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Compound Logics for Modification Problems
- Publication Year :
- 2021
-
Abstract
- We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or G-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle’s theorem does not apply.<br />LIPIcs, Vol. 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), pages 61:1-61:21
- Subjects :
- FOS: Computer and information sciences
Flat Wall theorem
Computer Science - Logic in Computer Science
Mathematics of computing → Graph algorithms
G.2.2
Theory of computation → Logic
Irrelevant vertex technique
05C83, 05C85, 68R10, 68Q19, 68Q27, 68Q25
Computer Science - Data Structures and Algorithms
FOS: Mathematics
Theory of computation → Parameterized complexity and exact algorithms
Mathematics - Combinatorics
Data Structures and Algorithms (cs.DS)
Graph minors
F.2.2
F.4.1
Logic in Computer Science (cs.LO)
Model-checking
TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES
Algorithmic meta-theorems
Monadic second-order logic
Combinatorics (math.CO)
First-order logic
Graph modification problems
MathematicsofComputing_DISCRETEMATHEMATICS
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....0f12af5fa1654ee16552e939224b7d36