On the Effective Enumerability of NP Problems

In the field of computational optimization, it is often the case that we are given an instance of an NP problem and asked to enumerate the first few ”best” solutions to the instance. Motivated by the recent research performed in these fields, we propose in this paper a new framework to measure the effective enumerability of NP optimization problems. More specifically, given an instance of an NP problem, we consider the problem of enumerating a given number of best solutions for the instance, and study its average complexity in terms of the number of solutions. Our framework is different from the previouslyproposed ones, which studied the counting complexity of a problem, or the complexity of enumerating all solutions to a given instance of the problem. For example, even though it was shown by Flum and Grohe that counting the number of k-paths in a graph is fixed-parameter intractable, we present a fixedparameter enumeration algorithm for the problem. The developed enumeration framework consists of two phases: the structure-generation phase and the solution-enumeration phase. We show that most algorithm-design techniques for fixed-parameter tractable problems, such as search trees, color coding, and bounded treewidth, can be transformed into techniques for the structure-generation phase. We design elegant enumeration techniques, and combine them with the use of effective data structures, to show how to generate small-size structures and enumerate them efficiently.