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Approximation schemes for firstorder definable optimisation problems
 In Proc. LICS’06
, 2006
"... Let ϕ(X) be a firstorder formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in ϕ(X). Then a natural minimisation problem associated with ϕ(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies ϕ(S). Similarly, ..."
Abstract

Cited by 13 (8 self)
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Let ϕ(X) be a firstorder formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in ϕ(X). Then a natural minimisation problem associated with ϕ(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies ϕ(S). Similarly, if X only occurs negatively in ϕ(X), then ϕ(X) defines a maximisation problem. Many wellknown optimisation problems are firstorder 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 firstorder 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. 1.