Karp and Smale Receive National Medals of Science

comm-medal.qxp 10/21/96 9:22 AM Page 1524

In June of this year President Clinton announced entist at the International Computer Science In- the recipients of the National Medal of Science, stitute in Berkeley. In 1995 he moved to the Uni- the nation’s highest honor in science and tech- versity of Washington in Seattle to take positions nology. Among the eight recipients are math- as professor of computer science and engi- ematician Stephen Smale and theoretical com- neering and adjunct professor of molecular puter scientist Richard Karp. Five National biotechnology. Medals of Technology were also awarded. The National Medal of Science, established The Contributions of Richard M. Karp to by Congress and administered by the National Computer Science Science Foundation, honors individuals for con- David B. Shmoys tributions to the present state of knowledge in Richard M. Karp has made fundamental contri- one of the following fields: physical, biological, butions to the foundations of computer science mathematical, engineering, or social and be- and over the past four decades has dramatically havioral sciences. The medal has now been extended our understanding of the nature of awarded to 344 distinguished scientists and en- efficient computation. He has made seminal gineers. contributions to a wide variety of areas within the field of theoretical computer science and Richard M. Karp has had a profound influence on the directions Richard M. Karp was awarded the National Medal in which this rapidly growing field has moved of Science for “linking advances in theoretical over this period, giving it both mathematical computer science to real-world problems.” Karp depth and practical relevance. was born January 3, 1935, in Boston, Massa- Karp’s most significant contribution, presented in his landmark 1972 paper entitled “Re- chusetts. He received his bachelor’s degree ducibility among Combinatorial Problems”, (1955), his master’s degree (1956), and his Ph.D. showed that twenty-one combinatorially defined in applied mathematics (1959) from Harvard computational problems are all -complete. University. He was a member of the research This provided concrete evidence NPthat a plethora staff at the IBM T. J. Watson Research Center of well-studied optimization problems, such as from 1959 until 1968, and he was on the faculty the traveling salesman problem and the graph of industrial and management engineering at coloring problem, were hard to solve. This work Columbia University during 1967–68. In 1968 he focused attention on the “ = ?” question took a position as professor of computer science as the central open problemP NP in our under- and operations research at the University of Cal- ifornia, Berkeley, and starting in 1980 he also David B. Shmoys is associate professor of operations re- held a joint position in the department of math- search and computer science, Cornell University. His e- ematics. He also held a position as research sci- mail address is [email protected].

1524 NOTICES OF THE AMS VOLUME 43, NUMBER 12 comm-medal.qxp 10/21/96 9:22 AM Page 1525

standing of efficient computation. For any set L, the membership problem for L is to decide, given an input x, whether x L. For some sets, ∈ it is trivial to decide if a given x is in L, whereas for others no efficient algorithm is known. The set L is in the complexity class if there exists P an algorithm that solves the membership problem for L and runs in time bounded by a polynomial in the length of the input; this is the leading theoretical characterization of an efficient algorithm, and is the set of all efficiently P solvable computational problems. The class NP is the set of problems L for which there exists L such that x L if and only if there ex- 0 ∈P ∈ ists a polynomial-length y such that (x, y) L ; ∈ 0 intuitively, is the class of problems L with NP the property that each x L has a succinct, ef- ∈ ficiently verifiable proof y of its membership. In 1971 Cook obtained a pivotal result: he showed that the satisfiability problem, i.e., de- ciding whether a given Boolean formula (in con- junctive normal form) has an assignment that makes the formula “true”, is complete for , NP in the sense that any problem in could be NP Richard M. Karp solved by a polynomial number of calls to a sub- routine for the satisfiability problem; further- mial algorithms, where in the latter case the more, he showed that this result might have im- running-time bound is a function of the size of plications for other combinatorial problems by the underlying combinatorial structure, but not showing, in essence, that the clique problem in the magnitude of the numbers that are part of graphs was also -complete. Karp’s paper re- the input. In joint work with Hopcroft, Karp gave NP fined this approach and showed that many of the the most efficient algorithm known for another most notoriously intractable computational basic problem in combinatorial optimization, problems were all -complete. As Karp noted the bipartite matching problem; the algorithm NP at the time, this suggests “that these problems, is quite natural, and their running-time analysis as well as many others, will remain intractable provides deep insight into the structure of this perpetually.” The theory of -completeness, well-studied problem. NP as developed by Cook and Karp, and indepen- The traveling salesman problem, another cen- dently by Levin, has had an impact well beyond tral problem in operations research, was a re- computer science, since -complete compu- curring subject of Karp’s investigations; in this NP tational problems arise in virtually every disci- problem the aim is to find a shortest tour that pline of engineering and the physical and social visits a collection of points, given the distances sciences. between points. In 1962, in joint work with Held, Karp also made important contributions to the he gave an elegant dynamic programming algo- design of efficient algorithms for a number of rithm for the traveling salesman problem. Al- combinatorial problems. In particular, his work though this algorithm takes exponential time, it with Edmonds gave polynomial-time algorithms was the most efficient approach known then to for the maximum flow and minimum-cost flow solve this problem. Nearly ten years later, in two problems, two of the most fundamental net- further papers with Held, he gave an important work optimization problems and a subject of in- way to quickly compute a lower bound on the tense investigation in operations research since length of the optimal tour. The Held-Karp lower the mid-50s. These algorithms introduced basic bound remains one of the most effective ways techniques, such as data scaling, that were later of efficiently estimating the optimal tour length; used by others in a wide variety of settings. Fur- years of testing have led to the general belief that thermore, if one considers the work that has it typically produces bounds within 1–2 percent followed in the design of algorithms for net- of optimal. Even more importantly, the bound work problems, there is scarcely a paper that is based on a “Lagrangian relaxation” technique; does not build on this work in some substantial due to its success in this context, this approach way. These results also highlighted what would was immediately applied to a wide range of com- become a central issue in this area: the differ- binatorial problems. For well more than a decade, ence between polynomial and strongly polyno- Lagrangian relaxation was the method of choice

