On the hardness of learning queries from tree structured data

Abstract

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 $\mathcal Q ^{\tiny /}$, $\mathcal Q ^{\tiny /,*}$, $\mathcal Q ^{\tiny /,//}$ and $\mathcal Q ^{\tiny /,[]}$, this paper studies the hardness of the corresponding tree query learning problems. For $\mathcal Q ^{\tiny /}$, a PTime algorithm is given. For $\mathcal Q ^{\tiny /,*}$ and $\mathcal Q ^{\tiny /,//}$, the NP-complete results are shown. For $\mathcal Q ^{\tiny /,[]}$, the problem is shown to be NP-hard by considering two constrained fragments of $\mathcal Q ^{\tiny /,[]}$. Also, for $\mathcal Q ^{\tiny /,*}$, $\mathcal Q ^{\tiny /,[]}$ and $\mathcal Q ^{\tiny /,//}$, it is shown that there are no $n^{1-\epsilon }$-approximation algorithms for any $\epsilon >0$.