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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. AkyildizX. Wang and W. Wang, Wireless mesh networks: A survey, Computer Networks, 47 (2005), 445-487.   Google Scholar

[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.   Google Scholar

[3]

F. BirlikÖ. 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.   Google Scholar

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V. E. BrimkovA. LeachJ. 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.  Google Scholar

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P. Cappanera, A survey on obnoxious facility location problems, Tech. Rep. TR-99-11, University of Pisa, 1999. Google Scholar

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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.  Google Scholar

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F. D. CroceA. 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.  Google Scholar

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Z. Drezner and H. W. Hamacher, Facility Location. Applications and Theory Berlin: Springer, 2002. doi: 10.1007/978-3-642-56082-8.  Google Scholar

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E. Erkut and S. Neuman, A multiobjective model for locating undesirable facilities, Annals of Operations Research, 40 (1992), 209-227.   Google Scholar

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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.  Google Scholar

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Ö. Gürbüz and H. Owen, Power control based QoS provisioning for multimedia in W-CDMA, ACM Wireless Networks (WINET), 8 (2002), 37-44.   Google Scholar

[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.  Google Scholar

[13]

P. T. KabambaS. 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.  Google Scholar

[14]

R. L. Keeney and H. Raiffa, Decisions with Multiple Objectives New York: John Wiley, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Melachrinoudis, Bicriteria location of a semi-obnoxious facility, Computers, 37 (1999), 581-593.  doi: 10.1016/S0360-8352(00)00022-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. RakasD. 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.  Google Scholar

[26]

M. RebaiM. L. berreH. SnoussiF. 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.  Google Scholar

[27]

O. SimeoneA. MaederM. PengO. 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.  Google Scholar

[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.  Google Scholar

[29]

J. L. WilliamsJ. 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.  Google Scholar

[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.  Google Scholar

[31]

IBM, IBM ILOG CPLEX Optimizer, www.ibm.com/software/commerce/optimization/cplex-optimizer/, Last retrieved April 2015. Google Scholar

[32]

The 5G Infrastructure Public Private Partnership, https://5g-ppp.eu/. Google Scholar

show all references

References:
[1]

I. F. AkyildizX. Wang and W. Wang, Wireless mesh networks: A survey, Computer Networks, 47 (2005), 445-487.   Google Scholar

[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.   Google Scholar

[3]

F. BirlikÖ. 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.   Google Scholar

[4]

V. E. BrimkovA. LeachJ. 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.  Google Scholar

[5]

P. Cappanera, A survey on obnoxious facility location problems, Tech. Rep. TR-99-11, University of Pisa, 1999. Google Scholar

[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.  Google Scholar

[7]

F. D. CroceA. 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.  Google Scholar

[8]

Z. Drezner and H. W. Hamacher, Facility Location. Applications and Theory Berlin: Springer, 2002. doi: 10.1007/978-3-642-56082-8.  Google Scholar

[9]

E. Erkut and S. Neuman, A multiobjective model for locating undesirable facilities, Annals of Operations Research, 40 (1992), 209-227.   Google Scholar

[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.  Google Scholar

[11]

Ö. Gürbüz and H. Owen, Power control based QoS provisioning for multimedia in W-CDMA, ACM Wireless Networks (WINET), 8 (2002), 37-44.   Google Scholar

[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.  Google Scholar

[13]

P. T. KabambaS. 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.  Google Scholar

[14]

R. L. Keeney and H. Raiffa, Decisions with Multiple Objectives New York: John Wiley, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Melachrinoudis, Bicriteria location of a semi-obnoxious facility, Computers, 37 (1999), 581-593.  doi: 10.1016/S0360-8352(00)00022-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. RakasD. 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.  Google Scholar

[26]

M. RebaiM. L. berreH. SnoussiF. 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.  Google Scholar

[27]

O. SimeoneA. MaederM. PengO. 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.  Google Scholar

[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.  Google Scholar

[29]

J. L. WilliamsJ. 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.  Google Scholar

[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.  Google Scholar

[31]

IBM, IBM ILOG CPLEX Optimizer, www.ibm.com/software/commerce/optimization/cplex-optimizer/, Last retrieved April 2015. Google Scholar

[32]

The 5G Infrastructure Public Private Partnership, https://5g-ppp.eu/. Google Scholar