DECEMBER 1996 NOTICES OF THE AMS 1525 comm-medal.qxp 10/21/96 9:22 AM Page 1526

for obtaining strong lower bounds for opti- the subsequent study of randomized approxi- mization problems. mation schemes for # -complete problems has P The traveling salesman problem was also the blossomed into one of the most exciting areas starting point for Karp’s investigation into al- of algorithm design for combinatorial problems. gorithms that work well with respect to inputs In the mid-80s Karp turned his attention to drawn from specified probability distributions. the design of parallel algorithms. This is an- Suppose that the input to the traveling salesman other setting in which randomization has turned problem consists of points that are selected in- out to be an important tool for algorithm design. dependently and uniformly at random in the One theoretical notion of efficiency for parallel unit square [0, 1]2 and the distance between computation is the class ; this is the set of NC them is measured in the `2 metric. Karp pro- problems solvable with a polynomial number of posed a natural divide-and-conquer strategy processors in time bounded by a polynomial in which partitions the points into clusters of the logarithm of the input size, with a shared nearby points, solves the induced problem for memory for interprocessor communication. In each cluster (using dynamic programming), and a quite surprising result, Karp and Wigderson then patches these subtours together. He then showed that the maximal independent set prob- showed that this algorithm produces a tour that lem is in ; prior to their work, this problem NC asymptotically tends to the optimum with prob- had been widely suspected to be one for which ability approaching 1. Many results on the prob- parallelism could not be used effectively. Their abilistic analysis of heuristics for -hard algorithm was based on first designing a ran- NP problems followed, and this continues to be a domized algorithm and then “derandomizing” thriving area of research. Karp made many sig- it by showing that the randomization could be nificant contributions to this area, giving inge- restricted to uniform sampling from a polyno- nious analyses of algorithms for such problems mial-size sample space and then making paral- as linear programming, bin-packing, and set par- lel runs, one for each sample. This subsequently titioning. became one of the most useful paradigms of par- A recurring theme in Karp’s research in the allel algorithm design. His algorithm for RN C past two decades is the use of randomization in the matching problem, obtained jointly with the design of algorithms. In this setting an al- Upfal and Wigderson, was another breakthrough gorithm is allowed to “toss coins” as one of its result; this has prompted interesting research on basic operations. Unlike the results on proba- techniques that might lead to a deterministic bilistic analysis mentioned above, in this set- algorithm, but no such matching algorithm NC ting one still takes a worst-case view: one wishes is currently known. to show that for any input, the algorithm’s out- Among the issues prevalent in his research put, which is a random variable, has the desired in the past decade, Karp has made leading con- property, e.g., is optimal with high probability. tributions to the study of online algorithms and In many settings, a problem can be solved using has worked towards bridging the gap between a randomized algorithm and yet no determinis- theoretical models of parallel computation and tic analogue is known. In joint work with Aleli- the realities of parallel machines. Most recently, unas, Lipton, Lovász, and Rackoff in 1979, he Karp has focused on computational issues in gave an algorithm to determine if two given ver- molecular biology. Although the motivation for tices are in the same connected component of these problems is different from those that he an undirected graph that, surprisingly, uses studied previously, this research nonetheless is space bounded by a logarithm of the size of the a natural extension of his investigations into ef- graph. Roughly speaking, such an algorithm is ficient algorithm design for combinatorial prob- limited to auxiliary storage consisting of a con- lems. stant number of node labels, and it is widely be- Karp has been awarded virtually every major lieved, but still unproven, that no deterministic prize in computer science and operations re- analogue exists. The randomized linear-time search, including the ACM Turing Award, the pattern-matching algorithm of Karp and Rabin ORSA/TIMS von Neumann Theory Prize, the Lan- is one of the most appealing results of the area: chester Prize, and the Fulkerson Prize; he is a it combines mathematical elegance with practi- member of the National Academy of Sciences and cal efficiency and generality. Luby and Karp con- the National Academy of Engineering. In addi- sidered the problem of computing the reliabil- tion to his gifts as a researcher, Karp has a phe- ity of planar networks, which is # -complete, nomenal talent for teaching. Having spent most P and hence believed to be substantially harder of his career at the University of California, than even the -complete problems discussed Berkeley, and now at the University of Wash- NP above. They gave a randomized approximation ington, he has inspired several generations of scheme for this problem, which was the first students with his lectures of laser-sharp clarity. known for any # -complete problem; once again, He has the uncanny ability to take the most P

