Approximately Counting Locally-Optimal Structures

A

locally-optimal

structure is a combinatorial structure that cannot be improved by certain (greedy) local moves, even though it may not be globally optimal. An example is a maximal independent set in a graph. 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.This situation is typical. 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

$$\#\text {P} $$

#

P

-complete. However, we highlight an interesting phenomenon that arises in approximate counting, where approximately counting locally-optimal structures is apparently more difficult than approximately counting globally-optimal structures. Specifically, we show that counting maximal independent sets is complete for

$$\#\text {P} $$

#

P

with respect to approximation-preserving reductions, whereas counting all independent sets, or counting maximum independent sets is complete for an apparently smaller class, #RH

$$\varPi _1$$

Π

1

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 counting problems involving minimal separators and minimal edge separators (which are also locally-optimal structures). 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 as hard as they could possibly be. All of the exact counting problems are

$$\#\text {P} $$

#

P

-complete, and all of the approximation problems are complete for

$$\#\text {P} $$

#

P

with respect to approximation-preserving reductions. A full version [

14

] containing detailed proofs is available at

http://arxiv.org/abs/1411.6829

. Theorem-numbering here matches the full version.