New Perspectives on Semiring Applications to Dynamic Programming

2025-12-03

AI Summary

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.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Semiring algebras have been shown to provide a suitable language to formalize many noteworthy combinatorial problems. For instance, the Shortest-Path problem can be seen as a special case of the Algebraic-Path problem when applied to the tropical semiring. The application of semirings typically makes it possible to solve extended problems without increasing the computational complexity. In this article we further exploit the idea of using semiring algebras to address and tackle several extensions of classical computational problems by dynamic programming. We consider a general approach which allows us to define a semiring extension of any problem with a reasonable notion of a certificate (e.g., an NP problem). This allows us to consider cost variants of these combinatorial problems, as well as their counting extensions where the goal is to determine how many solutions a given problem admits. The approach makes no particular assumptions (such as idempotence) on the semiring structure. We also propose a new associative algebraic operation on semirings, called $Delta$-product, which enables our dynamic programming algorithms to count the number of solutions of minimal costs. We illustrate the advantages of our framework on two well-known but computationally very different NP-hard problems, namely, Connected-Dominating-Set problems and finite-domain Constraint Satisfaction Problems (CSPs). In particular, we prove fixed parameter tractability (FPT) with respect to clique-width and tree-width of the input. This also allows us to count solutions of minimal cost, which is an overlooked problem in the literature.

Problem

• Extends combinatorial problems via semiring algebras
• Enables counting solutions and cost variants efficiently
• Applies to NP-hard problems with FPT guarantees

Innovation

• General semiring extension for NP problems
• Delta-product for counting minimal cost solutions
• Fixed parameter tractability for clique-width and tree-width