The NP-completeness column: An ongoing guide

doi:10.1016/0196-6774(84)90022-1

Journal of Algorithms

Volume 5, Issue 3, September 1984, Pages 433-447

https://doi.org/10.1016/0196-6774(84)90022-1 Get rights and content

Abstract

This is the twelfth edition of a quarterly column that provides continuing coverage of new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book “Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., New York 1979 (hereinafter referred to as “[G&J]”; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C-355, AT&T Bell Laboratories, Murray Hill, NJ 07974 (CSNET address: dsj.rabbit.btl@csnet-relay). Please include details, or at least sketches, of any new proofs (full papers are preferred). If the results are unpublished, please state explicitly that you would like them to be mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this Journal.

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We introduce new, elementary Monte Carlo methods to speed up and greatly simplify the manipulation of permutation groups (given by a list of generators). The methods are of a combinatorial character, using only elementary group theory. The key idea is that under certain conditions, "random subproducts" of the generators successfully emulate truly random elements of a group. We achieve a nearly optimal O(n³ log^cn) asymptotic running time for membership testing, where n is the size of the permutation domain. This is an improvement of two orders of magnitude compared to known elementary algorithms and one order of magnitude compared to algorithms which depend on heavy use of group theory. An even greater asymptotic speedup is achieved for normal closures, a key ingredient in group-theoretic computation, now constructible in Monte Carlo time O(n² log^cn), i.e., essentially linear time (as a function of the input length). Some of the new techniques are sufficiently general to allow polynomial-time implementations in the very general model of "black box groups" (group operations are performed by an oracle). In particular, the normal closure algorithm has a number of applications to matrix-group computation. It should be stressed that our randomized algorithms are not heuristic: the probability of error is guaranteed not to exceed a bound ϵ > 0, prescribed by the user. The cost of this requirement is a factor of |log ϵ| in the running time.
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Let F be a field of q=pⁿ elements, where p is prime. We present two new probabilisticalgorithms for factoring polynomials in F[X] that make particularly efficient use of random bits. They are easy to implement, and require no randomness beyond an initial seed whose length is proportional to the input size. The first algorithm is based on a procedure of Berlekamp; on input f in F[X] of degree d, it uses d log₂p random bits and produces in polynomial time a complete factorisation of f with a failure probability of no more than 1/p^(1−e)d/2. (Here ε denotes a fixed parameter between 0 and 1 that can be chosen by the implementer.) The second algorithm is based on a method of Cantor and Zassenhaus; it uses d log₂q random bits and fails to find a complete factorisation with probability no more than 1/q^(1−e)d/4. For both of these algorithms, the failure probability is exponentially small in the number of random bits used.
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The NP-completeness column: An ongoing guide

Abstract

Theor. Comput. Sci.

Inform. Process. Lett.

Inform. Process. Lett.

J. Comput. System Sci.

J. Number Theory

J. Comput. System Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Disc. Applied Math.

Information and Control

Two theorems on random polynomial time

Reducibility, randomness, and intractability

On distinguishing prime numbers from composite numbers

Annals Math.

Random walks, traversal sequences and the complexity of maze problems

On a primality test of Solovay and Strassen

SIAM J. Comput.

Monte-Carlo algorithms in graph isomorphism testing

Relativizations of the P =? NP question

SIAM J. Comput.

Relative to a random oracle A, PA ≠ NPA ≠ co-NPA with probability 1

SIAM J. Comput.

Analysis of a randomized data structure for representing ordered sets

Probabilistic machines can use less running time

A subexponential algorithm for trivalent graph isomorphism

An O(n3logn) deterministic and an O(n3) probabilistic isomorphism test for trivalent graphs

Computational complexity of probabilistic Turing machines

SIAM J. Comput.

On relativized exponential and probabilistic complexity classes

Topics in Probabilistic Complexity

Relative to a random oracle A, P^A ≠ NP^A ≠ co-NP^A with probability 1

An O(n³logn) deterministic and an O(n³) probabilistic isomorphism test for trivalent graphs