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April 2018, 14(2): 759-784. doi: 10.3934/jimo.2017074

## Dispersion with connectivity in wireless mesh networks

 1 R & D Center, Information Technologies Department, Migros T.A.Ş., 34758, Istanbul, Turkey 2 Faculty of Engineering and Natural Sciences, Sabanci University, 34956, Istanbul, Turkey

* Corresponding author: byuceoglu@migros.com.tr

Received  March 2016 Published  September 2017

We study a multi-objective access point dispersion problem, where the conflicting objectives of maximizing the distance and maximizing the connectivity between the agents are considered with explicit coverage (or Quality of Service) constraints. We model the problem first as a multi-objective model, and then, we consider the constrained single objective alternatives, which we propose to solve using three approaches: The first approach is an optimal tree search algorithm, where bounds are used to prune the search tree. The second approach is a beam search heuristic, which is also used to provide lower bound for the first approach. The third approach is a straightforward integer programming approach. We present an illustrative application of our solution approaches in a real wireless mesh network deployment problem.

Citation: Birol Yüceoǧlu, ş. ilker Birbil, özgür Gürbüz. Dispersion with connectivity in wireless mesh networks. Journal of Industrial & Management Optimization, 2018, 14 (2) : 759-784. doi: 10.3934/jimo.2017074
##### References:
 [1] I. F. Akyildiz, X. Wang and W. Wang, Wireless mesh networks: A survey, Computer Networks, 47 (2005), 445-487. [2] C. R. Anderson and T. S. Rappaport, In-building wideband partition loss measurements at 2.5 and 60 GHz, IEEE Transactions on Wireless Communications, 3 (2004), 922-928. [3] F. Birlik, Ö. Gürbüz and Ö. Erçetin, Iptv home networking via 802.11 wireless mesh networks: An implementation experience, IEEE Transactions on Consumer Electronics, 55 (2009), 1192-1199. [4] V. E. Brimkov, A. Leach, J. Wu and M. Mastroianni, Approximation algorithms for a geometric set cover problem, Discrete Applied Mathematics, 160 (2012), 1039-1052. doi: 10.1016/j.dam.2011.11.023. [5] P. Cappanera, A survey on obnoxious facility location problems, Tech. Rep. TR-99-11, University of Pisa, 1999. [6] R. L. Carraway and R. L. Schmidt, An improved discrete dynamic programming algorithm for allocating resources among interdependent projects, Management Science, 37 (1991), 1195-1200. doi: 10.1287/mnsc.37.9.1195. [7] F. D. Croce, A. Grosso and M. Locatelli, A heuristic approach for the max-min diversity problem based on max-clique, Computers & Operations Research, 36 (2009), 2429-2433. doi: 10.1016/j.cor.2008.09.007. [8] Z. Drezner and H. W. Hamacher, Facility Location. Applications and Theory Berlin: Springer, 2002. doi: 10.1007/978-3-642-56082-8. [9] E. Erkut and S. Neuman, A multiobjective model for locating undesirable facilities, Annals of Operations Research, 40 (1992), 209-227. [10] S. Fekete and H. Meijer, Maximum dispersion and geometric maximum weight cliques, in APPROX '00: Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization, (London, UK), Springer-Verlag, (2000), 132-143. doi: 10.1007/3-540-44436-X_14. [11] Ö. Gürbüz and H. Owen, Power control based QoS provisioning for multimedia in W-CDMA, ACM Wireless Networks (WINET), 8 (2002), 37-44. [12] R. Heydari and E. Melachrinoudis, Location of a semi-obnoxious facility with elliptic maximin and network minisum objectives, European Journal of Operational Research, 223 (2012), 452-460. doi: 