NPcompleteness A Retrosp ective

Christos H Papadimitriou

University of California Berkeley USA

Abstract For a quarter of a century now NPcompleteness has b een

computer sciences favorite paradigm fad punching bag buzzword alibi

and intellectual exp ort This pap er is a fragmentary commentary on its

origins its nature its impact and on the attributes that have made it

so p ervasive and contagious

A keyword search in Melvyl the University of Californias online library

reveals that ab out pap ers each year have the term NPcomplete on

their title abstract or list of keywords This is more than each of the terms

compiler database exp ert neural network and op erating system

Even more surprising is the diversity of the disciplines with pap ers referring to

NPcompleteness They range from statistics and articial life to automatic

control and nuclear engineering What is the nature and extent of the impact of

NPcompleteness on theoretical computer science computer science in general

computing practice as well as other domains of the natural sciences applied sci

ence and mathematics And why did NPcompleteness b ecome such a p ervasive

and inuential concept

One of the reasons of the immense impact of NPcompleteness has to b e

the app eal and elegance of the class P that is of the thesis that p olynomial

worstcase time is a plausible and pro ductive mathematical surrogate of the

empirical concept of practically solvable computational problem But obvi

ously NPcompleteness also draws on the imp ortance of NP as it rests on the

widely conjectured contradistinction b etween these two classes In this regard

it is crucial that NP captures vast domains of computational scientic and

mathematical endeavor and seems to roughly delimit what mathematicians and

scientists had b een aspiring to compute feasibly True there are domains such

as strategic analysis and counting which have b een within our computational

ambitions and still seem to lie outside NP but they are the exceptions rather

than the rule NPcompleteness has thus b ecome a valuable intermediary b e

tween the abstraction of computational mo dels and the reality of computational

problems grounding complexity theory to computational practice

Also crucial for the success of NPcompleteness has b een its surprising ubiq

uity and eectiveness as a classication to ol and the scarcity of problems in

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christoscsberkeleyedu Partially supp orted by the National Science Foundation

A version of this talk was given at a meeting in the Fall of celebrating the th birthday of Richard M Karp to whom this pap er is also aectionately dedicated

NP that resist classication as either p olynomialtime solvable or NPcomplete

Ladners result on intermediate degrees b etween P and NPcompleteness

had b een known almost as so on as NPcompleteness was introduced and thus

theoretically the world could b e full of mysterious intermediate problems In sev

eral o ccasions extremely broad classes of computational problems in NP have

b een dichotomized with surprising accuracy into p olynomially solvable and NP

complete see for two early examples

The founders of NPcompleteness app ear to have anticipated its

broad applicability and classication p ower Leonid Levin wrote in

The method described here clearly provides a means for readily obtaining re

sults of this type for the majority of important sequential search problems In

Karps pap er twenty one problems were proved NPcomplete showing b e

yond any doubt the surprisingly broad applicability of the metho d Signicantly

Karp seems annoyed and surprised that three other problems linear program

ming primality and graph isomorphism resisted at the time such classication

Primality and graph isomorphism were also mentioned by Co ok Knuth was

suciently convinced ab out the imp ortance and broad applicability of the new

concept to take early and delib erate action on the terminological front

NPcompleteness has had tremendous impact even in areas where in some

sense it should not have It is now common knowledge among computer sci

entists that NPcompleteness is largely irrelevant to publickey cryptography

since in that area one needs sophisticated cryptographic assumptions that go

b eyond NPcompleteness and worstcase p olynomialtime computation fur

thermore cryptographic proto cols based on NPcomplete problems have b een

illfated Fortunately the founders of mo dern cryptography did not know this

Die and Hellman base their famous pronouncement We stand today on the

brink of a revolution in cryptography on two facts Very fast hardware

and software and novel techniques for proving problems hard they cite

Karps pap er

NPcompleteness has also exhibited a great amount of versatility adapting

to contexts and computational asp ects b eyond its original scop e of worstcase

analysis of exact algorithms for decision and optimization problems For exam

ple it was used early on to show that certain optimization problems cannot b e

approximated satisfactorily and indeed in a most ingenious and compre

hensive way more recently By showing that even less ambitious goals than

worstcase p olynomial exact solution are unattainable NPcompleteness is thus

a most useful to ol for rep eatedly pruning unpromising research directions and

thus redirecting research to new ones in a manner reminiscent of the struggle

b etween Hercules and the monster Hydra

Let me illustrate this versatility of NPcompleteness by a technical interlude

on an asp ect of ecient computation that has interested me recently namely

output polynomial time Certain computational problems require an output f x

on input x that is in the worst case exp onential in the input For such problems

one would like to have algorithms that are p olynomial in jxj and jf xj The class

