Hard Optimization Problems in Learning Tree Contraction Patterns

Abstract

A tree contraction pattern (TC-pattern) is an unordered tree-structured pattern common to given unordered trees, which is obtained by merging every uncommon connected substructure into one vertex by edge contraction. In order to extract meaningful and hidden knowledge from tree structured documents, we consider a minimal language (MINL) problem for TC-patterns . The MINL problem for TC-patterns is to find a TC-pattern $t$ such that the language generated by $t$ is minimal among languages, generated by TC-patterns, which contain all given unordered trees. Recently, [8] showed that the MINL problem for TC-patterns is computable in polynomial time if there are infinitely many vertex labels. In this chapter, we discuss two optimization versions of the MINL problem, which are called MINL with Tree-size Maximization (MAX MINL) and MINL with Variable-size Minimization (MIN-MAX MINL). We show that MAX MINL is NP-complete and MIN-MAX MINL is MAX SNP-hard.