Abstract This paper introduces k -d PCPs – the class of probabilistic counter programs with $$k \in \mathbb {N}$$ k ∈ N counter variables inducing possibly infinite-state Markov chains. We show that the universal (positive) almost-sure termination problem is undecidable for k -d PCPs in general, yet decidable for 1-d PCPs. We present an efficient decision procedure for the latter leveraging the technique of Markov chain finitization . Moreover, we identify several classes of k -d PCPs that are reducible to 1-d PCPs – thus their termination properties can be inferred automatically. Experiments demonstrate that our decision procedure can certify (positive) almost-sure termination – without resorting to invariants or supermartingales – of non-trivial probabilistic programs beyond the scope of existing tools.

We present efficient combinatorial parameterized algorithms for several classical graph-based counting problems in computational chemistry, including (i) Kekulé structures, (ii) the Hosoya index, (iii) the Merrifield–Simmons index, and (iv) Graph entropy based on matchings and independent sets. All these problems were known to be #P-complete. Building on the intuition that molecular graphs are often sparse and tree-like, we provide fixed-parameter tractable (FPT) algorithms using treewidth as our parameter. We also provide extensive experimental results over the entire PubChem database of chemical compounds, containing more than 113 million real-world molecules. In our experiments, we observe that the molecules are indeed sparse and tree-like, with more than 99.9% of them having a treewidth of at most 5. Our experiments also illustrate considerable improvements over the previous approaches. Based on these results, we argue that parameterized algorithms, especially based on treewidth, should be adopted as the default approach for problems in computational chemistry that are defined over molecular graphs.