Reducibility Among Combinatorial Problems. by Richard M. Karp; Raymond E. Miller; JamesW. ThatcherReview by: Ronald V. BookThe Journal of Symbolic Logic, Vol. 40, No. 4 (Dec., 1975), pp. 618-619Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271828 .

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618 REVIEWS

RICHARD M. KARP. Reducibility among combinatorial problems. Complexity of computer computations, Proceedings of a Symposium on the Complexity of Computer Computations, held March 20-22,1972, at the IBM Thomas J. Watson Center, Yorktown Heights, New York, edited by Raymond E. Miller and James W. Thatcher, Plenum Press, New York and London 1972, pp. 85-103.

In the area of computational complexity, it is widely accepted that a computational problem in combinatorial mathematics, mathematical programming, or logic can be regarded as trac- table if and only if there is an algorithm for its solution whose running time is bounded by a polynomial in the size of the input (see Cobham XXXIV 657, and Edmunds, Canadian journal of mathematics, vol. 17 (1965), pp. 449-467). In this paper Karp considers some problems which are not known to be tractable but have the property that if they are tractable, then a large number of problems of independent interest are tractable, and any proof of their tractability is likely to yield some fundamental insights into the nature of computing.

An instance of a particular computational problem (viewed as the input to an algorithm) can be encoded as a set of strings, i.e., a language; an algorithm for its solution is then a lan- guage-theoretic recognition algorithm. Let P be the class of languages recognizable by poly- nomial time-bounded string recognition algorithms. Thus, a problem is tractable if and only if a natural string representation of the problem is in P.

Non-deterministic algorithms are defined by relaxing the condition that demands at each step a unique next step; rather, in a non-deterministic algorithm there are a bounded number of possible next steps. A non-deterministic algorithm recognizes an input string if there is some computation which begins on this string and ends in an accepting configuration. The time required to recognize an input string is the length of a shortest accepting computation. Let NP be the class of languages reconizable by polynomial time-bounded recognition algorithms which are non-deterministic. Both P and NP are invariant under a wide range of choices of models for algorithms; for example NP is the class of recognition problems solvable by back- track search algorithms of polynomial bounded depth, and i's the class of languages recognized in polynomial time by non-deterministic Turing-machines.

The question "does P equal NP?" is an important open question of computational com- plexity. It is an instance of a more general problem arising in automata theory and compu- tational complexity: Given a restricted model of computation, do the deterministic and non- deterministic modes of operation have the same power? Cook (Proceedings of the Third ACM Symposium on Theory of Computing (1971), pp. 151-158) called attention to this instance of the more general problem. It is known that if a recognition problem is in NP, then there is a deterministic algorithm for its solution which has running time of order 2nk for some k > 0. In the general case no better bound is known. In contrast, it is known that the class of languages accepted by deterministic algorithms which operate in polynomial space is equal to the class of languages accepted by non-deterministic algorithms which operate in polynomial space (Savitch XXXIX 346).

A language L1 is polynomially reducible to a language L2 if there is a function f which can be computed in time bounded by a polynomial in the length of the input string with the property that for all strings x, x E L1 if and only if f(x) E L2 . A language L is NP-complete if L E NP

and every language in NP is polynomially reducible to L. Thus, P = NP if and only if there is some NP-complete language L such that L e P. Cook showed that the set of strings representing satisfiable formulas in conjunctive normal form is NP-complete and that P = NP if and only if the set of strings representing tautologies in disjunctive normal form is in P.

Building on the work of Cook, Karp shows that a variety of computational problems arising in fields such as computational logic, graph theory, and mathematical programming are NP-complete. Examples of such problems are: clique (given a graph G and positive integer k, does G have a set of k mutually adjacent nodes?); knapsack (given (a,, - --, a, , b) E Zr+ 1, does there exist x E {O, I}r such that 2aixi = b?); set covering (given a finite family of finite sets {Sj}jEJ and a positive integer k, does there exist a subfamily {Th}hEH containing at most k sets such that UhEHTh = UjEJSj?). Thus a wide variety of problems from diverse fields are "computationally equivalent" in the sense that either all are in P (hence, tractable) or none are in P (hence, intractable). This fact has inspired researchers from many diverse areas to consider

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REVIEWS 619

whether P equals NP, and to study the computational complexity of other problems within this framework.

