Abstract
Product graphs arise naturally in formal verification and program analysis. For example, the analysis of two concurrent threads requires the product of two component controlflow 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 controlflow graphs of programs. We consider the algorithmic analysis of products of two constanttreewidth graphs with respect to three classic specification languages, namely, (a) algebraic properties, (b) meanpayoff properties, and (c) initial credit for energy properties.
Our main contributions are as follows. Consider a graph G that is the product of two constanttreewidth 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 meanpayoff objective, we present an O(n³)time algorithm for deciding whether the value of a starting state is nonnegative, 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 stronglybalanced tree decompositions of constanttreewidth graphs. Given a constanttreewidth 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}).
BibTeX  Entry
@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2021.42,
author = {Chatterjee, Krishnendu and IbsenJensen, Rasmus and Pavlogiannis, Andreas},
title = {{Quantitative Verification on Product Graphs of Small Treewidth}},
booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
pages = {42:142:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772150},
ISSN = {18688969},
year = {2021},
volume = {213},
editor = {Boja\'{n}czy, Miko{\l}aj and Chekuri, Chandra},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15553},
URN = {urn:nbn:de:0030drops155533},
doi = {10.4230/LIPIcs.FSTTCS.2021.42},
annote = {Keywords: graph algorithms, algebraic paths, meanpayoff, initial credit for energy}
}
Keywords: 

graph algorithms, algebraic paths, meanpayoff, initial credit for energy 
Collection: 

41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021) 
Issue Date: 

2021 
Date of publication: 

29.11.2021 