Graph-Theoretic Concepts in Computer Science
Universal augmentation schemes for network navigability
Theoretical Computer Science
Complexity and approximation results for the connected vertex cover problem
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
IWPEC'08 Proceedings of the 3rd international conference on Parameterized and exact computation
On brambles, grid-like minors, and parameterized intractability of monadic second-order logic
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Algorithmic meta-theorems for restrictions of treewidth
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On the approximability of some degree-constrained subgraph problems
Discrete Applied Mathematics
Finite model theory on tame classes of structures
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
The power of counting logics on restricted classes of finite structures
CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
Boundedness of monadic FO over acyclic structures
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Model theory makes formulas large
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Model checking lower bounds for simple graphs
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
An Optimal Gaifman Normal Form Construction for Structures of Bounded Degree
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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Let \varphi(X) be a first-order formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in \varphi(X). Then a natural minimisation problem associated with \varphi(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies \varphi(S). Similarly, if X only occurs negatively in \varphi(X), then \varphi(X) defines a maximisation problem. Many well-known optimisation problems are first-order definable in this sense, for example, MINIMUM DOMINATING SET or MAXIMUM INDEPENDENT SET. We prove that for each class C of graphs with excluded minors, in particular for each class of planar graphs, the restriction of a first-order definable optimisation problem to the class C has a polynomial time approximation scheme. A crucial building block of the proof of this approximability result is a version of Gaifman's locality theorem for formulas positive in a set variable. This result may be of independent interest.