Parameterized and approximation complexity of Partial VC Dimension

Abstract

We introduce the problem Partial VC Dimension that asks, given a hypergraph H = (X, E) and integers k and ℓ, whether one can select a set C ⊆ X of k vertices of H such that the set {e ∩ C, e ∈ E} of distinct hyperedge-intersections with C has size at least ℓ. The sets e ∩ C define equivalence classes over E. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case ℓ = 2 k , and of Distinguishing Transversal, which corresponds to the case ℓ = |E| (the latter is also known as Test Cover in the dual hypergraph). We also introduce the associated fixed-cardinality maximization problem Max Partial VC Dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices. We study the algorithmic complexity of Partial VC Dimension and Max Partial VC Dimension both on general hypergraphs and on more restricted instances, in particular, neighborhood hypergraphs of graphs.⋆ A preliminary version of this work containing the approximation complexity results appeared in [8].

Fixed-Parameter Tractable Algorithms for Tracking Set Problems

“…We first study d-Tracking Set, which is a restricted version of Tracking Set System where the size of subsets in the family is restricted to d. We show d-Tracking Set to be NP-hard by showing a correlation with the problems Identifying Vertex Cover and Packing [7], [8]. We then give a compression Tracking Set System is known to be related to Test Cover [10]. Test Cover requires finding a subfamily of sets in a set system, that can help identify each element in the universe uniquely by inclusion.…”

“…Some results by Crowston et al [11] on Test Cover have been summarized in the following theorem. Test Cover is known to be a dual of Tracking Set System [10], as explained in the following lemmas. Gutin et al [24] have shown that there does not exist a polynomial kernel for Test Cover when parameterized by solution size, under standard complexity theory assumptions.…”

“…Hitting Set was one of Karp's original NP-complete problems [6]. Tracking Set System is known to be related to Test Cover [10]. Test Cover requires finding a subfamily of sets in a set system, that can help identify each element in the universe uniquely by inclusion.…”

“…(iii) Determining whether there exists a test cover of size at most (n − k) is fixed-parameter tractable. (iv) Determining whether there exists a test cover of size at most (lg n + k) is hard for the parameterized complexity class W[2].Test Cover is known to be a dual of Tracking Set System[10], as explained in the following lemmas.Lemma 5. Let {X, S} where X = {1, • • • , n} and S = {S 1 , S 2 , ..., S m }, be an instance of Test Cover.…”

Parallel Machine Learning Algorithm Using Fine-Grained-Mode Spark on a Mesos Big Data Cloud Computing Software Framework for Mobile Robotic Intelligent Fault Recognition

“…SUPPORT VECTOR MACHINE LEARNING FOR ROLLER BEARING FAULT DIAGNOSIS PROCESSA. SUPPORT VECTOR MACHIMECorinna Cortes and Vapnik developed the currently used SVM technique to minimize the VC dimension[27][28][29][30][31]. The hyperplane, which separates data (…”