In this chapter we consider some problems that are non-approximable by NC algorithms unless P=NC. The basic technique used to prove non-approximability is the design of a reduction that generates instances in such a way that a gap in the value of the objective function is created.

First we concentrate on the Induced Subgraph of High Weight type problems studied in Chapter 3. Anderson and Mayr [AM86] studied this problem when the weight considered is the minimum degree of the graph. They provide a reduction from the Circuit Value problem to the High Degree Subgraph problem that creates a 1/2 gap in the weight of the so obtained instance. Kirousis, Serna and Spirakis studied this problem for two new weight functions, namely, vertex connectivity and edge connectivity, (see [SS89], [Ser90], [KSS93]). Kirousis and Thilikos analyzed the graph linkage [KT96] whose approximability presents the same kind of threshold behavior. All non-approximability results follow the same technique, that is, they create a graph, translating each circuit component or connection into a particular subgraph, in order to generate a graph that satisfies the required gap properties.

Our second result corresponds to problems that are non-approximable in NC, which means that for any e we are able to generate an instance in which an e-gap appears. Serna [Ser91] showed that the linear programming problem cannot be approximated in NC for any ratio, other non-approximable problems can be found in [SS89], [KS88], [Ser90].

To complement the non-approximability view we want to consider also the possible paralellization of heuristics used to deal with NP-hard problems.