexity text-book with an emphasis on recent re-search directions.5

The Foundations and Trends in Theo-retical Computer Science journal and theComputational Complexity columns othe Bulletin of the European Associationof Theoretical Computer Science and SI-GACT News have many wonderul sur-

veys on various topics in theory includ-

ing those mentioned in this article.Read the blog Computational Com-

plexity and you will be among the rst toknow about any updates o the status othePversusNPproblem.13

aknwldgnts

Thanks to Rahul Santhanam or manyuseul discussions and comments. JoshGrochow wrote an early drat. The anon-

ymous reerees provided critical advice.Some o the material in this article hasappeared in my earlier surveys and my

blog.13

References

1. Aar, s. I P veru nP rmall idepedet?Bulletin of the European Association for TheoreticalComputer Science 81 (oct. 2003).

2. Agrawal, M., Kaal, n., ad saxea, n. PRIMEs. IAnnals of Mathematics 160, 2 (2004) 781793.

3. Applegate, D., Bixb, R., Chvtal, V., ad Ck, W. o theluti travelig alema prblem. DocumentaMathematica, Extra Vlume ICM III (1998), 645656.

4. Arra, s. Plmial time apprximati cheme rEuclidea travelig alema ad ther gemetricprblem. J. ACM 45, 5 (sept. 1998), 753782.

5. Arra, s. ad Barak, B. Complexity Theory: A ModernApproach. Cambridge Uiverit Pre, Cambridge,2009.

6. Baker, T., Gill, j., ad slva, R. Relativizati the P= nP queti. SIAM Journal on Computing 4, 4 (1975),

431442.7. Catr, G. Ueber eie Eigechat de Ibegrie

aller reelle algebraiche Zahle. Crelles Journal 77(1874), 258262.

8. Cipra, B. Thi Iig mdel i nP-cmplete. SIAM News33, 6 (jul/Aug. 2000).

9. Citzer, V. ad sadhlm, T. new cmplexit reultabut nah equilibria. Games and Economic Behavior63, 2 (jul 2008), 621641.

10. Ck, s. The cmplexit therem-prvigprcedure. I Proceedings of the 3rd ACM Symposiumon the Theory of Computing, ACM, ny, 1971, 151158.

11. Dwe, R. ad Fellw, M. Parameterized Complexity.spriger, 1999.

12. Edmd, j. Path, tree ad wer. Canadian Journalof Mathematics 17, (1965), 449467.

13. Frtw, L. ad Gaarch, W. Cmputatial cmplexit;http://weblg.rtw.cm.

14. Frtw, L. ad Hmer, s. A hrt hitr

cmputatial cmplexit. Bulletin of the EuropeanAssociation for Theoretical Computer Science 80,

Documents

Fortnow-The Status of the P vs NP Problem