A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics

We present a unified polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space, whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be connected for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known. Our unified PTAS is based on the previous dynamic programming frameworks proposed in [Talwar STOC 2004] and [Bartal, Gottlieb, Krauthgamer STOC 2012]. However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

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

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

[3]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[4]  T.-H. Hubert Chan,et al.  A PTAS for the Steiner Forest Problem in Doubling Metrics , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[5]  Lee-Ad Gottlieb,et al.  The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme , 2011, STOC '12.

[6]  Kunal Talwar,et al.  Bypassing the embedding: algorithms for low dimensional metrics , 2004, STOC '04.

[7]  Mohammad Taghi Hajiaghayi,et al.  Prize-collecting Steiner problems on planar graphs , 2011, SODA '11.

[8]  Robert Krauthgamer,et al.  Bounded geometries, fractals, and low-distortion embeddings , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[9]  Mohammad Taghi Hajiaghayi,et al.  Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[10]  T.-H. Hubert Chan,et al.  Reducing Curse of Dimensionality , 2016, SODA.

[11]  Ken-ichi Kawarabayashi,et al.  Contraction decomposition in h-minor-free graphs and algorithmic applications , 2011, STOC '11.

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

[13]  Ittai Abraham,et al.  Advances in metric embedding theory , 2006, STOC '06.

[14]  Li Ning,et al.  New Doubling Spanners: Better and Simpler , 2013, SIAM J. Comput..

[15]  Mohammad Taghi Hajiaghayi,et al.  Euclidean Prize-Collecting Steiner Forest , 2011, Algorithmica.

[16]  Philip N. Klein,et al.  A Polynomial-Time Approximation Scheme for Euclidean Steiner Forest , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[17]  Kenneth L. Clarkson,et al.  Nearest Neighbor Queries in Metric Spaces , 1999, Discret. Comput. Geom..

[18]  T.-H. Hubert Chan,et al.  A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics , 2020, ACM Trans. Algorithms.

[19]  P. Assouad.  Plongements lipschitziens dans Rn , 2003 .

[20]  Michel X. Goemans.  Combining Approximation Algorithms for the Prize-Collecting TSP , 2009, ArXiv.