On the hardness of learning queries from tree structured data

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 , and applying a query on will return a set which is a subset of nodes in . For a tree-node pair where is a node in , is called to accept the pair if , and reject the pair if . For some query class , given tree-node pair sets and , the tree query learning problem is to find a query such that (1) rejects all pairs in , and (2) the size of pairs in accepted by is maximized. On four different query classes , , and , this paper studies the hardness of the corresponding tree query learning problems. For , a PTime algorithm is given. For and , the NP-complete results are shown. For , the problem is shown to be NP-hard by considering two constrained fragments of . Also, for , and , it is shown that there are no -approximation algorithms for any epsilon > 0.