Efficient Counting of Maximal Independent Sets in Sparse Graphs
There are a number of problems that require the counting or the enumeration of all occurrences of a certain structure within a given data set. We consider one such problem, namely that of counting the number of maximal independent sets (MISs) in a graph. Along with its complement problem of counting all maximal cliques, this is a well studied problem with applications in several research areas.
We present a new efficient algorithm for counting all MISs suitable for sparse graphs. Similar to previous algorithms for this problem, our algorithm is based on branching and exhaustively considering vertices to be either in or out of the current MIS. What is new is that we consider the vertices in a predefined order so that it is likely that the graph will decompose into multiple connected components. When this happens, we show that it is sufficient to solve the problem for each connected component, thus considerably speeding up the algorithm. We have performed extensive experiments comparing our algorithm with the previous best algorithms for this problem using both real world as well as synthetic input graphs. The results from this show that our algorithm outperforms the other algorithms and that it enables the solution of graphs where other approaches are clearly infeasible.
As there is a one-to-one correspondence between the MISs of a graph and the maximal cliques of its complement graph, it follows that our algorithm also solves the problem of counting the number of maximal cliques in a dense graph. To our knowledge, this is the first algorithm that can handle this problem.
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