NP-Completeness: A Retrospective

For a quarter of a century now NP completeness has been computer science s favorite paradigm fad punching bag buzzword alibi and intellectual export This paper is a fragmentary commentary on its origins its nature its impact and on the attributes that have made it so pervasive and contagious A keyword search in Melvyl the University of California s on line library reveals that about papers each year have the term NP complete on their title abstract or list of keywords This is more than each of the terms compiler database expert neural network and operating system Even more surprising is the diversity of the disciplines with papers referring to NP completeness They range from statistics and arti cial life to automatic control and nuclear engineering What is the nature and extent of the impact of NP completeness 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 NP completeness become such a pervasive and in uential concept One of the reasons of the immense impact of NP completeness has to be the appeal and elegance of the class P that is of the thesis that polynomial worst case time is a plausible and productive mathematical surrogate of the empirical concept of practically solvable computational problem But obvi ously NP completeness also draws on the importance of NP as it rests on the widely conjectured contradistinction between these two classes In this regard it is crucial that NP captures vast domains of computational scienti c and mathematical endeavor and seems to roughly delimit what mathematicians and scientists had been aspiring to compute feasibly True there are domains such as strategic analysis and counting which have been within our computational ambitions and still seem to lie outside NP but they are the exceptions rather than the rule NP completeness has thus become a valuable intermediary be tween the abstraction of computational models and the reality of computational problems grounding complexity theory to computational practice Also crucial for the success of NP completeness has been its surprising ubiq uity and e ectiveness as a classi cation tool and the scarcity of problems in christos cs berkeley edu Partially supported 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 paper is also a ectionately dedicated NP that resist classi cation as either polynomial time solvable or NP complete Ladner s result on intermediate degrees between P and NP completeness had been known almost as soon as NP completeness was introduced and thus theoretically the world could be full of mysterious intermediate problems In sev eral occasions extremely broad classes of computational problems in NP have been dichotomized with surprising accuracy into polynomially solvable and NP complete see for two early examples The founders of NP completeness appear to have anticipated its broad applicability and classi cation power 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 Karp s paper twenty one problems were proved NP complete showing be yond any doubt the surprisingly broad applicability of the method Signi cantly Karp seems annoyed and surprised that three other problems linear program ming primality and graph isomorphism resisted at the time such classi cation Primality and graph isomorphism were also mentioned by Cook Knuth was su ciently convinced about the importance and broad applicability of the new concept to take early and deliberate action on the terminological front NP completeness 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 NP completeness is largely irrelevant to public key cryptography since in that area one needs sophisticated cryptographic assumptions that go beyond NP completeness and worst case polynomial time computation fur thermore cryptographic protocols based on NP complete problems have been ill fated Fortunately the founders of modern cryptography did not know this Di e 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 Karp s paper NP completeness has also exhibited a great amount of versatility adapting to contexts and computational aspects beyond its original scope of worst case analysis of exact algorithms for decision and optimization problems For exam ple it was used early on to show that certain optimization problems cannot be approximated satisfactorily and indeed in a most ingenious and compre hensive way more recently By showing that even less ambitious goals than worst case polynomial exact solution are unattainable NP completeness is thus a most useful tool for repeatedly pruning unpromising research directions and thus redirecting research to new ones in a manner reminiscent of the struggle between Hercules and the monster Hydra Let me illustrate this versatility of NP completeness by a technical interlude on an aspect of e cient 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 exponential in the input For such problems one would like to have algorithms that are polynomial in jxj and jf x j The class of problems thus solvable can be called output polynomial time One can use NP completeness to prove that certain functions are not in output polynomial time unless P NP For example consider the function MIN which maps a regular expression to the minimum state equivalent deterministic nite state automaton MIN can be computed by rst designing a nondeterministic automatonM then an equivalent deterministic automaton M and next minimizing the states of M to obtain the nal output the problem is of course that the intermediate result M could be exponential in both the input and the output It is rather straightforward to use traditional NP completeness techniques to show the following Theorem Unless P NP MIN is not in output polynomial time In fact we cannot even compute in output polynomial time a deterministic au tomaton that has at most polynomially more states than the minimum unless of course P NP Often the required output f x is a set fy ykg of strings that are related to x via an NP mapping for example if G is a graph let AMIS G be the set of all maximal independent sets of G AMIS is known to be in output polynomial time see for an exposition and strengthening of this result and an early discussion of output polynomial time For such problems we have an elegant alternative de nition of output polynomial time A function f is in output polynomial time if the following problem is solvable in polynomial 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 polynomial time algorithm for f and vice versa if an output polynomial time algorithm exists for f it can be used to produce an element of y f x For example AMIS is in output polynomial time its generalization to hypergraphs is open but was recently shown to be in output n logn time see for an extensive discussion of the hypergraph generalization of AMIS One can use again traditional NP completeness to show that the following generalization is not in output polynomial time unless P NP Given a monotone circuit compute the set of all minimal with respect to the set of true inputs satisfying truth assignments But sometimes traditional NP completeness techniques do not seem to suf ce to bring out the intractability of a problem because this problem belongs to a class or computational mode that appears to be between P and NP In such cases NP completeness has acted as an open ended research paradigm spawn ing variants that are appropriate for the computational context being studied examples are classes that capture local search the parity argument loga rithmic nondeterminism the related concept of xed parameter tractability and approximability Complexity classes introduced this way as abstractions of natural compu tational problems of mysteriously intermediate complexity are in some precise sense well motivated 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 occasionally happens recall and its brilliant follow up NP completeness is of course a valuable tool for demonstrating the di culty of computational problems However NP completeness is often used allegori cally a problem is shown NP complete that is not strictly speaking a natural computational problem but an arti cial problem created to capture a mathe matical concept NP completeness in this context suggests that a problem area or approach is mathematically nasty Because if we believe that e cient algo rithms are the natural out ow of the mathematical structure of a problem a view shared by all computer scientists with the possible exception of researchers in metaphor based algorithmic paradigms such as neural nets in which algorith mic behavior is thought to be emergent then contrapositively complexity must be the manifestation of mathematical poverty lack of structure See for an early example of such a use of NP completeness in the theory of relational databases Beyond mathematics NP completeness and complexity in general can also be applied allegorically in other disciplines It can be 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 political science for cognitive implausibility in arti cial intelligence for genetic indeterminism in genetics and so on see for references NP completeness is thus an important intellectual export of computer science to other disciplines And it does ll a void in the interdisciplinary intel lectual trade It seems to me that the concept of lower bounds and negative results in general is particular to computer science and has no well developed counterpart in other disciplines True one sees isolated results in other sciences such as Heisenberg s uncertainty principle in quantum mechanics Arrow s im possibility theorem in economics and Carnot s theorem in thermodynamics which are arguably negative however nowhere else in science does one nd such a comprehensive methodology for obtaining negative results with the exception of complexity s own precursor mathematical logic with its many incomplete ness undecidability and inexpressibility results NP completeness is therefore valuable for another reason It is one of the few precious features which give our science its special character which set it apart from the other sciences see for another development of this argument In science successful ideas are those that are pervasive and invasive are invitingly elegant and methodical are open to extensions and variants and cap ture an objective necessity answer a widespread but di use sense of dissatisfac tion in the scienti c community in the case of NP completeness 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 about the nature and history of NP completeness could give us useful hints about computer science s next great paradigm which for all I know has started being articulated somewhere else in this volume