10.1016/j.ejor.2012.06.039. [13] P. T. Kabamba, S. M. Meerkov and C. Y. Tang, Optimal, suboptimal, and adaptive threshold policies for power efficiency of wireless networks, IEEE Transactions on Information Theory, 51 (2005), 1359-1376. doi: 10.1109/TIT.2005.844074. [14] R. L. Keeney and H. Raiffa, Decisions with Multiple Objectives New York: John Wiley, 1976. [15] A. M. Khedr and W. Osamy, Mobility-assisted minimum connected cover in a wireless sensor network, Journal of Parallel and Distributed Computing, 72 (2012), 827-837. doi: 10.1016/j.jpdc.2012.03.009. [16] A. H. Land and A. G. Doig, An automatic method of solving discrete programming problems, Econometrica, 28 (1960), 497-520. doi: 10.2307/1910129. [17] E. Macambira, An application of tabu search heuristic for the maximum edge-weighted subgraph problem, Annals of Operations Research, 117 (2002), 175-190. doi: 10.1023/A:1021525624027. [18] C. Malandraki and R. B. Dial, A restricted dynamic programming heuristic algorithm for the time dependent traveling salesman problem, European Journal of Operational, 90 (1996), 45-55. [19] R. E. Marsten and T. L. Morin, A hybrid approach to discrete mathematical programming, Mathematical Programming, 14 (1978), 21-40. doi: 10.1007/BF01588949. [20] E. Melachrinoudis and Z. Xanthopulos, Semi-obnoxious single facility location in euclidean space, Computers, 30 (2003), 2191-2209. doi: 10.1016/S0305-0548(02)00140-5. [21] E. Melachrinoudis, Bicriteria location of a semi-obnoxious facility, Computers, 37 (1999), 581-593. doi: 10.1016/S0360-8352(00)00022-X. [22] A. T. Murray and R. L. Church, Solving the anti-covering location problem using lagrangian relaxation, Computers, 24 (1997), 127-140. doi: 10.1016/S0305-0548(96)00048-2. [23] Y. Ohsawa, Bicriteria euclidean location associated with maximin and minimax criteria, Naval Research Logistics (NRL), 47 (2000), 581-592. doi: 10.1002/1520-6750(200010)47:7<581::AID-NAV3>3.0.CO;2-R. [24] Y. Ohsawa and K. Tamura, Efficient location for a semi-obnoxious facility, Annals of Operations Research, 123 (2003), 173-188. doi: 10.1023/A:1026127430341. [25] J. Rakas, D. Teodorovic and T. Kim, Multi-objective modeling for determining location of undesirable facilities, Transportation Research Part D: Transport and Environment, 9 (2004), 125-138. doi: 10.1016/j.trd.2003.09.002. [26] M. Rebai, M. L. berre, H. Snoussi, F. Hnaien and L. Khoukhi, Sensor deployment optimization methods to achieve both coverage and connectivity in wireless sensor networks, Computers, 59 (2015), 11-21. doi: 10.1016/j.cor.2014.11.002. [27] O. Simeone, A. Maeder, M. Peng, O. Sahin and W. Yu, Cloud radio access network: Virtualizing wireless access for dense heterogeneous systems, Journal of Communications and Networks, 18 (2016), 135-149. doi: 10.1109/JCN.2016.000023. [28] A. J. V. Skriver and K. A. Andersen, The bicriterion semi-obnoxious location (bsl) problem solved by an [epsilon]-approximation, European Journal of Operational Research, 146 (2003), 517-528. doi: 10.1016/S0377-2217(02)00271-0. [29] J. L. Williams, J. W. Fisher and A. S. Willsky, Approximate dynamic programming for communication-constrained sensor network management, IEEE Transactions on Signal Processing, 55 (2007), 4300-4311. doi: 10.1109/TSP.2007.896099. [30] D. Wood, An algorithm for finding a maximum clique in a graph, Operation Research Letters, 21 (1997), 211-217. doi: 10.1016/S0167-6377(97)00054-0. [31] IBM, IBM ILOG CPLEX Optimizer, www.ibm.com/software/commerce/optimization/cplex-optimizer/, Last retrieved April 2015. [32] The 5G Infrastructure Public Private Partnership, https://5g-ppp.eu/.

