{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T16:42:53Z","timestamp":1783701773273,"version":"3.55.0"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,19]],"date-time":"2024-01-19T00:00:00Z","timestamp":1705622400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Dicyt","award":["062313AS"],"award-info":[{"award-number":["062313AS"]}]},{"name":"Dicyt","award":["11230129"],"award-info":[{"award-number":["11230129"]}]},{"name":"Dicyt","award":["LPR21-02"],"award-info":[{"award-number":["LPR21-02"]}]},{"name":"ANID\/FONDECYT Iniciaci\u00f3n","award":["062313AS"],"award-info":[{"award-number":["062313AS"]}]},{"name":"ANID\/FONDECYT Iniciaci\u00f3n","award":["11230129"],"award-info":[{"award-number":["11230129"]}]},{"name":"ANID\/FONDECYT Iniciaci\u00f3n","award":["LPR21-02"],"award-info":[{"award-number":["LPR21-02"]}]},{"name":"Competition for Research Regular Projects","award":["062313AS"],"award-info":[{"award-number":["062313AS"]}]},{"name":"Competition for Research Regular Projects","award":["11230129"],"award-info":[{"award-number":["11230129"]}]},{"name":"Competition for Research Regular Projects","award":["LPR21-02"],"award-info":[{"award-number":["LPR21-02"]}]},{"name":"Universidad Tecnol\u00f3gica Metropolitana","award":["062313AS"],"award-info":[{"award-number":["062313AS"]}]},{"name":"Universidad Tecnol\u00f3gica Metropolitana","award":["11230129"],"award-info":[{"award-number":["11230129"]}]},{"name":"Universidad Tecnol\u00f3gica Metropolitana","award":["LPR21-02"],"award-info":[{"award-number":["LPR21-02"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, the problem of achieving the minimum backbone connectivity cost while simultaneously maximizing user coverage for 5G millimeter-wave (mmWave)-based networks is considered. Let G=(N,E) be an input graph instance with a set of nodes N (base stations) and a set of edges E. It is assumed that G represents a wireless backbone network. Let M represent a set of users to be covered by G. Note that mmWave technology has been considered in the literature as an important candidate solution for 5G networks due to its low latency. However, there remain some problems to be addressed before using this technology. A serious one is that millimeter waves cannot cover large transmission distances. In this article, the proposed methodology consists of formulating mixed-integer programming models to deal with the problem from a management point of view. Our models allow the determination of which of the nodes of G should be active and connected while simultaneously maximizing the total number of covered users. The models are solved with the CPLEX solver using its branch and cut and automatic Benders decomposition algorithms. For this purpose, symmetric complete and sparse graphs are considered. Using the symmetry concept, it is considered that the distances between base stations and users and between base stations themselves are symmetrical. Finally, an efficient local search meta-heuristic is proposed that allows for finding near-optimal solutions. Our numerical experiments indicate that the problem is hard to solve optimally. Thus, instances with up to 40 nodes and 500 users have been solved to optimality so far. In particular, it is observed that one of the models presents slightly better performance in terms of CPU time. Finally, the heuristic approach allows us to obtain tight solutions with less computational effort when dealing with even larger instances of the problem.<\/jats:p>","DOI":"10.3390\/sym16010123","type":"journal-article","created":{"date-parts":[[2024,1,19]],"date-time":"2024-01-19T11:04:57Z","timestamp":1705662297000},"page":"123","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Optimizing Connectivity and Coverage for Millimeter-Wave-Based Networks"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2500-3294","authenticated-orcid":false,"given":"Pablo","family":"Adasme","sequence":"first","affiliation":[{"name":"Centre for Multidisciplinary Research in Telecommunication Technologies (CIMTT), Department of Electrical Engineering, Faculty of Engineering, Universidad de Santiago de Chile, Avenida V\u00edctor Jara 3519, Estaci\u00f3n Central, Santiago 7800002, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6391-6863","authenticated-orcid":false,"given":"Ali Dehghan","family":"Firoozabadi","sequence":"additional","affiliation":[{"name":"Department of Electricity, Universidad Tecnol\u00f3gica Metropolitana, Av. Jose Pedro Alessandri 1242, Santiago 7800002, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4178-5591","authenticated-orcid":false,"given":"Sergio","family":"Cordero","sequence":"additional","affiliation":[{"name":"Centre for Multidisciplinary Research in Telecommunication Technologies (CIMTT), Department of Electrical Engineering, Faculty of Engineering, Universidad de Santiago de Chile, Avenida V\u00edctor Jara 3519, Estaci\u00f3n Central, Santiago 7800002, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,19]]},"reference":[{"key":"ref_1","unstructured":"(2023, May 30). IEEE 5G and Beyond Technology Road Map. Available online: https:\/\/futurenetworks.ieee.org\/images\/files\/pdf\/ieee-5g-roadmap-white-paper.pdf."