On the Parallel Approximability of Some Classes of Quadratic Programming

In this paper we analyze the parallel approximability of two special classes of Quadratic Programming. First, we consider Convex Quadratic Programming. We show that the problem of Approximating Convex Quadratic Programming is P-complete. We also consider two approximation problems related to it, Solution Approximation and Value Approximation and show both of these cannot be solved in NC, unless P=NC. Secondly, on the positive side, we show that we have an NC Approximation Scheme for those instances of Quadratic Programming that are \smooth" and \positive." Then we show how to extend the result for positive instances of bounded degree Smooth Integer Programming problems. Finally, we formulate several combinatorial problems as positive QP (or positive Integer Programs) in packing/covering form and show that the presented techniques can be used to obtain NC Approximation Schemes for \dense" instances of such problems.