show all references

##### References:
 [1] I. F. Akyildiz, X. Wang and W. Wang, Wireless mesh networks: A survey, Computer Networks, 47 (2005), 445-487. [2] C. R. Anderson and T. S. Rappaport, In-building wideband partition loss measurements at 2.5 and 60 GHz, IEEE Transactions on Wireless Communications, 3 (2004), 922-928. [3] F. Birlik, Ö. Gürbüz and Ö. Erçetin, Iptv home networking via 802.11 wireless mesh networks: An implementation experience, IEEE Transactions on Consumer Electronics, 55 (2009), 1192-1199. [4] V. E. Brimkov, A. Leach, J. Wu and M. Mastroianni, Approximation algorithms for a geometric set cover problem, Discrete Applied Mathematics, 160 (2012), 1039-1052. doi: 10.1016/j.dam.2011.11.023. [5] P. Cappanera, A survey on obnoxious facility location problems, Tech. Rep. TR-99-11, University of Pisa, 1999. [6] R. L. Carraway and R. L. Schmidt, An improved discrete dynamic programming algorithm for allocating resources among interdependent projects, Management Science, 37 (1991), 1195-1200. doi: 10.1287/mnsc.37.9.1195. [7] F. D. Croce, A. Grosso and M. Locatelli, A heuristic approach for the max-min diversity problem based on max-clique, Computers & Operations Research, 36 (2009), 2429-2433. doi: 10.1016/j.cor.2008.09.007. [8] Z. Drezner and H. W. Hamacher, Facility Location. Applications and Theory Berlin: Springer, 2002. doi: 10.1007/978-3-642-56082-8. [9] E. Erkut and S. Neuman, A multiobjective model for locating undesirable facilities, Annals of Operations Research, 40 (1992), 209-227. [10] S. Fekete and H. Meijer, Maximum dispersion and geometric maximum weight cliques, in APPROX '00: Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization, (London, UK), Springer-Verlag, (2000), 132-143. doi: 10.1007/3-540-44436-X_14. [11] Ö. Gürbüz and H. Owen, Power control based QoS provisioning for multimedia in W-CDMA, ACM Wireless Networks (WINET), 8 (2002), 37-44. [12] R. Heydari and E. Melachrinoudis, Location of a semi-obnoxious facility with elliptic maximin and network minisum objectives, European Journal of Operational Research, 223 (2012), 452-460. doi: 10.1016/j.ejor.2012.06.039. [13] P. T. Kabamba, S. M. Meerkov and C. Y. Tang, Optimal, suboptimal, and adaptive threshold policies for power efficiency of wireless networks, IEEE Transactions on Information Theory, 51 (2005), 1359-1376. doi: 10.1109/TIT.2005.844074. [14] R. L. Keeney and H. Raiffa, Decisions with Multiple Objectives New York: John Wiley, 1976. [15] A. M. Khedr and W. Osamy, Mobility-assisted minimum connected cover in a wireless sensor network, Journal of Parallel and Distributed Computing, 72 (2012), 827-837. doi: 10.1016/j.jpdc.2012.03.009. [16] A. H. Land and A. G. Doig, An automatic method of solving discrete programming problems, Econometrica, 28 (1960), 497-520. doi: 10.2307/1910129. [17] E. Macambira, An application of tabu search heuristic for the maximum edge-weighted subgraph problem, Annals of Operations Research, 117 (2002), 175-190. doi: 10.1023/A:1021525624027. [18] C. Malandraki and R. B. Dial, A restricted dynamic programming heuristic algorithm for the time dependent traveling salesman problem, European Journal of Operational, 90 (1996), 45-55. [19] R. E. Marsten and T. L. Morin, A hybrid approach to