--- _id: '10629' abstract: - lang: eng text: "Product graphs arise naturally in formal verification and program analysis. For example, the analysis of two concurrent threads requires the product of two component control-flow graphs, and for language inclusion of deterministic automata the product of two automata is constructed. In many cases, the component graphs have constant treewidth, e.g., when the input contains control-flow graphs of programs. We consider the algorithmic analysis of products of two constant-treewidth graphs with respect to three classic specification languages, namely, (a) algebraic properties, (b) mean-payoff properties, and (c) initial credit for energy properties.\r\nOur main contributions are as follows. Consider a graph G that is the product of two constant-treewidth graphs of size n each. First, given an idempotent semiring, we present an algorithm that computes the semiring transitive closure of G in time Õ(n⁴). Since the output has size Θ(n⁴), our algorithm is optimal (up to polylog factors). Second, given a mean-payoff objective, we present an O(n³)-time algorithm for deciding whether the value of a starting state is non-negative, improving the previously known O(n⁴) bound. Third, given an initial credit for energy objective, we present an O(n⁵)-time algorithm for computing the minimum initial credit for all nodes of G, improving the previously known O(n⁸) bound. At the heart of our approach lies an algorithm for the efficient construction of strongly-balanced tree decompositions of constant-treewidth graphs. Given a constant-treewidth graph G' of n nodes and a positive integer λ, our algorithm constructs a binary tree decomposition of G' of width O(λ) with the property that the size of each subtree decreases geometrically with rate (1/2 + 2^{-λ})." alternative_title: - LIPIcs article_number: '42' article_processing_charge: No author: - first_name: Krishnendu full_name: Chatterjee, Krishnendu id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87 last_name: Chatterjee orcid: 0000-0002-4561-241X - first_name: Rasmus full_name: Ibsen-Jensen, Rasmus id: 3B699956-F248-11E8-B48F-1D18A9856A87 last_name: Ibsen-Jensen orcid: 0000-0003-4783-0389 - first_name: Andreas full_name: Pavlogiannis, Andreas id: 49704004-F248-11E8-B48F-1D18A9856A87 last_name: Pavlogiannis orcid: 0000-0002-8943-0722 citation: ama: 'Chatterjee K, Ibsen-Jensen R, Pavlogiannis A. Quantitative verification on product graphs of small treewidth. In: 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Vol 213. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2021. doi:10.4230/LIPIcs.FSTTCS.2021.42' apa: 'Chatterjee, K., Ibsen-Jensen, R., & Pavlogiannis, A. (2021). Quantitative verification on product graphs of small treewidth. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (Vol. 213). Virtual: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.FSTTCS.2021.42' chicago: Chatterjee, Krishnendu, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. “Quantitative Verification on Product Graphs of Small Treewidth.” In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, Vol. 213. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.FSTTCS.2021.42. ieee: K. Chatterjee, R. Ibsen-Jensen, and A. Pavlogiannis, “Quantitative verification on product graphs of small treewidth,” in 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, Virtual, 2021, vol. 213. ista: 'Chatterjee K, Ibsen-Jensen R, Pavlogiannis A. 2021. Quantitative verification on product graphs of small treewidth. 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. FSTTCS: Foundations of Software Technology and Theoretical Computer Science, LIPIcs, vol. 213, 42.' mla: Chatterjee, Krishnendu, et al. “Quantitative Verification on Product Graphs of Small Treewidth.” 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, vol. 213, 42, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:10.4230/LIPIcs.FSTTCS.2021.42. short: K. Chatterjee, R. Ibsen-Jensen, A. Pavlogiannis, in:, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. conference: end_date: 2021-12-17 location: Virtual name: 'FSTTCS: Foundations of Software Technology and Theoretical Computer Science' start_date: 2021-12-15 date_created: 2022-01-16T23:01:28Z date_published: 2021-11-29T00:00:00Z date_updated: 2022-01-17T10:39:40Z day: '29' ddc: - '000' department: - _id: KrCh doi: 10.4230/LIPIcs.FSTTCS.2021.42 file: - access_level: open_access checksum: 71141acdeffa9056f24d6dbef952d254 content_type: application/pdf creator: cchlebak date_created: 2022-01-17T10:36:08Z date_updated: 2022-01-17T10:36:08Z file_id: '10633' file_name: 2021_LIPIcs_Chatterjee.pdf file_size: 891566 relation: main_file success: 1 file_date_updated: 2022-01-17T10:36:08Z has_accepted_license: '1' intvolume: ' 213' language: - iso: eng month: '11' oa: 1 oa_version: Published Version publication: 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science publication_identifier: isbn: - 978-3-9597-7215-0 issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: Quantitative verification on product graphs of small treewidth tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 volume: 213 year: '2021' ...