Approximating k-forest with resource augmentation: A primal-dual approach

In this paper, we study the k-forest problem in the model of resource augmentation. In the k-forest problem, given an edge-weighted graph G(V, E), a parameter k, and a set of m demand pairs \(\subseteq V \times V\), the objective is to construct a minimum-cost subgraph that connects at least k demands. The problem is hard to approximate—the best-known approximation ratio is \(O(\min \{\sqrt{n}, \sqrt{k}\})\). Furthermore, k-forest is as hard to approximate as the notoriously-hard densest k-subgraph problem.

[1]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[2]  Philip N. Klein,et al.  Approximation algorithms for NP-hard optimization problems , 2010 .

[3]  R. Ravi,et al.  Dial a Ride from k-forest , 2007, TALG.

[4]  Aditya Bhaskara,et al.  Detecting high log-densities: an O(n¼) approximation for densest k-subgraph , 2010, STOC '10.

[5]  Silvio Micali,et al.  An O(v|v| c |E|) algoithm for finding maximum matching in general graphs , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[6]  K. Nguyen.  Lagrangian Duality in Online Scheduling with Resource Augmentation and Speed Scaling , 2013, ESA.

[7]  Amit Kumar,et al.  Resource augmentation for weighted flow-time explained by dual fitting , 2012, SODA.

[8]  Michael Langberg,et al.  Approximation Algorithms for Maximization Problems Arising in Graph Partitioning , 2001, J. Algorithms.

[9]  Pierre Fraigniaud,et al.  Online computation with advice , 2009, Theor. Comput. Sci..

[10]  Danny Segev,et al.  Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing , 2008, Algorithmica.

[11]  Kirk Pruhs,et al.  The Online Transportation Problem: On the Exponential Boost of One Extra Server , 2008, LATIN.

[12]  Avi Wigderson,et al.  Improving the performance guarantee for approximate graph coloring , 1983, JACM.

[13]  Kamesh Munagala,et al.  Competitive Flow Time Algorithms for Polyhedral Scheduling , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[14]  Nikhil R. Devanur,et al.  Primal Dual Gives Almost Optimal Energy-Efficient Online Algorithms , 2014, ACM Trans. Algorithms.

[15]  Denis Trystram,et al.  Online Non-Preemptive Scheduling in a Resource Augmentation Model Based on Duality , 2016, ESA.

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

[17]  Benjamin E. Birnbaum,et al.  An Improved Analysis for a Greedy Remote-Clique Algorithm Using Factor-Revealing LPs , 2007, Algorithmica.

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

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

[20]  Amit Kumar,et al.  Rejecting jobs to minimize load and maximum flow-time , 2014, J. Comput. Syst. Sci..

[21]  Moses Charikar,et al.  The finite capacity dial-a-ride problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[22]  Giorgio Lucarelli,et al.  Primal-Dual and Dual-Fitting Analysis of Online Scheduling Algorithms for Generalized Flow Time Problems , 2015, ESA.

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

[24]  Anand Srivastav,et al.  Finding Dense Subgraphs with Semidefinite Programming , 1998, APPROX.

[25]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1989, 30th Annual Symposium on Foundations of Computer Science.

[26]  Avrim Blum,et al.  New approximation algorithms for graph coloring , 1994, JACM.

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

[28]  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.

[29]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[30]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[31]  Michael E. Saks,et al.  Clustering is difficult only when it does not matter , 2012, ArXiv.

[32]  Kirk Pruhs,et al.  SelfishMigrate: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[33]  David R. Karger,et al.  An Õ(n^{3/14})-Coloring Algorithm for 3-Colorable Graphs , 1997, Information Processing Letters.

[34]  Uriel Feige.  A threshold of ln n for approximating set cover (preliminary version) , 1996, STOC '96.

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

[36]  Alexander H. G. Rinnooy Kan,et al.  Bounds and Heuristics for Capacitated Routing Problems , 1985, Math. Oper. Res..

[37]  James B. Orlin,et al.  Max flows in O(nm) time, or better , 2013, STOC '13.

[38]  Chul E. Kim,et al.  Approximation algorithms for some routing problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[39]  Neal E. Young,et al.  On-Line Paging Against Adversarially Biased Random Inputs , 2000, J. Algorithms.

[40]  Bala Kalyanasundaram,et al.  Speed is as powerful as clairvoyance , 2000, JACM.

[41]  Allan Borodin,et al.  Competitive paging with locality of reference , 1991, STOC '91.

[42]  Bala Kalyanasundaram,et al.  The Online Transportation Problem , 2000, SIAM J. Discret. Math..

[43]  Hisao Tamaki,et al.  Greedily Finding a Dense Subgraph , 2000, J. Algorithms.

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

[45]  Christos H. Papadimitriou,et al.  Beyond competitive analysis [on-line algorithms] , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[46]  Peter J. Stuckey,et al.  Propagation via lazy clause generation , 2009, Constraints.

[47]  Cynthia A. Phillips,et al.  Optimal Time-Critical Scheduling via Resource Augmentation , 1997, STOC '97.

[48]  Subhash Khot,et al.  Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[49]  Kamesh Munagala,et al.  Competitive algorithms from competitive equilibria: non-clairvoyant scheduling under polyhedral constraints , 2014, STOC.

[50]  Ojas Parekh,et al.  A Unified Approach to Approximating Partial Covering Problems , 2006, Algorithmica.