Che Cheng | Searcharxiv

arXiv2026

On the Impact of Stability and the Helly Property on the Dominating Set Problem

We extend the algorithmic framework of progressive exploration [Fabia\'nski et al., STACS 2019], which yields simple, yet surprisingly general and efficient parameterized algorithms for Dominating Set, Independent Set, and some of their variants. While they identified stability and the Helly property as necessary for their approach, we show that -- with a simple change -- in the case of Dominating Set, one can get rid of the stability requirement. This yields a fixed-parameter tractable algorithm on exactly those graph classes which do not contain long co-matchings or double-ladders as semi-induced subgraphs. Lifting one of these two restrictions makes Dominating Set W[1]-hard on these classes. Our algorithm generalizes results on weakly $\gamma$-closed graphs, and results from Sparsity theory, e.g., nowhere dense and biclique-free classes. At the same time, we match the time complexity of the previously known algorithms on those classes. We demonstrate that this technique can easily be applied to the Distance-$r$ Dominating Set and the Set Cover problem.

cs.DS↗

arXiv2025

Model Counting for Dependency Quantified Boolean Formulas

Dependency Quantified Boolean Formulas (DQBF) generalize QBF by explicitly specifying which universal variables each existential variable depends on, instead of relying on a linear quantifier order. The satisfiability problem of DQBF is NEXP-complete, and many hard problems can be succinctly encoded as DQBF. Recent work has revealed a strong analogy between DQBF and SAT: k-DQBF (with k existential variables) is a succinct form of k-SAT, and satisfiability is NEXP-complete for 3-DQBF but PSPACE-complete for 2-DQBF, mirroring the complexity gap between 3-SAT (NP-complete) and 2-SAT (NL-complete). Motivated by this analogy, we study the model counting problem for DQBF, denoted #DQBF. Our main theoretical result is that #2-DQBF is #EXP-complete, where #EXP is the exponential-time analogue of #P. This parallels Valiant's classical theorem stating that #2-SAT is #P-complete. As a direct application, we show that first-order model counting (FOMC) remains #EXP-complete even when restricted to a PSPACE-decidable fragment of first-order logic and domain size two. Building on recent successes in reducing 2-DQBF satisfiability to symbolic model checking, we develop a dedicated 2-DQBF model counter. Using a diverse set of crafted instances, we experimentally evaluated it against a baseline that expands 2-DQBF formulas into propositional formulas and applies propositional model counting. While the baseline worked well when each existential variable depends on few variables, our implementation scaled significantly better to larger dependency sets.

cs.LO↗