Karp's treatment of NP (in terms of NP-complete sets and polynomial time reducibilities) has motivated work on the structure of NP and its generalizations. This paper and that of Cook have caused workers in theoretical computer science to return to the problems of comparing the relative power of the deterministic and non-deterministic modes of operation for restricted classes of algorithms.

This paper is an extremely important contribution to the study of computational complexity and, more broadly, to modern applied mathematics. RONALD V. BOOK

ANN YASUHARA. Recursive function theory and logic. Academic Press, New York and London 1971, xv + 338 pp.

This brave undertaking is intended as a textbook in a year's course for computer scientists with limited mathematical background. Its avowed prerequisites are slight, but parts of it will frustrate the unsophisticated reader. Its eight chapters on recursive function theory precede its six on logic, both because they are deemed "easier to grasp for the reader whose mathe- matical background may not be very strong" and because "it is the computable and non- computable aspects of (logic) which are particularly important to the computer scientist." Thus the quantifier does not appear until page 164.

Given that much description, a logician's estimate of what the book covers might be 80. accurate. He or she would be surprised to find inclusion of the Grzegorczyk hierarchy and the Friedberg-Muchnik theorem (though no more so than the beginning student), and might be disappointed that the treatment of Gddel's theorem and undecidable theories runs for less than nineteen pages. But it should be no surprise at all to find chapters on Turing machines, semi-Thue and Thue systems, the equivalence of recursive and Turing-computable functions, recursively enumerable sets and relations, and axiomatizations, semantics and standard metatheorems for propositional calculus and first-order logic; and it should not be much more surprising to find sections on Herbrand's theorem, mechanical theorem-proving, Presburger arithmetic and elimination of quantifiers, and finite automata and the weak monadic second- order theory of successor.

The author tries very earnestly, especially early, to inject more than formal life into the topics treated and to render them intuitively palatable. But some of her informal remarks are aimed too low, lapsing into looseness and vagueness, while there is little compensating trace of such expository genius as distinguishes a book on recursive functions like Rogers XXXVI 141. The proofs themselves are rarely as crisp as one would wish; they are forged somewhat heavily. Arguments like those given for the recursion theorem and for Theorem 6.19, to pick two among many, might have been made much clearer. In fact the student could easily come away from the former believing it to depend on Church's thesis. And proofs of a large number of important theorems are not given at all, but are wholly relegated to the status of "exercises." This seems too short shrift for results like the compactness theorem for first-order logic. There are numerous other exercises, many of them meaty; very few are solved in the book.

Several rather lengthy sections with especially promising titles (" Kalmar-elementary func- tions and the Grzegorczyk Hierarchy"; "The predictably computable functions, or the Ritchie Hierarchy"; "Complexity of partial recursive functions"; "Friedberg's Theorem") turn out to be mainly direct synopses of parts of research papers (as of Grzegorczyk XX 71; Ritchie XXVIII 252; Blum XXXIV 657 and Helm and Young (this JOURNAL, vol. 36 (1971), pp. 21-27); Friedberg XXIII 225). The author urges the reader to study the original publications, and since the sections in question give the material no fresh illumination, the exhortations alone should have been allowed to suffice.

Most of the book's formal deficiencies are minor, but there are enough of them to make one regard the book as somewhat less than tight. A sample follows. The axiom of choice is never mentioned, but it is needed to prove Exercise 13.9, to which the subsequent Theorem 13.4 appeals, and to prove (K6nig's) Lemma 13.1. A needed case, clause, or qualification is omitted in each of these: definition at top of page 26; definition of 'A#' on page 65; the penultimate para- graph of page 150; item 2 of page 174; example (3) of page 207; 9 lines from bottom of page

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