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On the hardness of learning queries from tree structured data. (English) Zbl 1321.90144
Summary: The problem of learning queries from tree structured data is studied by this paper. A tree structured data is modeled as a node-labeled tree \(T\), and applying a query \(q\) on \(T\) will return a set \(q(T)\) which is a subset of nodes in \(T\). For a tree-node pair \((T,t)\) where \(t\) is a node in \(T\), \(q\) is called to accept the pair if \(t\in {q(T)}\), and reject the pair if \(t\notin {q(T)}\). For some query class \(\mathcal{L}\), given tree-node pair sets \(E_p\) and \(E_n\), the tree query learning problem is to find a query \(q\in \mathcal{L}\) such that (1) \(q\) rejects all pairs in \(E_n\), and (2) the size of pairs in \(E_p\) accepted by \(q\) is maximized. On four different query classes \(\mathcal Q^{/}\), \(\mathcal Q^{/,*}\), \(\mathcal Q^{/,//}\) and \(\mathcal Q^{/,[]}\), this paper studies the hardness of the corresponding tree query learning problems. For \(\mathcal Q^{/}\), a PTIME algorithm is given. For \(\mathcal Q^{,*}\) and \(\mathcal Q^{/,//}\), the NP-complete results are shown. For \(\mathcal Q^{/,[]}\), the problem is shown to be NP-hard by considering two constrained fragments of \(\mathcal Q^{/,[]}\). Also, for \(\mathcal Q^{/,*}\), \(\mathcal Q^{/,[]}\) and \(\mathcal Q^{/,//}\), it is shown that there are no \(n^{1-\epsilon}\)-approximation algorithms for any \(\epsilon >0\).
MSC:
90C35 Programming involving graphs or networks
Software:
XPath; XQuery
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