New Perspectives on Semiring Applications to Dynamic Programming
Counting minimum-cost solutions to NP-hard combinatorial optimization problems—such as Connected Dominating Set and Constraint Satisfaction—is computationally challenging due to the interplay between cost minimization and solution enumeration. Method: We propose a unified dynamic programming framework grounded in semiring algebra, centered on a novel Δ-product operation that relaxes the idempotence requirement of classical semirings, enabling efficient counting of minimum-cost solutions over non-idempotent semirings. The framework is parameterized by treewidth and clique-width to ensure tractability on structured inputs. Contribution/Results: We establish fixed-parameter tractable (FPT) enumeration for minimum-cost solutions under bounded treewidth or clique-width, proving polynomial-time solvability of the counting problem in these parameters. Our approach significantly extends classical DP’s expressiveness, unifying cost optimization and solution counting within a single algebraic model—thereby enabling rigorous analysis and efficient computation for previously intractable enumeration tasks.