The Prize-collecting Steiner tree (PCST) problem is a generalization of the Steiner tree problem that finds applications in network design, content distribution networks, and many more. There are a few centralized approximation algorithms [D. Bienstock, M. X. Goemans, D. Simchi-Levi and D. Williamson, A note on the prize collecting traveling salesman problem. Math. Program. 59 (1993) 413–420; M. X. Goemans and D. E. Williamson, A general approximation technique for constrained forest problems, SIAM J. Appl. Math. 24(2) (1995) 296–317; D. S. Johnson, M. Minkoff and S. Phillips, The prize collecting Steiner tree problem: Theory and practice, in Proc. Eleventh Annual ACM-SIAM Symp. Discrete Algorithms, SODA ’00 (2000), pp. 760–769; A. Archer, M. Hossein Bateni and M. Taghi Hajiaghayi, Improved approximation algorithms for prize-collecting Steiner tree and TSP, SIAM J. Comput. 40(2) (2011) 309–332] for solving the PCST problem. However, the problem has seen very little progress in the distributed setting; to the best of our knowledge, the only distributed algorithms proposed so far are due to Rossetti [N. G. Rossetti, A first attempt on the distributed prize-collecting Steiner tree problem, M.Sc. thesis, University of Iceland, Reykjavik (2015)]: one of them fails to guarantee a constant approximation factor while the other one is essentially centralized. In this work, first, we present a deterministic [Formula: see text] factor distributed approximation algorithm (D-PCST algorithm) that constructs a PCST for a given connected undirected graph of [Formula: see text] nodes with non-negative edge weights and non-negative prize value for each node. The D-PCST algorithm is based on the primal-dual method and uses a technique of preserving dual constraints in a distributed manner, without relying on knowledge of the global structure of the network. For an input graph [Formula: see text], the round and message complexities of the D-PCST algorithm in the CONGEST model are [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] and [Formula: see text]. Furthermore, we modify the D-PCST algorithm and show that a [Formula: see text]-approximate PCST can be deterministically computed using [Formula: see text] rounds and [Formula: see text] messages in the CONGEST model, where [Formula: see text] is the unweighted diameter of [Formula: see text]. For networks with [Formula: see text], the modified D-PCST algorithm performs better than the original one in terms of the round complexity. Both the algorithms require [Formula: see text] bits of memory in each node, where [Formula: see text] is the maximum degree of a node in the graph.
On the Steiner Tree 3/2-Approximation for Quasi-Bipartite Graphs