Survivable network design with degree or order constraints

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv+3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv +3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain’s iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost λ-edge-connected subgraph with at least k vertices, which we call the (k, λ)-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost λ-edgeconnected subgraph problems. We give a polylogarithmic approximation for the (k, 2)-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the (k, λ)-subgraph problem might be hard to approximate for arbitrary λ.

[1]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[2]  Ariel Orda,et al.  Maximum-Lifetime Routing in Wireless Networks : System Optimization & Game-Theoretic Perspectives , 2006 .

[3]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[4]  Santosh S. Vempala,et al.  A Constant-Factor Approximation Algorithm for the k-MST Problem , 1999, J. Comput. Syst. Sci..

[5]  Mohammad Taghi Hajiaghayi,et al.  Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees , 2006, APPROX-RANDOM.

[6]  Chandra Chekuri,et al.  Pruning 2-Connected Graphs , 2008, FSTTCS.

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

[8]  Universitat-Gesamthochschule Paderborn.  BOUNDED DEGREE SPANNING TREES , 1997 .

[9]  Martin Fürer,et al.  Approximating the Minimum-Degree Steiner Tree to within One of Optimal , 1994, J. Algorithms.

[10]  Rajeev Motwani,et al.  K-connected Spanning Subgraphs of Low Degree , 2006, Electron. Colloquium Comput. Complex..

[11]  Satish Rao,et al.  What Would Edmonds Do? Augmenting Paths and Witnesses for Degree-Bounded MSTs , 2009, Algorithmica.

[12]  R. Ravi,et al.  Primal-Dual Meets Local Search: Approximating MSTs With Nonuniform Degree Bounds , 2005, SIAM J. Comput..

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

[14]  R. Ravi,et al.  Approximation algorithms for finding low-degree subgraphs , 2004, Networks.

[15]  Robert Krauthgamer,et al.  Hardness of Approximation for Vertex-Connectivity Network Design Problems , 2002, SIAM J. Comput..

[16]  Mohit Singh,et al.  Delegate and Conquer: An LP-Based Approximation Algorithm for Minimum Degree MSTs , 2006, ICALP.

[17]  Michel X. Goemans,et al.  Minimum Bounded Degree Spanning Trees , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[18]  Nikhil Bansal,et al.  Additive guarantees for degree bounded directed network design , 2008, SIAM J. Comput..

[19]  Mohammad R. Salavatipour,et al.  A Constant Factor Approximation for Minimum lambda-Edge-Connected k-Subgraph with Metric Costs , 2008, APPROX-RANDOM.

[20]  Chandra Chekuri,et al.  Min-Cost 2-Connected Subgraphs With k Terminals , 2008, ArXiv.

[21]  David P. Williamson,et al.  Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems , 2006, J. Comput. Syst. Sci..

[22]  Harold N. Gabow,et al.  On the L∞-norm of extreme points for crossing supermodular directed network LPs , 2007, Math. Program..

[23]  Santosh S. Vempala,et al.  New Approximation Guarantees for Minimum-Weight k-Trees and Prize-Collecting Salesmen , 1999, SIAM J. Comput..

[24]  Mohit Singh,et al.  Degree bounded matroids and submodular flows , 2008, Comb..

[25]  R. Ravi,et al.  Quasi-polynomial Time Approximation Algorithm for Low-Degree Minimum-Cost Steiner Trees , 2003, FSTTCS.

[26]  Kamal Jain.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 2001, Comb..

[27]  Éva Tardos,et al.  Algorithms for a network design problem with crossing supermodular demands , 2004, Networks.

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

[29]  Mohammad Taghi Hajiaghayi,et al.  Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees , 2007, Algorithmica.

[30]  Santosh S. Vempala,et al.  Network Design via Iterative Rounding of Setpair Relaxations , 2022 .

[31]  R. Ravi,et al.  Primal-dual meets local search: approximating MST's with nonuniform degree bounds , 2003, STOC '03.

[32]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[33]  Mohammad Taghi Hajiaghayi,et al.  Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[34]  R. Ravi,et al.  A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees , 2000, STOC '00.

[35]  Ran Raz.  A Parallel Repetition Theorem , 1998, SIAM J. Comput..

[36]  Lachezar Krumov,et al.  Approximation Algorithm for the Minimum Degree Spanning Tree Problem , 2007 .

[37]  Nimrod Megiddo,et al.  Combinatorial optimization with rational objective functions , 1978, Math. Oper. Res..

[38]  R. Ravi,et al.  Spanning trees short or small , 1994, SODA '94.

[39]  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).

[40]  Uriel Feige,et al.  The Dense k -Subgraph Problem , 2001, Algorithmica.

[41]  Naveen Garg,et al.  A 3-approximation for the minimum tree spanning k vertices , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[42]  R. Ravi,et al.  Many birds with one stone: multi-objective approximation algorithms , 1993, STOC '93.

[43]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[44]  D. H. Younger,et al.  Integer flows , 1983, J. Graph Theory.

[45]  R. Ravi,et al.  Approximation Through Local Optimality: Designing Networks with Small Degree , 1992, FSTTCS.

[46]  Naveen Garg,et al.  Saving an epsilon: a 2-approximation for the k-MST problem in graphs , 2005, STOC '05.

[47]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .