Robust algorithms for restricted domains

• Authors: Vijay Raghavan;Jeremy Spinrad
• Venue: SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
• Year: 2001

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.The more stringent requirement of robustness in algorithms gives an important benefit: such algorithms are amenable to manipulation as building blocks of more general algorithms; in other words, composition of robust algorithms preserves robustness. In contrast, promise algorithms cannot be so composed.There exist problems which have a polynomial time promise solution, while being NP-hard if required to be robust. We show the perhaps surprising result that 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.Graph classes provide interesting natural restricted domains; there are many problems which are efficiently solvable given a special and natural representation of a graph (i.e., a “model”), but which are open with respect to time complexity if the graph is given in a general form such as an adjacency list or an adjacency matrix. One such open problem is that of finding a maximum clique in unit disk graphs here, we give a polynomial time robust algorithm for this problem, 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: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.Finding a maximum clique in an input graph given in general form is NP-hard.