},{"key":"ref_2","unstructured":"Palizban, N. (2017). Millimeter Wave Small Cell Network Planning for Outdoor Line-of-Sight Coverage Millimeter Wave Small Cell Network Planning for Outdoor. [Ph.D. Thesis, Carleton University]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1109\/ACCESS.2013.2260813","article-title":"Millimeter wave mobile communications for 5G cellular: It will work!","volume":"1","author":"Rappaport","year":"2013","journal-title":"IEEE Access"},{"key":"ref_4","unstructured":"(2023, May 30). Rakuten, Reimagining the End-to-End Mobile Network in the 5G Era. Available online: https:\/\/www.cisco.com\/c\/dam\/en\/us\/products\/collateral\/cloud-systems-management\/elastic-services-controller-esc\/reimagining-mobile-network-white-paper.pdf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2657","DOI":"10.1007\/s11276-015-0942-z","article-title":"A survey of millimeter wave communications (mmWave) for 5G: Opportunities and challenges","volume":"21","author":"Niu","year":"2015","journal-title":"Wirel. Netw."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.asoc.2018.03.008","article-title":"p-Median based formulations with backbone facility locations","volume":"67","author":"Adasme","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_7","unstructured":"(2023, June 10). IBM ILOG, CPLEX High-Performance Mathematical Programming Engine, V20.1.0, (2020). Available online: http:\/\/www.ibm.com\/software\/integration\/optimization\/cplex\/."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.ejor.2016.12.005","article-title":"The Benders decomposition algorithm: A literature review","volume":"259","author":"Rahmaniani","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Adasme, P., Cordero, S., Soto, I., and Firoozabadi, A.D. (2022, January 20\u201322). New Formulations for Optimal User Coverage using 5G Millimeter-Wave-based Networks. Proceedings of the 2022 13th International Symposium on Communication Systems, Networks and Digital Signal Processing (CSNDSP), Porto, Portugal.","DOI":"10.1109\/CSNDSP54353.2022.9907932"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7968","DOI":"10.1109\/JSEN.2023.3245063","article-title":"Unobtrusive Human Fall Detection System Using mmWave Radar and Data Driven Methods","volume":"23","author":"Rezaei","year":"2023","journal-title":"IEEE Sensors J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3525","DOI":"10.1109\/TVT.2021.3065221","article-title":"Base Station and Passive Reflectors Placement for Urban mmWave Networks","volume":"70","author":"Anjinappa","year":"2021","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8316","DOI":"10.1109\/TWC.2021.3091815","article-title":"Association and Caching in Relay-Assisted mmWave Networks: A Stochastic Geometry Perspective","volume":"12","author":"Gu","year":"2021","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1109\/LWC.2016.2552168","article-title":"Automated Placement of Individual Millimeter-Wave Wall-Mounted Base Stations for Line-of-Sight Coverage of Outdoor Urban Areas","volume":"5","author":"Szyszkowicz","year":"2016","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Sandoval, C., Adasme, P., Juan, E.S., Soto, I., and Firoozabadi, A.D. (2021, January 11\u201312). Novel p-Median-based Formulations for Maximizing Coverage in 5G\/6G Wireless Networks. Proceedings of the 2021 Third South American Colloquium on Visible Light Communications (SACVLC), Toledo, Brazil.","DOI":"10.1109\/SACVLC53127.2021.9652216"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2699","DOI":"10.1109\/JSAC.2019.2947940","article-title":"Fast Inter-Base Station Ring (FIBR): A New Millimeter Wave Cellular Network Architecture","volume":"37","author":"Koutsaftis","year":"2019","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1662","DOI":"10.1109\/LWC.2021.3076463","article-title":"3D Placement and Orientation of mmWave-Based UAVs for Guaranteed LoS Coverage","volume":"10","author":"Sabzehali","year":"2021","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_17","first-page":"17","article-title":"Applications de l\u00e1lgebre de boole en recherche operationelle","volume":"4","author":"Fortet","year":"1960","journal-title":"Rev. Fr. Rech. Oper."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.ejor.2013.01.044","article-title":"Dirk Oliver Theis: Compact formulations of the Steiner Traveling Salesman Problem and related problems","volume":"228","author":"Letchford","year":"2013","journal-title":"Eur. J. Oper. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1002\/net.21610","article-title":"Stochastic maximum weight forest problem","volume":"65","author":"Adasme","year":"2015","journal-title":"Networks"},{"key":"ref_20","unstructured":"Cormen, T.H., Leiserson, C.E., Rivest, R.L., and Stein, C. (2009). Introduction to Algorithms, MIT Press and McGraw-Hill."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/123\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:46:11Z","timestamp":1760103971000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,19]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["sym16010123"],"URL":"https:\/\/doi.org\/10.3390\/sym16010123","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,19]]}}}