Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity
This paper settles the computational complexity of model checking of several extensions of the monadic second order (MSO) logic on two classes of graphs: graphs of bounded treewidth and graphs of bounded neighborhood diversity. A classical theorem of Courcelle states that any graph property defina...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V.
2019-12-01
|
| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/5017/pdf |
| _version_ | 1827322776809111552 |
|---|---|
| author | Dušan Knop Martin Koutecký Tomáš Masařík Tomáš Toufar |
| author_facet | Dušan Knop Martin Koutecký Tomáš Masařík Tomáš Toufar |
| author_sort | Dušan Knop |
| collection | DOAJ |
| description | This paper settles the computational complexity of model checking of several
extensions of the monadic second order (MSO) logic on two classes of graphs:
graphs of bounded treewidth and graphs of bounded neighborhood diversity.
A classical theorem of Courcelle states that any graph property definable in
MSO is decidable in linear time on graphs of bounded treewidth. Algorithmic
metatheorems like Courcelle's serve to generalize known positive results on
various graph classes. We explore and extend three previously studied MSO
extensions: global and local cardinality constraints (CardMSO and MSO-LCC) and
optimizing the fair objective function (fairMSO).
First, we show how these extensions of MSO relate to each other in their
expressive power. Furthermore, we highlight a certain "linearity" of some of
the newly introduced extensions which turns out to play an important role.
Second, we provide parameterized algorithm for the aforementioned structural
parameters. On the side of neighborhood diversity, we show that combining the
linear variants of local and global cardinality constraints is possible while
keeping the linear (FPT) runtime but removing linearity of either makes this
impossible. Moreover, we provide a polynomial time (XP) algorithm for the most
powerful of studied extensions, i.e. the combination of global and local
constraints. Furthermore, we show a polynomial time (XP) algorithm on graphs of
bounded treewidth for the same extension. In addition, we propose a general
procedure of deriving XP algorithms on graphs on bounded treewidth via
formulation as Constraint Satisfaction Problems (CSP). This shows an alternate
approach as compared to standard dynamic programming formulations. |
| first_indexed | 2024-04-25T01:34:16Z |
| format | Article |
| id | doaj.art-34331dbfa1ed49a093f248c8577e3444 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:34:16Z |
| publishDate | 2019-12-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-34331dbfa1ed49a093f248c8577e34442024-03-08T10:29:38ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742019-12-01Volume 15, Issue 410.23638/LMCS-15(4:12)20195017Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood DiversityDušan Knophttps://orcid.org/0000-0003-2588-5709Martin Kouteckýhttps://orcid.org/0000-0002-7846-0053Tomáš Masaříkhttps://orcid.org/0000-0001-8524-4036Tomáš Toufarhttps://orcid.org/0000-0003-0007-9508This paper settles the computational complexity of model checking of several extensions of the monadic second order (MSO) logic on two classes of graphs: graphs of bounded treewidth and graphs of bounded neighborhood diversity. A classical theorem of Courcelle states that any graph property definable in MSO is decidable in linear time on graphs of bounded treewidth. Algorithmic metatheorems like Courcelle's serve to generalize known positive results on various graph classes. We explore and extend three previously studied MSO extensions: global and local cardinality constraints (CardMSO and MSO-LCC) and optimizing the fair objective function (fairMSO). First, we show how these extensions of MSO relate to each other in their expressive power. Furthermore, we highlight a certain "linearity" of some of the newly introduced extensions which turns out to play an important role. Second, we provide parameterized algorithm for the aforementioned structural parameters. On the side of neighborhood diversity, we show that combining the linear variants of local and global cardinality constraints is possible while keeping the linear (FPT) runtime but removing linearity of either makes this impossible. Moreover, we provide a polynomial time (XP) algorithm for the most powerful of studied extensions, i.e. the combination of global and local constraints. Furthermore, we show a polynomial time (XP) algorithm on graphs of bounded treewidth for the same extension. In addition, we propose a general procedure of deriving XP algorithms on graphs on bounded treewidth via formulation as Constraint Satisfaction Problems (CSP). This shows an alternate approach as compared to standard dynamic programming formulations.https://lmcs.episciences.org/5017/pdfcomputer science - computational complexitycomputer science - logic in computer sciencef.2.2, g.2.2f.2.2g.2.2 |
| spellingShingle | Dušan Knop Martin Koutecký Tomáš Masařík Tomáš Toufar Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity Logical Methods in Computer Science computer science - computational complexity computer science - logic in computer science f.2.2, g.2.2 f.2.2 g.2.2 |
| title | Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity |
| title_full | Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity |
| title_fullStr | Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity |
| title_full_unstemmed | Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity |
| title_short | Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity |
| title_sort | simplified algorithmic metatheorems beyond mso treewidth and neighborhood diversity |
| topic | computer science - computational complexity computer science - logic in computer science f.2.2, g.2.2 f.2.2 g.2.2 |
| url | https://lmcs.episciences.org/5017/pdf |
| work_keys_str_mv | AT dusanknop simplifiedalgorithmicmetatheoremsbeyondmsotreewidthandneighborhooddiversity AT martinkoutecky simplifiedalgorithmicmetatheoremsbeyondmsotreewidthandneighborhooddiversity AT tomasmasarik simplifiedalgorithmicmetatheoremsbeyondmsotreewidthandneighborhooddiversity AT tomastoufar simplifiedalgorithmicmetatheoremsbeyondmsotreewidthandneighborhooddiversity |