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# Faster algorithms for quantitative verification in constant treewidth graphs

Chatterjee K, Ibsen-Jensen R, Pavlogiannis A. 2015. Faster algorithms for quantitative verification in constant treewidth graphs, IST Austria, 31p.

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*Technical Report*|

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*English*

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IST Austria Technical Report

Abstract

We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean- payoff property, the ratio property, 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 constant treewidth, and it is well-known that the control-flow graphs of most programs have constant treewidth. Let n denote the number of nodes of a graph, m the number of edges (for constant treewidth graphs m = O ( n ) ) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for constant treewidth graphs we present an algorithm that approximates the mean-payoff value within a mul- tiplicative factor of ∊ in time O ( n · log( n/∊ )) and linear space, as compared to the classical algorithms that require quadratic time. Second, for the ratio property we present an algorithm that for constant treewidth graphs works in time O ( n · log( | a · b · n | )) = O ( n · log( n · W )) , when the output is a b , as compared to the previously best known algorithm with running time O ( n 2 · log( n · W )) . Third, for the minimum initial credit problem we show that (i) for general graphs the problem can be solved in O ( n 2 · m ) time and the associated decision problem can be solved in O ( n · m ) time, improving the previous known O ( n 3 · m · log( n · W )) and O ( n 2 · m ) bounds, respectively; and (ii) for constant treewidth graphs we present an algorithm that requires O ( n · log n ) time, improving the previous known O ( n 4 · log( n · W )) bound. We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks.

Publishing Year

Date Published

2015-02-10

Page

31

ISSN

IST-REx-ID

### Cite this

Chatterjee K, Ibsen-Jensen R, Pavlogiannis A.

*Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs*. IST Austria; 2015. doi:10.15479/AT:IST-2015-319-v1-1Chatterjee, K., Ibsen-Jensen, R., & Pavlogiannis, A. (2015).

*Faster algorithms for quantitative verification in constant treewidth graphs*. IST Austria. https://doi.org/10.15479/AT:IST-2015-319-v1-1Chatterjee, Krishnendu, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis.

*Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs*. IST Austria, 2015. https://doi.org/10.15479/AT:IST-2015-319-v1-1.K. Chatterjee, R. Ibsen-Jensen, and A. Pavlogiannis,

*Faster algorithms for quantitative verification in constant treewidth graphs*. IST Austria, 2015.Chatterjee, Krishnendu, et al.

*Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs*. IST Austria, 2015, doi:10.15479/AT:IST-2015-319-v1-1.**All files available under the following license(s):**

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