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We study three closely related problems motivated by the prediction of RNA secondary structures with arbitrary pseudoknots: the problem 2-lnterval Pattern proposed by Vialette, the problem Maximum Base Pair Stackings proposed by Leong et al., and the problem Maximum Stacking Base Pairs proposed by Lyngso. For the 2-lnterval Pattern, we present polynomial-time approximation algorithms for the problem over the preceding-and-crossing model and on input with the unitary restriction. For Maximum Base Pair Stackings and Maximum Stacking Base Pairs, we present polynomial-time approximation algorithms for the two problems on explicit input of candidate base pairs. We also propose a new problem called Length-Weighted Balanced 2-lnterval Pattern, which is natural in the context of RNA secondary structure prediction.