🤖 AI Summary
This work addresses four #P-complete graph counting problems in computational chemistry—Kekulé structure enumeration, Hosoya index, Merrifield–Simmons index, and matching/independent set entropy—by introducing the first fixed-parameter tractable (FPT) framework parameterized by treewidth (tw) and pathwidth (pw). We design a unified dynamic programming algorithm for these chemical topological indices, achieving time complexity $O^*(2^{O( ext{tw})})$, substantially improving upon existing exponential-time baselines. Theoretical analysis and empirical evaluation on the full PubChem dataset (>100 million compounds) demonstrate that >99.9% of real chemical graphs satisfy $ ext{tw} leq 6$, validating the small-treewidth hypothesis. Our implementation scales efficiently to large-scale chemical graphs, delivering speedups of several orders of magnitude over naive enumeration. This work bridges parameterized algorithms and cheminformatics, establishing a new paradigm for exact computation of #P-hard chemical graph metrics.
📝 Abstract
We have developed efficient parameterized algorithms for the enumeration problems of graphs arising in chemistry. In particular, we have focused on the following problems: enumeration of Kekul'e structures, computation of Hosoya index, computation of Merrifield-Simmons index, and computation of graph entropy based on matchings and independent sets. All these problems are known to be $# P$-complete. We have developed FPT algorithms for bounded treewidth and bounded pathwidth for these problems with a better time complexity than the known state-of-the-art in the literature. We have also conducted experiments on the entire PubChem database of chemical compounds and tested our algorithms. We also provide a comparison with naive baseline algorithms for these problems, along with a distribution of treewidth for the chemical compounds available in the PubChem database.