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 ? 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