A locally-optimal structure is a combinatorial structure such as a maximal
independent set that cannot be improved by certain (greedy) local moves, even
though it may not be globally optimal. It is trivial to construct an
independent set in a graph. It is easy to (greedily) construct a maximal
independent set. However, it is NP-hard to construct a globally-optimal
(maximum) independent set. In general, constructing a locally-optimal structure
is somewhat more difficult than constructing an arbitrary structure, and
constructing a globally-optimal structure is more difficult than constructing a
locally-optimal structure. The same situation arises with listing. The
differences between the problems become obscured when we move from listing to
counting because nearly everything is #P-complete. However, we highlight an
interesting phenomenon that arises in approximate counting, where the situation
is apparently reversed. Specifically, we show that counting maximal independent
sets is complete for #P with respect to approximation-preserving reductions,
whereas counting all independent sets, or counting maximum independent sets is
complete for an apparently smaller class, #RHΠ1​ which has a
prominent role in the complexity of approximate counting. Motivated by the
difficulty of approximately counting maximal independent sets in bipartite
graphs, we also study the problem of approximately counting other
locally-optimal structures that arise in algorithmic applications, particularly
problems involving minimal separators and minimal edge separators. Minimal
separators have applications via fixed-parameter-tractable algorithms for
constructing triangulations and phylogenetic trees. Although exact
(exponential-time) algorithms exist for listing these structures, we show that
the counting problems are #P-complete with respect to both exact and
approximation-preserving reductions.Comment: Accepted to JCSS, preliminary version accepted to ICALP 2015 (Track
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