This is the nineteenth edition of a (usually) quarterly column that covers 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

In the past 20 years there has been treftlendous progress in developing and analyzing parallel algorithftls. Researchers have developed efficient parallel algorithms to solve most problems for which efficient sequential solutions are known. Although some ofthese algorithms are efficient only in a theoretical framework, many are quite efficient in practice or have key ideas that have been used in efficient implementations. This research on parallel algorithms has not only improved our general understanding ofparallelism but in several cases has led to improvements in sequential algorithms. Unf:ortunately there has been less success in developing good languages f:or prograftlftling parallel algorithftls, particularly languages that are well suited for teaching and prototyping algorithms. There has been a large gap between languages

This paper considers the ultimate impact of fundamental physical limitations---notably, speed of light and device size---on parallel computing machines. Although we fully expect an innovative and very gradual evolution to the limiting situation, we take here the provocative view of exploring the consequences of the accomplished attainment of the physical bounds. The main result is that scalability holds only for neighborly interconnections, such as the square mesh, of bounded-size synchronous modules, presumably of the area-universal type. We also discuss the ultimate infeasibility of latencyhiding, the violation of intuitive maximal speedups, and the emerging novel processor-time tradeoffs.

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the class of sequentially polynomial-time problems. Our algorithm can be extended to compute in parallel one irreducible complex factor of a multivariate integral polynomial. However, the coefficients of the computed factor are only represented modulo a not necessarily irreducible polynomial specifying a splitting field. A consequence of our algoithm is that multivariate polynomials over finite fields can be tested for absolute irreducibility in deterministic sequential polynomial time in the size of the input. We also obtain a sharp bound for the last prime p for which, when taking an absolutely irreducible integral polynomial modulo p, the polynomial's irreducibility in the algebraic closure of the finite field order p is not preserved.

A new parallel algorithm is given to evaluate a straight line program. The algorithm evaluates a program over a commutative semi-ring R of degree d and size n in time O(log n(log nd)) using M(n) processors, where M(n) is the number of processors required for multiplying n \Theta n matrices over the semi-ring R in O(log n) time. Appears in SIAM J. Comput., 17/4, pp. 687--695 (1988). Preliminary version of this paper appeared in [6]. y Research supported in part by National Science Foundation Grant MCS-800756 A01. z Research supported by NSF under ECS-8404866, the Semiconductor Research Corporation under RSCH 84-06-049-6, and by an IBM Faculty Development Award. x Research Supported in part by NSF Grant DCR-8504391 and by an IBM Faculty Development Award. 1 INTRODUCTION 1 1 Introduction In this paper we consider the problem of dynamic evaluation of a straight line program in parallel. This is a generalization of the result of Valiant et al [10]. They consider the problem of ta...

We identify and study a natural and frequently occurring subclass of Concurrent-Read, Exclusive-Write Parallel Random Access Machines (CREW-PRAMs). Called Concurrent-Read, Owner-Write, or CROW-PRAMs, these are machines in which each global memory location is assigned a unique "owner" processor, which is the only processor allowed to write into it. Considering the difficulties that would be involved in physically realizing a full CREW-PRAM model, it is interesting to observe that in fact, most known CREW-PRAM algorithms satisfy the CROW restriction or can be easily modified to do so. This paper makes three main contributions. First, we formally define the CROW-PRAM model and demonstrate its stability

this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...

This paper presents two new parallel algorithms QSP1 and QSP2 based on sequential quicksort for sorting data on a mesh multicomputer, and analyzes their scalability using the isoefficiency metric. We show that QSP2 matches the lower bound on the isoefficiency function for mesh multicomputers. The isoefficiency of QSP1 is also fairly close to optimal. Lang et al. and Schnorr et al. have developed parallel sorting algorithms for the mesh architecture that have either optimal (Schnorr) or close to optimal (Lang) run-time complexity for the one-element-per-processor case. Both QSP1 and QSP2 have worse performance than these algorithms for the one-element-perprocessor case. But QSP1 and QSP2 have better scalability than the scaled-down variants of these algorithms (for the case in which there are more elements than processors). As a result, our new parallel formulations are better than these scaled-down variants in terms of speedup w.r.t the best sequential algorithms. We also present a dif...

this paper, except Figures 4.8 and 4.9, are two dimensional, representing polyhedra as polygons. 1 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 4 * vp Figure 4.1: Visibility ordering of the cells of a mesh relative to viewpoint vp. can be computed and stored in a preprocessing step. The MPVO algorithm can be extended to order many nonconvex meshes; this is described in detail in Section 4.4.2. 4.3 Preliminary Definitions A convex polyhedron in E

foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that "Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-completeness. The ensuing exploration of the boundaries and nature of the NP-complete class of problems has been one of the most active and important research activities in computer science for the last decade. Cook is well known for his influential results in fundamental areas of computer science. He has made significant contributions to complexity theory, to time-space tradeoffs in computation, and to logics for programming languages. His work is characterized by elegance and insights and has illuminated the very nature of computation." During 1970-1979, Cook did extensive work under grants from the