Robust algorithms for restricted domains

Abstract

We introduce a new definition of efficient algorithms for restricted domains. Under this definition, an algorithm is required to be “robust,” i.e., it must produce correct output regardless of whether the input actually belongs to the restricted domain or not. This is to be contrasted with the “promise” version of solving problems on restricted domains, in which there is a guarantee that the input is in the class, and an algorithm to “solve” the problem need not function correctly or even terminate if this guarantee is not met. There exist problems that have a polynomial time promise solution, while being NP-hard if required to be robust. We show perhaps the surprising result that robustly finding a maximum independent set in a well-covered graph (i.e., a graph in which every maximal independent set is of the same size) is NP-hard. An argument can be made that this hardness result is more meaningful than the trivial polynomial time promise algorithm. We give a polynomial time robust algorithm for the maximum clique problem in unit disk graphs, i.e., given an input graph G in general form, the output is either a maximum clique for G or a certificate that G is not a unit disk graph. The existence of this algorithm is to be reconciled with the apparent contradiction posed by the facts: (1) Recognizing whether an input graph given in general form is a unit disk graph is NP-hard; in fact, it is not even known to be in NP. (2) Finding a maximum clique in an input graph given in general form is NP-hard.