1526 NOTICES OF THE AMS VOLUME 43, NUMBER 12 comm-medal.qxp 10/21/96 9:22 AM Page 1527

opaque just-proven result, to understand its essence, and then to present it in a perfectly transparent way. Throughout his career, Dick Karp has re- peatedly initiated an important path of research, obtained elegant and significant results to start the area, and stimulated the research commu- nity to continue in his footsteps. It is difficult to imagine any individual having a more profound impact on any field.

Stephen Smale Stephen Smale received the National Medal of Sci- ence for “four decades of pioneering work on basic research questions which have led to major advances in pure and applied mathematics.” Smale was born on July 15, 1930, in Flint, Michi- gan. He received his bachelor’s degree (1952), his master’s degree (1953), and his Ph.D. in mathematics (1956) from the University of Michigan, Ann Arbor. He was an instructor at the Univer- sity of Chicago from 1956 until 1958, when he became a member of the Institute for Advanced Study. In 1961 he went to Columbia University, Stephen Smale and in 1964 to the University of California, Berke- ley. He retired from Berkeley in 1995 and is cur- their winding numbers. In his thesis Smale gen- rently a professor of mathematics at the City Uni- eralized the result to regular closed curves liv- versity of Hong Kong. His many honors include ing on an n-manifold. The classification is pro- the Fields Medal, awarded in 1966. vided by a bijection to the fundamental group of the unit tangent bundle to the manifold. To The Mathematical Work of Stephen Smale establish the link, Smale skillfully employed al- Steve Batterson gebraic topology, analysis, and especially the Steve Smale has made profoundly original con- theory of fiber spaces. The thesis did not at- tributions to a stunning array of mathematical tract a great deal of attention. specialities. In 1990 the Smalefest conference cel- A few months into his postdoctoral career at ebrated Steve’s sixtieth birthday with a program the University of Chicago, Smale obtained his entitled “From Topology to Computation: Unity first famous result. He succeeded in pushing and Diversity in the Mathematical Sciences”. The the fiber space techniques to immersions of proceedings of this conference (edited by M. W. spheres. Smale classified C2 immersions of S2 Hirsch et al., Springer-Verlag, 1993) include in- into Rn, up to regular homotopy, by the second dividual survey articles on Smale’s work in dif- homotopy group of the Stiefel manifold, Vn,2 . ferential topology, economic theory, dynamical Since π2(V3,2) is trivial, an immediate corollary systems, computation, nonlinear analysis, and is the regular homotopy equivalence of all im- mechanics. See this volume for a more sub- mersions of S2 into R3 . In particular, there ex- stantive review of the remarkable depth and ists a regular homotopy of the inclusion to the breadth of Smale’s mathematics. antipodal immersion, in effect turning the sphere Smale completed his Ph.D. thesis in 1956 inside out. The result was counter to the pre- under the direction of Raoul Bott at the Univer- vailing intuition that placed these immersions in sity of Michigan. Two curves are regularly ho- different classes. The fact that Smale’s proof of- motopic provided that there exists a homotopy fered little insight into a comprehensible real- through regular curves for which the tangent vec- ization of the homotopy added to its mystique. tor varies continuously as a function of the curve In the summer of 1958 Mauricio Peixoto in- and homotopy parameters. The 1937 Whitney- troduced Steve to the Andronov-Pontryagin con- Graustein Theorem classified regular closed cept of structurally stable vector fields (the tra- curves in the plane, up to regular homotopy, by jectory dynamics are preserved under small perturbations to the vector field). Peixoto sought Steve Batterson is associate professor of mathematics in the Department of Mathematics and Computer Sci- a characterization of structural stability that ex- ence at Emory University. He is presently writing a bi- tended beyond the surface setting. Smale quickly ography of Stephen Smale. His e-mail address is saw the relevance of transversality and topology [email protected]. to the higher-dimensional situation. He conjec-

