[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86143-en":3,"doc-seo-86143-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86143,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Proving Optimality for the Bandwidth Multicoloring Problem via SAT","The Bandwidth Multicoloring Problem (BMCP) is an NPhard extension of the Bandwidth Coloring Problem (BCP) with key uses in telecommunications, resource allocation, and scheduling. While metaheuristics find high-quality solutions, they cannot certify global optimality. Existing exact CP/IP methods often require heavy computation and still lag in solution quality, leaving many benchmarks without certificates. This paper introduces a first SAT-based exact framework with an efficient encoding, strong domain reduction, and incremental SAT solving. Experiments on GEOM and MS-CAP prove improved performance and expand optimality certification.","RAIRO Operations Research  \nWill be set by the publisher  \ne-mail: [23021533@vnu.edu.vn](23021533@vnu.edu.vn) e-mail: [khanhtv@vnu.edu.vn](khanhtv@vnu.edu.vn)  \narXiv :2607 . 1 1 12 1v 1 [ cs .LO] 13 Jul 2026  \nPROVING OPTIMALITY FOR THE BANDWIDTH MULTICOLORING PROBLEM VIA SAT  \nDuc Trung Kim Nguyen 1 and Khanh Van To 1  \nAbstract. The Bandwidth Multicoloring Problem (BMCP) is an NPhard extension of the Bandwidth Coloring Problem (BCP) with important applications in telecommunications, resource allocation, and scheduling. While state-of-the-art metaheuristics can efficiently produce high-quality solutions, they cannot certify global optimality. Existing exact approaches based on Constraint Programming (CP) and Integer Programming (IP) provide such guarantees but typically require extensive computation and still lag behind metaheuristics in solution quality, leaving many benchmark instances without optimality certificates. In this paper, we present the first SAT-based exact framework for the BMCP. Our main contribution is an efficient SAT encoding that compactly models both intra-vertex and inter-vertex color distance constraints. Combined with tight color domain reduction and an incremental SAT-solving strategy, the proposed formulation significantly prunes the search space and enables efficient exact optimization. Experimental results on the GEOM and MS-CAP benchmark suites demonstrate substantial improvements over previous exact approaches. On the challenging GEOM benchmark, the proposed framework proves optimality for more instances within only one hour of computation than the previous CP/IP approach, which required a 48-hour time limit, while also verifying the optimality of several previously reported bestknown solutions. These results demonstrate that SAT-based reasoning provides an effective exact optimization framework for the BMCP and substantially expands the range of benchmark instances whose optimality can be certified.  \nKeywords: Channel assignment, Graph coloring problem, Bandwidth coloring, Bandwidth Multicoloring, SAT solving  \nMathematics Subject Classification. 05C15, 90C27, 68R07  \n1 VNU University of Engineering and Technology, Hanoi, Vietnam  \n© EDP Sciences 2001  \n2 TITLE WILL BE SET BY THE PUBLISHER  \n1. Introduction  \nThe Bandwidth Coloring Problem (BCP) is a combinatorial optimization problem that extends the classical graph coloring problem (GCP), with numerous practical applications in areas such as scheduling, resource allocation, and network design. The problem can be formally defined as follows: given an edge-weighted undirected graph G = (V, E), a weight function d : E → Z+ assigns a positive integer weight to each edge e = {u, v} ∈ E. The objective of the BCP is to find a coloring c : V → N such that ∀e = {u, v} ∈ E : |c(u) − c(v)| ≥ d ({u, v}), while minimizing the span value, defined as k = maxv∈V c (v) .  \nThe Bandwidth Multicoloring Problem (BMCP) generalizes the BCP by allowing each vertex to be assigned multiple colors. Formally, let G = (V, E) be an undirected graph equipped with two weight functions: a vertex weight function w : V → Z+ specifying the number of colors required for each vertex, and an edge weight function d : E → Z+ defining the minimum color separation between adjacent vertices. The objective of the BMCP is to find a multicoloring assignment c : V → P (Z+ ) that minimizes the span value k = maxu∈V,m∈c(u) m while satisfying the following conditions: first, each vertex u ∈ V is assigned exactly w (u) distinct colors; second, any two distinct colors m, n ∈ c (u) assigned to the same vertex must satisfy |m − n| ≥ d ({u, u}); and finally, for any edge e = {u, v} ∈ E , every color m ∈ c (u) and n ∈ c (v) must satisfy |m − n| ≥ d ({u, v}) .  \nBoth the BCP and the BMCP arise naturally from frequency assignment problems (FAP) [1] in the field of telecommunications, where the graph’s vertices represent radio transmitters and the edge weights dictate the minimum frequency separatio","cbCaiiqf8aCeEYnp","https://ap.wps.com/l/cbCaiiqf8aCeEYnp","pdf",372416,4,1,23,"English","en",105,"# Introduction\n## Bandwidth Coloring Problem (BCP)\n## Bandwidth Multicoloring Problem (BMCP)\n## Motivation for exact optimality","[{\"question\":\"What is the Bandwidth Multicoloring Problem (BMCP) and how does it extend the BCP?\",\"answer\":\"BMCP generalizes BCP by assigning each vertex multiple distinct colors. It minimizes a span value while enforcing minimum separation constraints between colors on the same vertex and across adjacent vertices via edge-weighted requirements.\"},{\"question\":\"Why do metaheuristics not provide sufficient results for BMCP?\",\"answer\":\"Metaheuristics can efficiently produce high-quality solutions, but they do not certify global optimality. As a result, benchmark instances may lack provable optimality certificates.\"},{\"question\":\"What is the main contribution of the proposed work using SAT for BMCP?\",\"answer\":\"The work presents an SAT-based exact framework with an efficient SAT encoding that compactly models intra-vertex and inter-vertex distance constraints. It also uses tight color domain reduction and an incremental SAT-solving strategy to prune the search and enable efficient exact optimization.\"}]",1784208886,58,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"proving-optimality-for-the-bandwidth-multicoloring-problem-via-sat","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/proving-optimality-for-the-bandwidth-multicoloring-problem-via-sat/86143/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the Bandwidth Multicoloring Problem (BMCP) and how does it extend the BCP?","Question",{"text":75,"@type":76},"BMCP generalizes BCP by assigning each vertex multiple distinct colors. It minimizes a span value while enforcing minimum separation constraints between colors on the same vertex and across adjacent vertices via edge-weighted requirements.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do metaheuristics not provide sufficient results for BMCP?",{"text":80,"@type":76},"Metaheuristics can efficiently produce high-quality solutions, but they do not certify global optimality. As a result, benchmark instances may lack provable optimality certificates.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main contribution of the proposed work using SAT for BMCP?",{"text":84,"@type":76},"The work presents an SAT-based exact framework with an efficient SAT encoding that compactly models intra-vertex and inter-vertex distance constraints. It also uses tight color domain reduction and an incremental SAT-solving strategy to prune the search and enable efficient exact optimization.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]