# Faster algorithms for quantitative verification in bounded treewidth graphs

Chatterjee K, Ibsen-Jensen R, Pavlogiannis A. 2021. Faster algorithms for quantitative verification in bounded treewidth graphs. Formal Methods in System Design.

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*English*

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Abstract

We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean-payoff, the ratio, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with bounded treewidthβa class that contains the control flow graphs of most programs. Let n denote the number of nodes of a graph, m the number of edges (for bounded treewidth π=π(π)) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for the minimum initial credit problem we show that (1) for general graphs the problem can be solved in π(π2β
π) time and the associated decision problem in π(πβ
π) time, improving the previous known π(π3β
πβ
log(πβ
π)) and π(π2β
π) bounds, respectively; and (2) for bounded treewidth graphs we present an algorithm that requires π(πβ
logπ) time. Second, for bounded treewidth graphs we present an algorithm that approximates the mean-payoff value within a factor of 1+π in time π(πβ
log(π/π)) as compared to the classical exact algorithms on general graphs that require quadratic time. Third, for the ratio property we present an algorithm that for bounded treewidth graphs works in time π(πβ
log(|πβ
π|))=π(πβ
log(πβ
π)), when the output is ππ, as compared to the previously best known algorithm on general graphs with running time π(π2β
log(πβ
π)). We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks.

Publishing Year

Date Published

2021-04-29

Journal Title

Formal Methods in System Design

Acknowledgement

The research was partly supported by Austrian Science Fund (FWF) Grant No P23499- N23, FWF NFN Grant No S11407-N23 (RiSE/SHiNE), ERC Start Grant (279307: Graph Games), and Microsoft faculty fellows award.

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### Cite this

Chatterjee K, Ibsen-Jensen R, Pavlogiannis A. Faster algorithms for quantitative verification in bounded treewidth graphs.

*Formal Methods in System Design*. 2021. doi:10.1007/s10703-021-00373-5Chatterjee, K., Ibsen-Jensen, R., & Pavlogiannis, A. (2021). Faster algorithms for quantitative verification in bounded treewidth graphs.

*Formal Methods in System Design*. Springer. https://doi.org/10.1007/s10703-021-00373-5Chatterjee, Krishnendu, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. βFaster Algorithms for Quantitative Verification in Bounded Treewidth Graphs.β

*Formal Methods in System Design*. Springer, 2021. https://doi.org/10.1007/s10703-021-00373-5.K. Chatterjee, R. Ibsen-Jensen, and A. Pavlogiannis, βFaster algorithms for quantitative verification in bounded treewidth graphs,β

*Formal Methods in System Design*. Springer, 2021.Chatterjee, Krishnendu, et al. βFaster Algorithms for Quantitative Verification in Bounded Treewidth Graphs.β

*Formal Methods in System Design*, Springer, 2021, doi:10.1007/s10703-021-00373-5.