Prize-Collecting Steiner Networks via Iterative Rounding

In this paper we design an iterative rounding approach for the classic prize-collecting Steiner forest problem and more generally the prize-collecting survivable Steiner network design problem. We show as an structural result that in each iteration of our algorithm there is an LP variable in a basic feasible solution which is at least one-third-integral resulting a 3-approximation algorithm for this problem. In addition, we show this factor 3 in our structural result is indeed tight for prize-collecting Steiner forest and thus prize-collecting survivable Steiner network design. This especially answers negatively the previous belief that one might be able to obtain an approximation factor better than 3 for these problems using a natural iterative rounding approach. Our structural result is extending the celebrated iterative rounding approach of Jain [13] by using several new ideas some from more complicated linear algebra. The approach of this paper can be also applied to get a constant factor (bicriteria-)approximation algorithm for degree constrained prize-collecting network design problems. We emphasize that though in theory we can prove existence of only an LP variable of at least one-third-integral, in practice very often in each iteration there exists a variable of integral or almost integral which results in a much better approximation factor than provable factor 3 in this paper (see patent application [11]). This is indeed the advantage of our algorithm in this paper over previous approximation algorithms for prize-collecting Steiner forest with the same or slightly better provable approximation factors.

[1]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[2]  R. Ravi,et al.  An efficient cost-sharing mechanism for the prize-collecting Steiner forest problem , 2007, SODA '07.

[3]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[4]  Tim Roughgarden,et al.  Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation , 2001, Math. Program..

[5]  Yossi Azar,et al.  Buy-at-bulk network design , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[6]  Tim Roughgarden,et al.  Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[7]  Mohit Singh,et al.  Additive approximation for bounded degree survivable network design , 2008, SIAM J. Comput..

[8]  David R. Karger,et al.  Building Steiner trees with incomplete global knowledge , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[9]  Tim Roughgarden,et al.  A constant-factor approximation algorithm for the multicommodity rent-or-buy problem , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[10]  Shai Gutner Elementary Approximation Algorithms for Prize Collecting Steiner Tree Problems , 2008, COCOA.

[11]  Mohit Singh,et al.  Survivable network design with degree or order constraints , 2007, STOC '07.

[12]  Nikhil Bansal,et al.  Additive Guarantees for Degree-Bounded Directed Network Design , 2009, SIAM J. Comput..

[13]  Luca Becchetti,et al.  Sharing the cost more efficiently: improved approximation for multicommodity rent-or-buy , 2005, SODA '05.

[14]  Kamal Jain,et al.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[15]  David P. Williamson,et al.  A note on the prize collecting traveling salesman problem , 1993, Math. Program..

[16]  R. Ravi,et al.  Approximating the Single-Sink Link-Installation Problem in Network Design , 2001, SIAM J. Optim..

[17]  Mohammad Taghi Hajiaghayi,et al.  The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema , 2006, SODA '06.

[18]  David S. Johnson,et al.  The prize collecting Steiner tree problem: theory and practice , 2000, SODA '00.

[19]  Egon Balas,et al.  The prize collecting traveling salesman problem , 1989, Networks.

[20]  Chaitanya Swamy,et al.  Approximation algorithms for prize collecting forest problems with submodular penalty functions , 2007, SODA '07.