Figure 3.  The Pareto curve with $n=5$ and with different values of $e_{min}$
Figure 1.  An illustration of the TSWB approach
Figure 2.  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
Figure 4.  The solution of problem 2 for $n=10$ with $c_{\min}=40$
Figure 5.  The solutions of problem 2 for $n=5$
Figure 6.  The solutions of problem 3 for $n=5$
Figure 7.  The solutions of problem 3 for $n=10$
Figure 8.  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 $\binom{70}{10} \approx 3.967\times 10^{11}$
Figure 9.  The performance of the tree search algorithm for the problem in Figure 4. The cardinality of the set of all solutions is $\binom{70}{10} \approx 3.967\times 10^{11}$
Figure 10.  The performance of the tree search algorithm for the problem in Figure 7(a)
Table3 
00.420.330.250.860.480.480.370.810.29
0.4200.640.740.100.540.920.610.680.45
0.330.6400.230.410.250.510.720.400.55
0.250.740.2300.420.480.750.550.540.44
0.860.100.410.4200.770.520.400.240.24
0.480.540.250.480.7700.330.780.630.92
0.480.920.510.750.520.3300.420.310.41
0.370.610.720.550.400.780.4200.470.55
0.810.680.400.540.240.630.310.4700.40
0.290.450.550.440.240.920.410.550.400
00.420.330.250.860.480.480.370.810.29
0.4200.640.740.100.540.920.610.680.45
0.330.6400.230.410.250.510.720.400.55
0.250.740.2300.420.480.750.550.540.44
0.860.100.410.4200.770.520.400.240.24
0.480.540.250.480.7700.330.780.630.92
0.480.920.510.750.520.3300.420.310.41
0.370.610.720.550.400.780.4200.470.55
0.810.680.400.540.240.630.310.4700.40
0.290.450.550.440.240.920.410.550.400
Table4 
0.060.400.270.310.830.730.890.04
0.510.180.930.890.990.870.940.15
0.990.660.980.580.400.990.790.77
0.680.310.660.300.760.910.880.52
0.320.260.350.630.060.480.540.89
0.530.730.770.680.430.100.340.03
0.340.910.420.010.100.810.130.40
0.720.520.470.970.180.750.480.95
0.440.680.700.060.890.820.330.73
0.820.750.520.880.210.620.740.67
Bounds
0.990.660.980.580.830.990.890.77
0.990.910.980.970.890.990.890.95
0.060.400.270.310.830.730.890.04
0.510.180.930.890.990.870.940.15
0.990.660.980.580.400.990.790.77
0.680.310.660.300.760.910.880.52
0.320.260.350.630.060.480.540.89
0.530.730.770.680.430.100.340.03
0.340.910.420.010.100.810.130.40
0.720.520.470.970.180.750.480.95
0.440.680.700.060.890.820.330.73
0.820.750.520.880.210.620.740.67
Bounds
0.990.660.980.580.830.990.890.77
0.990.910.980.970.890.990.890.95
Table 1.  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
Table 2.  Results of the computational study for the single objective problems
Instance Beam Search TSWB CPLEX
Model$n$ConstraintGapTimeTime
25$c(\mathbf{x}) \geq 8$0.00%1129
25$c(\mathbf{x}) \geq 9$50.00%1561
210$c(\mathbf{x}) \geq 35$0.00%1551566
210$c(\mathbf{x}) \geq 40$50.00%231695
35$d(\mathbf{x}) \geq 15$0.10%128
35$d(\mathbf{x}) \geq 20$0.00%17
310$d(\mathbf{x}) \geq 10$0.18%22154
310$d(\mathbf{x}) \geq 15$3.54%7912
Instance Beam Search TSWB CPLEX
Model$n$ConstraintGapTimeTime
25$c(\mathbf{x}) \geq 8$0.00%1129
25$c(\mathbf{x}) \geq 9$50.00%1561
210$c(\mathbf{x}) \geq 35$0.00%1551566
210$c(\mathbf{x}) \geq 40$50.00%231695
35$d(\mathbf{x}) \geq 15$0.10%128
35$d(\mathbf{x}) \geq 20$0.00%17
310$d(\mathbf{x}) \geq 10$0.18%22154
310$d(\mathbf{x}) \geq 15$3.54%7912
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