On two open problems of 2-interval patterns

The 2-interval pattern problem, introduced in [Stéphane Vialette, On the computational complexity of 2-interval pattern matching problems Theoret. Comput. Sci. 312 (2–3) (2004) 223–249], models general problems with biological structures such as protein contact maps and macroscopic describers of secondary structures of ribonucleic acids. Given a set of 2-intervals D and a model R, the problem is to find a maximum cardinality subset D′ of D such that any two 2-intervals in D′ satisfy R, where R is a subset of relations on disjoint 2-intervals: precedence (<), nest (⊏), and cross (≬). The problem left unanswered at present is that of whether there is a polynomial time solution for the 2-interval pattern problem, when R={<,≬} and all the support intervals of D are disjoint. In this paper, we present a reduction from the clique problem to show that, in this case, the problem is NP-hard.The disjoint 2-interval pattern matching problem is to decide whether a disjoint 2-interval pattern (called the pattern) is a substructure of another disjoint 2-interval pattern (called the target). In general, the problem is NP-hard, but when there are restrictions on the form of the pattern, the problem can, in some cases, be solved in polynomial time. In particular, a polynomial time algorithm has been proposed (Gramm, WABI 2004 and IEEE/ACM TCBB 2004) for the case where the patterns are so-called crossing contact maps. In this paper we show that the problem is actually NP-hard and point out an error in the analysis of the above algorithm. 1