discrete mathematical programming, Mathematical Programming, 14 (1978), 21-40. doi: 10.1007/BF01588949. [20] E. Melachrinoudis and Z. Xanthopulos, Semi-obnoxious single facility location in euclidean space, Computers, 30 (2003), 2191-2209. doi: 10.1016/S0305-0548(02)00140-5. [21] E. Melachrinoudis, Bicriteria location of a semi-obnoxious facility, Computers, 37 (1999), 581-593. doi: 10.1016/S0360-8352(00)00022-X. [22] A. T. Murray and R. L. Church, Solving the anti-covering location problem using lagrangian relaxation, Computers, 24 (1997), 127-140. doi: 10.1016/S0305-0548(96)00048-2. [23] Y. Ohsawa, Bicriteria euclidean location associated with maximin and minimax criteria, Naval Research Logistics (NRL), 47 (2000), 581-592. doi: 10.1002/1520-6750(200010)47:7<581::AID-NAV3>3.0.CO;2-R. [24] Y. Ohsawa and K. Tamura, Efficient location for a semi-obnoxious facility, Annals of Operations Research, 123 (2003), 173-188. doi: 10.1023/A:1026127430341. [25] J. Rakas, D. Teodorovic and T. Kim, Multi-objective modeling for determining location of undesirable facilities, Transportation Research Part D: Transport and Environment, 9 (2004), 125-138. doi: 10.1016/j.trd.2003.09.002. [26] M. Rebai, M. L. berre, H. Snoussi, F. Hnaien and L. Khoukhi, Sensor deployment optimization methods to achieve both coverage and connectivity in wireless sensor networks, Computers, 59 (2015), 11-21. doi: 10.1016/j.cor.2014.11.002. [27] O. Simeone, A. Maeder, M. Peng, O. Sahin and W. Yu, Cloud radio access network: Virtualizing wireless access for dense heterogeneous systems, Journal of Communications and Networks, 18 (2016), 135-149. doi: 10.1109/JCN.2016.000023. [28] A. J. V. Skriver and K. A. Andersen, The bicriterion semi-obnoxious location (bsl) problem solved by an [epsilon]-approximation, European Journal of Operational Research, 146 (2003), 517-528. doi: 10.1016/S0377-2217(02)00271-0. [29] J. L. Williams, J. W. Fisher and A. S. Willsky, Approximate dynamic programming for communication-constrained sensor network management, IEEE Transactions on Signal Processing, 55 (2007), 4300-4311. doi: 10.1109/TSP.2007.896099. [30] D. Wood, An algorithm for finding a maximum clique in a graph, Operation Research Letters, 21 (1997), 211-217. doi: 10.1016/S0167-6377(97)00054-0. [31] IBM, IBM ILOG CPLEX Optimizer, www.ibm.com/software/commerce/optimization/cplex-optimizer/, Last retrieved April 2015. [32] The 5G Infrastructure Public Private Partnership, https://5g-ppp.eu/.
The Pareto curve with $n=5$ and with different values of $e_{min}$
An illustration of the TSWB approach
The candidate locations on the floor plan. The dashed lines in the middle of the building is a meshed glass and solid lines around the rooms (colored areas) are walls
The solution of problem 2 for $n=10$ with $c_{\min}=40$
The solutions of problem 2 for $n=5$
The solutions of problem 3 for $n=5$
The solutions of problem 3 for $n=10$
The performance of the tree search algorithm for the multi-objective problem ($n=10$, $e_{min}=0.8$). The cardinality of the set of all solutions is $\binom{70}{10} \approx 3.967\times 10^{11}$
The performance of the tree search algorithm for the problem in Figure 4. The cardinality of the set of all solutions is $\binom{70}{10} \approx 3.967\times 10^{11}$
