On the complexity of finding emerging patterns

Emerging patterns have been studied as a useful type of pattern for the diagnosis and understanding of diseases based on the analysis of gene expression profiles. They are useful for capturing interactions among genes (or other biological entities), for capturing signature patterns for disease subtypes, and deriving potential disease treatment plans, etc. We study the complexity of finding emerging patterns (with the highest frequency). We first show that the problem is MAX SNP-hard. This implies that polynomial time approximation schemes do not exist for the problem unless P = NP. We then prove that for any constant δ < 1, the emerging pattern problem cannot be approximated within ratio 2^logδ in polynomial time unless NP ⊆ DTIME[2^polylogn], where n is the number of positions in a pattern.