DECEMBER 1996 NOTICES OF THE AMS 1527 comm-medal.qxp 10/21/96 9:22 AM Page 1528

tured that a class of vector fields, now known nonlinear maps on infinite-dimensional mani- as Morse-Smale, were exactly the structurally folds that satisfy what is now known as the stable ones. Additionally, he asked if the Morse- Palais-Smale condition. Next he obtained an in- Smale systems were dense in the C1 topology. finite-dimensional generalization of the Morse- This was the first approximation in Smale’s bold Sard theorem. vision for dynamical systems, a qualitative study Smale returned to dynamical systems in 1965, of differential equations that transcended the al- showing that structural stability was not dense. gebraic form of the equation. Smale’s original vision for dynamical systems Smale began a two-year National Science Foun- could not be realized, but his approximations dation Postdoctoral Fellowship in 1958. Follow- were converging to something exciting. In his ing 1-1/2 years at the Institute for Advanced landmark 1967 Bulletin survey article, Smale Study, Steve moved to Rio de Janeiro to complete presented his program for hyperbolic systems the final six months at the Instituto de Math- and stability, complete with a superb collection ematica Pura e Aplicada. During this period in of problems. The major theorem of the paper early 1960 he obtained two sensational results. was the -Stability Theorem, whose proof was Shortly after his arrival in Brazil, Smale received a tour de force in the new methods. a letter from Norman Levinson asserting that By theΩ late sixties Smale had moved into ap- there were structurally stable systems that were plications. He modeled physical processes by dy- not Morse-Smale. This led to Smale’s construc- namical systems, opening new lines of inquiry. tion of the horseshoe map and his early study The n-body problem and electric circuit theory of chaotic phenomena. were among the applications that Smale framed Next, Steve resumed his work on the Poincaré in the language of dynamical systems. For much Conjecture. The problem, a compact n-manifold of the seventies Steve focused on economics, in- with the algebraic topology of the n-sphere is jecting topology and dynamics into the study of homeomorphic to the n-sphere, had attracted general economic equilibria. Having established Smale since his graduate student days. The sit- the nature of the equilibria, Smale began to think uations n =3and n 3 are known respectively about algorithms for their computation. While ≥ as the Poincaré Conjecture and Generalized Poin- traditional approaches to the convergence the- caré Conjecture. At the time there was an over- ory of algorithms were local, Smale introduced whelming conventional wisdom that difficulty in- a global perspective to the problems. Was the al- creased with dimension. The three-dimensional gorithm reasonably reliable, and how many it- problem was so daunting that larger n appeared erations were to be expected? Newton’s method out of reach, at least until dimension three was and the simplex algorithm gained new meaning resolved. Undeterred, Smale conceived a Morse from this perspective. Theory approach to the Generalized Poincaré In recent years Smale, in collaboration with Conjecture. As he developed these ideas, an ex- Lenore Blum and Mike Shub, has sought to unify traordinary element emerged. His proof was the fields of theoretical computer science and valid for all dimensions n 5, but failed in di- numerical analysis. Practical numerical algo- ≥ mensions three and four. Smale had proved the rithms involve the computation of real num- Higher Dimensional Poincaré Conjecture. In the bers, while the classical Turing machines ma- eighties Michael Freedman established the four- nipulate discrete sets. The Blum-Shub-Smale dimensional theorem, but n =3 remains un- model for computation operates on a ring, thus solved. encompassing the 0–1 world of Turing machines In 1961 Smale proved the h-Cobordism The- and the real-complex number setting required orem. This seminal result provides algebraic for numerical analysis. The result is a general- topological criteria for establishing that higher- ization of the classical theory that provides a the- dimensional manifolds are diffeomorphic. Hav- oretical foundation for numerical analysis. ing set a new agenda for differential topology, So many brilliant threads pervade Smale’s Smale then abruptly shifted his attention back mathematics that providing a summary is partly to dynamical systems. Steve was renewing his a matter of personal taste. Throughout his career Smale has approached mathematical prob- quest for a generic structurally stable collection lems with the scholarship to learn from others, of dynamical systems. His original Morse-Smale the audacity to be unconstrained by conven- candidate had been disqualified by the horse- tional wisdom, and the power and vision to em- shoe. As Smale sought a second approximation ploy new methods and construct original frame- he unearthed other essential elements of his de- works. After the fact, a Smale development veloping program for dynamical systems. The seems so natural, yet no one else thought of it. project was put on hold from 1962 to 1964 as Steve’s interests detoured into infinite-dimensional analysis. There he, independently with Richard Palais, extended Morse Theory to those

1528 NOTICES OF THE AMS VOLUME 43, NUMBER 12