Tractable and intractable problems for parallel computers

Complete problems A computational problem is complete for a complexity class when it is, in a formal sense, one of the "hardest" or "most expressive" problems in the complexity class. Complexity classes are sets of all of the problems that can possibly be solved with at most a certain amount of some computational resource, and may include problems that actually require far less resources. The complete problems, however, are the most resource-intensive problems in the class.

Complete problems “encode” all of the difficulty of the complexity class Besides being very resource-intensive, complete problems are very expressive; in some sense, they "encode" all of the expressiveness or difficulty of the complexity class. This is because, given a method of efficiently solving a complete problem, we can efficiently solve any other problem in the complexity class. So, for example, many of the problems in EXPTIME, the exponential time class, are very hard, and we could never hope to find efficient solutions for them. But if we were given a magic box that quickly solved any EXPTIME-complete problem (called an oracle machine for that problem), we could quickly solve any other problem in EXPTIME

Hard and complete problems When a problem has the property that it allows you to quickly solve any problem in a complexity class, we say that it is hard for that class. Equivalently, we can say that there is a reduction from any problem in the class to the hard problem. When a problem is both hard for a class, and a member of the class, it is complete for that class.

P (complexity) In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. P is known to contain many natural problems, including linear programming, calculating the greatest common divisor, and finding a maximum matching. In 2002, it was shown that the problem of determining if a number is prime is in P.

P-complete class In complexity theory, the complexity class P-complete is a set of decision problems and is useful in the analysis of which problems can be efficiently solved on parallel computers. A decision problem is in P-complete if it is complete for P, meaning that it is in P, and that every problem in P can be reduced to it in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem A is in P-complete if, for each problem B in P, there are constants c and k such that B can be reduced to A in time O((log n)c) using O(nk) parallel processors.

Motivation The class P, typically taken to consist of all the "tractable" problems for a sequential computer, contains the class NC, which consists of those problems which can be efficiently solved on a parallel computer. This is because parallel computers can be simulated on a sequential machine. It is not known whether NC=P. In other words, it is not known whether there are any tractable problems that are inherently sequential. Just as it is widely suspected that P does not equal NP, so it is widely suspected that NC does not equal P.

P-complete problems The most basic P-complete problem is this: Given a Turing machine, an input for that machine, and a number T (written in unary), does that machine halt on that input within the first T steps? It is clear that this problem is P-complete: if we can parallelize a general simulation of a sequential computer, then we will be able to parallelize any program that runs on that computer. If this problem is in NC, then so is every other problem in P.

This problem illustrates a common trick in the theory of P-completeness. We aren't really interested in whether a problem can be solved quickly on a parallel machine. We're just interested in whether a parallel machine solves it much more quickly than a sequential machine. Therefore, we have to reword the problem so that the sequential version is in P. That is why this problem required T to be written in unary. If a number T is written as a binary number (a string of n ones and zeros, where n=log(T)), then the obvious sequential algorithm can take time 2n. On the other hand, if T is written as a unary number (a string of n ones, where n=T), then it only takes time n. By writing T in unary rather than binary, we have reduced the obvious sequential algorithm from exponential time to linear time. That puts the sequential problem in P. Then, it will be in NC if and only if it is parallelizable.

P-complete problems Many other problems have been proved to be P-complete, and therefore are widely believed to be inherently sequential. These include the following problems, either as given, or in a decision-problem form: In order to prove that a given problem is P-complete, one typically tries to reduce a known P-complete problem to the given one, using an efficient parallel algorithm.

Examples of P-complete problems Circuit Value Problem (CVP) - Given a circuit, the inputs to the circuit, and one gate in the circuit, calculate the output of that gate Game of Life - Given an initial configuration of Conway's Game of Life, a particular cell, and a time T (in unary), is that cell alive after T steps? Depth First Search Ordering - Given a graph with fixed ordered adjacency lists, and nodes u and v, is vertex u visited before vertex v in a depth-first search?

Problems not known to be P-complete Some problems are not known to be either NP-complete or P. These problems (e.g. factoring) are suspected to be difficult. Similarly there are problems that are not known to be either P-complete or NC, but are thought to be difficult to parallelize. Examples include the decision problem forms of finding the greatest common divisor of two binary numbers, and determining what answer the extended Euclidean algorithm would return when given two binary numbers.

Conclusion Just as the class P can be thought of as the tractable problems, so NC can be thought of as the problems that can be efficiently solved on a parallel computer. NC is a subset of P because parallel computers can be simulated by sequential ones. It is unknown whether NC = P, but most researchers suspect this to be false, meaning that there are some tractable problems which are probably "inherently sequential" and cannot significantly be sped up by using parallelism The class P-Complete can be thought of as "probably not parallelizable" or "probably inherently sequential".