of problems thus solvable can b e called output polynomial time One can use NP

completeness to prove that certain functions are not in outputp olynomial time

unless PNP For example consider the function MIN which maps a regular

expression to the minimumstate equivalent deterministic nitestate automaton

MIN can b e computed by rst designing a nondeterministic automaton M then

0

an equivalent deterministic automaton M and next minimizing the states of

0

M to obtain the nal output the problem is of course that the intermediate

0

result M could b e exp onential in both the input and the output It is rather

straightforward to use traditional NPcompleteness techniques to show the

following

Theorem Unless PNP MIN is not in output polynomial time

In fact we cannot even compute in outputp olynomial time a deterministic au

tomaton that has at most polynomially more states than the minimum unless

of course PNP

Often the required output f x is a set fy y g of strings that are related

k

to x via an NP mapping for example if G is a graph let AMISG b e the set of

all maximal indep endent sets of G AMIS is known to b e in outputp olynomial

time see for an exp osition and strengthening of this result and an early

discussion of output p olynomial time For such problems we have an elegant





is alternative denition of output p olynomial time A function f

in output p olynomial time if the following problem is solvable in p olynomial



time Given x and y either decide that y f x or nd a string in

y f x It is easy to see that if such an algorithm exists then its iteration

starting with S gives an output p olynomial time algorithm for f and vice

versa if an output p olynomial time algorithm exists for f it can b e used to

pro duce an element of y f x For example AMIS is in output p olynomial

time its generalization to hypergraphs is op en but was recently shown to b e

c log n

in output n time see for an extensive discussion of the hypergraph

generalization of AMIS One can use again traditional NPcompleteness to

show that the following generalization is not in output p olynomial time unless

PNP Given a monotone circuit compute the set of all minimal with resp ect

to the set of true inputs satisfying truth assignments

But sometimes traditional NPcompleteness techniques do not seem to suf

ce to bring out the intractability of a problem b ecause this problem b elongs to

a class or computational mo de that app ears to b e b etween P and NP In such

cases NPcompleteness has acted as an op enended research paradigm spawn

ing variants that are appropriate for the computational context b eing studied

examples are classes that capture lo cal search the parity argument loga

rithmic nondeterminism the related concept of xedparameter tractability

and approximability

Complexity classes introduced this way as abstractions of natural compu

tational problems of mysteriously intermediate complexity are in some precise

sense wellmotivated indeed necessary they are discovered not invented as they

have always existed by dint of their natural complete problems The only way to

make them go away is to collapse them with P or NP as o ccasionally happ ens

recall and its brilliant followup

NPcompleteness is of course a valuable to ol for demonstrating the diculty

of computational problems However NPcompleteness is often used allegori

cally a problem is shown NPcomplete that is not strictly sp eaking a natural

computational problem but an articial problem created to capture a mathe

matical concept NPcompleteness in this context suggests that a problem area

or approach is mathematically nasty Because if we b elieve that ecient algo

rithms are the natural outow of the mathematical structure of a problem a view

shared by all computer scientists with the p ossible exception of researchers in

metaphorbased algorithmic paradigms such as neural nets in which algorith

mic b ehavior is thought to b e emergent then contrapositively complexity

must b e the manifestation of mathematical p overty lack of structure See for

an early example of such a use of NPcompleteness in the theory of relational

databases

Beyond mathematics NPcompleteness and complexity in general can also

b e applied allegorically in other disciplines It can b e used as a metaphor

for chaos in dynamical systems for unbounded rationality in game theory for

unfairness in economics for integrity of electoral systems in p olitical science

for cognitive implausibility in articial intelligence for genetic indeterminism in

genetics and so on see for references

NPcompleteness is thus an imp ortant intellectual exp ort of computer

science to other disciplines And it do es ll a void in the interdisciplinary intel

lectual trade It seems to me that the concept of lower b ounds and negative

results in general is particular to computer science and has no welldeveloped

counterpart in other disciplines True one sees isolated results in other sciences

such as Heisenbergs uncertainty principle in quantum mechanics Arrows im

p ossibility theorem in economics and Carnots theorem in thermo dynamics

which are arguably negative however nowhere else in science do es one nd such

a comprehensive metho dology for obtaining negative results with the exception

of complexitys own precursor mathematical logic with its many incomplete

ness undecidability and inexpressibility results NPcompleteness is therefore

valuable for another reason It is one of the few precious features which give our

science its sp ecial character which set it apart from the other sciences see

for another development of this argument

In science successful ideas are those that are p ervasive and invasive are

invitingly elegant and metho dical are op en to extensions and variants and cap

ture an ob jective necessity answer a widespread but diuse sense of dissatisfac

tion in the scientic community in the case of NPcompleteness the widespread

feeling among computer scientists in the s that automata theory the previ

ous great paradigm had run its course as a useful abstraction of computation Thinking ab out the nature and history of NPcompleteness could give us useful

hints ab out computer sciences next great paradigm which for all I know has

started b eing articulated somewhere else in this volume

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