In this paper, we introduce the study of prize-collecting network design problems having general connectivity requirements. Prior work considered only 0-1 or very limited connectivity requirements. We introduce general connectivity requirements in the prize-collecting generalized Steiner tree framework of Hajiaghayi and Jain [9], and consider penalty functions linear in the violation of the connectivity requirements. Using Jain's iterated rounding algorithm [11] as a black box, and ideas from Goemans [7] and Levi, Lodi, Sviridenko [14], we give a 2.54-factor approximation algorithm for the problem. We also generalize the 0-1 requirements of PCF problem introduced by Sharma, Swamy, and Williamson [15] to include general connectivity requirements. Here we assume that the monotone submodular penalty function of Sharma et al. is generalized to a multiset function that can be decomposed into functions in the same form as that of Sharma et al. Using ideas from Goemans and Berstimas [6], we give an (α log K)-approximation algorithm for the resulting problem, where K is the maximum connectivity requirement, and α = 2.54.
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