The Hardness of Sampling Connected Subgraphs

Abstract

We consider the problem of sampling connected induced subgraphs of a given input graph G. Our first result is that an efficient algorithm to approximately sample connected induced subgraphs of a given size (the size is specified in the input) does not exist unless RP = NP. We then focus on the problem of approximately sampling connected induced subgraphs with a bias, more precisely we consider a distribution where the probability of a connected subgraph induced by S⊆ V(G) is proportional to λ|S|. When the input graph G has maximum degree d we identify a threshold λd=(d-1)(d-1)dd. For 0 < λ< λd there exists a trivial efficient sampler for the problem, and for λd< λ< 1 an efficient approximate sampler does not exist unless RP = NP. Finally, we show local Markov chains are unlikely to be effective at approximately sampling connected subgraphs.