The performance of the tree search algorithm for the problem in Figure 7(a)
 0 0.42 0.33 0.25 0.86 0.48 0.48 0.37 0.81 0.29 0.42 0 0.64 0.74 0.1 0.54 0.92 0.61 0.68 0.45 0.33 0.64 0 0.23 0.41 0.25 0.51 0.72 0.4 0.55 0.25 0.74 0.23 0 0.42 0.48 0.75 0.55 0.54 0.44 0.86 0.1 0.41 0.42 0 0.77 0.52 0.4 0.24 0.24 0.48 0.54 0.25 0.48 0.77 0 0.33 0.78 0.63 0.92 0.48 0.92 0.51 0.75 0.52 0.33 0 0.42 0.31 0.41 0.37 0.61 0.72 0.55 0.4 0.78 0.42 0 0.47 0.55 0.81 0.68 0.4 0.54 0.24 0.63 0.31 0.47 0 0.4 0.29 0.45 0.55 0.44 0.24 0.92 0.41 0.55 0.4 0
 0 0.42 0.33 0.25 0.86 0.48 0.48 0.37 0.81 0.29 0.42 0 0.64 0.74 0.1 0.54 0.92 0.61 0.68 0.45 0.33 0.64 0 0.23 0.41 0.25 0.51 0.72 0.4 0.55 0.25 0.74 0.23 0 0.42 0.48 0.75 0.55 0.54 0.44 0.86 0.1 0.41 0.42 0 0.77 0.52 0.4 0.24 0.24 0.48 0.54 0.25 0.48 0.77 0 0.33 0.78 0.63 0.92 0.48 0.92 0.51 0.75 0.52 0.33 0 0.42 0.31 0.41 0.37 0.61 0.72 0.55 0.4 0.78 0.42 0 0.47 0.55 0.81 0.68 0.4 0.54 0.24 0.63 0.31 0.47 0 0.4 0.29 0.45 0.55 0.44 0.24 0.92 0.41 0.55 0.4 0
 0.06 0.40 0.27 0.31 0.83 0.73 0.89 0.04 0.51 0.18 0.93 0.89 0.99 0.87 0.94 0.15 0.99 0.66 0.98 0.58 0.40 0.99 0.79 0.77 0.68 0.31 0.66 0.30 0.76 0.91 0.88 0.52 0.32 0.26 0.35 0.63 0.06 0.48 0.54 0.89 0.53 0.73 0.77 0.68 0.43 0.10 0.34 0.03 0.34 0.91 0.42 0.01 0.10 0.81 0.13 0.40 0.72 0.52 0.47 0.97 0.18 0.75 0.48 0.95 0.44 0.68 0.70 0.06 0.89 0.82 0.33 0.73 0.82 0.75 0.52 0.88 0.21 0.62 0.74 0.67 Bounds 0.99 0.66 0.98 0.58 0.83 0.99 0.89 0.77 0.99 0.91 0.98 0.97 0.89 0.99 0.89 0.95
 0.06 0.40 0.27 0.31 0.83 0.73 0.89 0.04 0.51 0.18 0.93 0.89 0.99 0.87 0.94 0.15 0.99 0.66 0.98 0.58 0.40 0.99 0.79 0.77 0.68 0.31 0.66 0.30 0.76 0.91 0.88 0.52 0.32 0.26 0.35 0.63 0.06 0.48 0.54 0.89 0.53 0.73 0.77 0.68 0.43 0.10 0.34 0.03 0.34 0.91 0.42 0.01 0.10 0.81 0.13 0.40 0.72 0.52 0.47 0.97 0.18 0.75 0.48 0.95 0.44 0.68 0.70 0.06 0.89 0.82 0.33 0.73 0.82 0.75 0.52 0.88 0.21 0.62 0.74 0.67 Bounds 0.99 0.66 0.98 0.58 0.83 0.99 0.89 0.77 0.99 0.91 0.98 0.97 0.89 0.99 0.89 0.95
The problem parameters
 $d_0 = 1$ meter $\lambda = (3\times 10^8)/(2.4 \times 10^9)$ $\alpha = 2$ $B= 20$ GHz $P_t = 4$ Watt $N_0=3.98\times10^{-17}$ $G_t = 1$ $t^{wall}=5.4$ dB, $t^{glass}=7.7$ dB
 $d_0 = 1$ meter $\lambda = (3\times 10^8)/(2.4 \times 10^9)$ $\alpha = 2$ $B= 20$ GHz $P_t = 4$ Watt $N_0=3.98\times10^{-17}$ $G_t = 1$ $t^{wall}=5.4$ dB, $t^{glass}=7.7$ dB
Results of the computational study for the single objective problems
 Instance Beam Search TSWB CPLEX Model $n$ Constraint Gap Time Time 2 5 $c(\mathbf{x}) \geq 8$ 0.00% 1 129 2 5 $c(\mathbf{x}) \geq 9$ 50.00% 1 561 2 10 $c(\mathbf{x}) \geq 35$ 0.00% 155 1566 2 10 $c(\mathbf{x}) \geq 40$ 50.00% 23 1695 3 5 $d(\mathbf{x}) \geq 15$ 0.10% 1 28 3 5 $d(\mathbf{x}) \geq 20$ 0.00% 1 7 3 10 $d(\mathbf{x}) \geq 10$ 0.18% 22 154 3 10 $d(\mathbf{x}) \geq 15$ 3.54% 79 12
 Instance Beam Search TSWB CPLEX Model $n$ Constraint Gap Time Time 2 5 $c(\mathbf{x}) \geq 8$ 0.00% 1 129 2 5 $c(\mathbf{x}) \geq 9$ 50.00% 1 561 2 10 $c(\mathbf{x}) \geq 35$ 0.00% 155 1566 2 10 $c(\mathbf{x}) \geq 40$ 50.00% 23 1695 3 5 $d(\mathbf{x}) \geq 15$ 0.10% 1 28 3 5 $d(\mathbf{x}) \geq 20$ 0.00% 1 7 3 10 $d(\mathbf{x}) \geq 10$ 0.18% 22 154 3 10 $d(\mathbf{x}) \geq 15$ 3.54% 79 12
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