Parallel approximation of optimization problems

TLDR: This survey will focus its attention on PRAM efficient algorithms for combinatorial optimization problems, which identify the inherent parallelism of the problem independently of what parallel computational model one chooses to use.

Abstract: The Parallel Random Access Machine (PRAM) is an abstract model of parallel computation consisting of a set of processors, i.e. random-access machines, that share a potentially infinite common memory and hence communicate via it. The simplicity and generality of this model has motivated the design of a large number of PRAM e~icient algorithms, that is, algorithms which run in poly-logarithmic time (i.e. the parallel computation time is bounded by a polynomial of the logarithm of the input size) and which use a polynomial number of processors (with respect to the input size). An extensive survey of basic techniques for designing PRAM efficient algorithms is contained in [21]. We will not distinguish between PRAM models with different restrictions on memory access since these models do not differ very widely in their computational power. In particular, the less restrictive model (i.e. the one where multiple processors may read or write to any memory location) can be simulated by the most restrictive model (i.e. the one where at most one processor may read or write to a particular memory location) with the parallel time increased only by a logarithmic factor (see [25] for a comparison of different conflict-resolution rules). The PRAM cannot be considered a physically realizable model since a multiported memory shared by a large number of processors is infeasible. However, PRAM efficient algorithms are interesting for two main reasons. On the one hand, they identify the inherent parallelism of the problem independently of what parallel computational model one chooses to use. On the other hand, several techniques have been developed in order to simulate a PRAM on a realistic parallel machine, namely one with distributed memory and an interconnection network of fixed degree, with "reasonable" slowdown and memory blow-up (see [16]). In this survey we will focus our attention on PRAM efficient algorithms for combinatorial optimization problems. The basic ingredients of an optimization problem are: the set of instances or input objects, the set of feasible solutions or output objects associated with any instance, and the measure defined for any feasible solution. The problem is specified as a maximization problem or a minimization problem depending whether its goal is to find a solution whose measure is maximum or minimum, respectively. An approximation PRAM efficient algorithm for an optimization problem receives as input an instance of the problem and returns a feasible solution of the instance. The quality of the returned solution can be measured in several ways