Optimal relay node placement in delay constrained wireless sensor network design

The Delay Constrained Relay Node Placement Problem (DCRNPP) frequently arises in the Wireless Sensor Network (WSN) design. In WSN, Sensor Nodes are placed across a target geographical region to detect relevant signals. These signals are communicated to a central location, known as the Base Station, for further processing. The DCRNPP aims to place the minimum number of additional Relay Nodes at a subset of Candidate Relay Node locations in such a manner that signals from various Sensor Nodes can be communicated to the Base Station within a pre-specified delay bound. In this paper, we study the structure of the projection polyhedron of the problem and develop valid inequalities in form of the node-cut inequalities. We also derive conditions under which these inequalities are facet defining for the projection polyhedron. We formulate a branch-and-cut algorithm, based upon the projection formulation, to solve DCRNPP optimally. A Lagrangian relaxation based heuristic is used to generate a good initial solution for the problem that is used as an initial incumbent solution in the branch-and-cut approach. Computational results are reported on several randomly generated instances to demonstrate the efficacy of the proposed algorithm.

[1]  Luís Gouveia,et al.  On Formulations and Methods for the Hop-Constrained Minimum Spanning Tree Problem , 2006, Handbook of Optimization in Telecommunications.

[2]  R. Gomory,et al.  A Linear Programming Approach to the Cutting-Stock Problem , 1961 .

[3]  Abdullah Konak,et al.  Network design problem with relays: A genetic algorithm with a path-based crossover and a set covering formulation , 2012, Eur. J. Oper. Res..

[4]  S. N. Maheshwari,et al.  Analysis of Preflow Push Algorithms for Maximum Network Flow , 1988, SIAM J. Comput..

[5]  Matteo Fischetti,et al.  Facets of two Steiner arborescence polyhedra , 1991, Math. Program..

[6]  Michel Minoux,et al.  Graphs and Algorithms , 1984 .

[7]  Gregory J. Pottie,et al.  Self-organizing distributed sensor networks , 1999, Defense, Security, and Sensing.

[8]  Y. Aneja,et al.  Using branch-and-price approach to solve the directed network design problem with relays , 2012 .

[9]  Mauricio G. C. Resende,et al.  Strong lower bounds for the prize collecting Steiner problem in graphs , 2004, Discret. Appl. Math..

[10]  Gen-Huey Chen,et al.  The steiner problem in distributed computing systems , 1993, Inf. Sci..

[11]  Moshe Dror,et al.  Note on the Complexity of the Shortest Path Models for Column Generation in VRPTW , 1994, Oper. Res..

[12]  Dushyant Sharma,et al.  Modeling and solving the rooted distance-constrained minimum spanning tree problem , 2008, Comput. Oper. Res..

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

[14]  R. Ravi,et al.  A nearly best-possible approximation algorithm for node-weighted Steiner trees , 1993, IPCO.

[15]  S. Voß,et al.  The Steiner tree problem with hop constraints , 1999, Ann. Oper. Res..

[16]  Guohui Lin,et al.  Steiner Tree Problem with Minimum Number of Steiner Points and Bounded Edge-Length , 1999, Inf. Process. Lett..

[17]  Ian F. Akyildiz,et al.  Wireless sensor networks: a survey , 2002, Comput. Networks.

[18]  Yuval Rabani,et al.  Approximation Algorithms for Constrained Node Weighted Steiner Tree Problems , 2007, SIAM J. Comput..

[19]  Lusheng Wang,et al.  Relay sensor placement in wireless sensor networks , 2008, Wirel. Networks.

[20]  A. Volgenant,et al.  Some generalizations of the steiner problem in graphs , 1987, Networks.

[21]  Matteo Fischetti,et al.  Solving the Prize-Collecting Steiner Tree Problem to Optimality , 2005, ALENEX/ANALCO.

[22]  Philip N. Klein,et al.  Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs , 2009, ICALP.

[23]  Celso C. Ribeiro,et al.  Local search with perturbations for the prize‐collecting Steiner tree problem in graphs , 2001, Networks.

[24]  M. R. Rao,et al.  The Steiner tree problem I: Formulations, compositions and extension of facets , 1994, Math. Program..

[25]  Jan Remy,et al.  Approximation Schemes for Node-Weighted Geometric Steiner Tree Problems , 2007, Algorithmica.

[26]  Martin W. P. Savelsbergh,et al.  A Branch-and-Price Algorithm for the Generalized Assignment Problem , 1997, Oper. Res..

[27]  Anurag Kumar,et al.  Delay constrained optimal relay placement for planned wireless sensor networks , 2010, 2010 IEEE 18th International Workshop on Quality of Service (IWQoS).

[28]  BERNARD M. WAXMAN,et al.  Routing of multipoint connections , 1988, IEEE J. Sel. Areas Commun..

[29]  Alysson M. Costa,et al.  Fast heuristics for the Steiner tree problem with revenues, budget and hop constraints , 2008, Eur. J. Oper. Res..

[30]  Arie Segev,et al.  The node-weighted steiner tree problem , 1987, Networks.

[31]  Raymond A. Patterson,et al.  The network design problem with relays , 2007, Eur. J. Oper. Res..

[32]  Yogesh Agarwal.  Design of Survivable Networks Using Three- and Four-Partition Facets , 2013, Oper. Res..

[33]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[34]  M. R. Rao,et al.  The Steiner tree problem II: Properties and classes of facets , 1994, Math. Program..

[35]  Peter Värbrand,et al.  A strong lower bound for the Node Weighted Steiner Tree Problem , 1998 .

[36]  J. Orlin,et al.  Distance-Directed Augmenting Path Algorithms for Maximum Flow and Parametric Maximum Flow Problems , 1991 .

[37]  J. E. Beasley.  An SST-based algorithm for the steiner problem in graphs , 1989, Networks.

[38]  Alex Zelikovsky,et al.  Tighter Bounds for Graph Steiner Tree Approximation , 2005, SIAM J. Discret. Math..

[39]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[40]  Satyajayant Misra,et al.  Constrained Relay Node Placement in Wireless Sensor Networks to Meet Connectivity and Survivability Requirements , 2008, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications.