Algorithmic Meta-Theorems for Monotone Submodular Maximization
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We consider a monotone submodular maximization problem whose constraint is described by a logic formula on a graph. Formally, we prove the following three `algorithmic metatheorems.' (1) If the constraint is specified by a monadic second-order logic on a graph of bounded treewidth, the problem is solved in n^O(1) time with an approximation factor of O( n). (2) If the constraint is specified by a first-order logic on a graph of low degree, the problem is solved in O(n^1 + ϵ) time for any ϵ > 0 with an approximation factor of 2. (3) If the constraint is specified by a first-order logic on a graph of bounded expansion, the problem is solved in n^O( k) time with an approximation factor of O( k), where k is the number of variables and O(·) suppresses only constants independent of k.
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