[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-505110-105":59,"doc-detail-505110-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","a-new-branch-and-filter-exact-algorithm-for-binary-constraint-satisfaction-problems","A new branch-and-filter exact algorithm for binary constraint satisfaction problems","","A binary constraint satisfaction problem (BCSP) seeks an assignment of variable values that satisfies pairwise constraints, forming a cornerstone challenge in constraint programming with broad practical relevance. The work introduces an exact method reformulating BCSP as a k-clique problem on the microstructure graph. The algorithm combines state-of-the-art maximum-clique branching, domain-reducing filtering phases using coloring and a related Boolean SAT solving step, and a microstructure vertex reordering initialization that often simplifies reformulation. Extensive experiments on nearly 2000 instances show strong overall speedups versus SAT-based and general CP solvers.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/a-new-branch-and-filter-exact-algorithm-for-binary-constraint-satisfaction-problems/505110/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/a-new-branch-and-filter-exact-algorithm-for-binary-constraint-satisfaction-problems/505110.png","ImageObject",300,407,{"name":92,"@type":93},"adawds","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-19","2026-10-04",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the binary constraint satisfaction problem (BCSP) and why is it important?","Question",{"text":112,"@type":113},"BCSP asks for an assignment to variables that satisfies constraints involving only pairs of variables. It is a fundamental problem in constraint programming and appears in many real-world applications.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the proposed algorithm solve BCSP exactly?",{"text":117,"@type":113},"It reformulates the BCSP as a k-clique problem on the microstructure graph, then applies an advanced branching scheme for maximum clique along with two filtering phases that reduce variable domains.",{"name":119,"@type":110,"acceptedAnswer":120},"What techniques drive the filtering phases and how is their effectiveness assessed?",{"text":121,"@type":113},"The filtering phases are based on coloring methods and on heuristically solving an associated Boolean SAT problem. Effectiveness is evaluated through a large benchmark campaign of nearly 2000 instances, comparing against SAT-based and general-purpose CP solvers.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},505110,1791309753,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},3985747872983,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","A new branch-and-􀀌lter exact algorithm for binary constraint satisfaction problems  \nPablo San Segundo 1  \nUniversidad Polit􀀓ecnica de Madrid (UPM), Centre for Automation and Robotics (CAR), Madrid, Spain,  \n[pablo.sansegundo@upm.es](pablo.sansegundo@upm.es)  \nFabio Furini  \nIstituto di Analisi dei Sistemi ed Informatica \\Antonio Ruberti\" { Consiglio Nazionale delle Ricerche  \n(IASI-CNR), Rome, [Italy. fabio.furini@cnr.it](Italy. fabio.furini@cnr.it)  \nRafael Le􀀓on  \nUniversidad Camilo Jos􀀓e Cela (UCJC), Science and Technology School, Madrid, Spain. rleon@ucjc. edu  \n\n| Abstract\u003Cbr>A binary constraint satisfaction problem (BCSP) consist in determining an assignment of values to variables which is compatible with a set of constraints. The problem is called binary because the constraints involve only pairs of variables. The BCSP is a cornerstone problem in Constraint Programming (CP), appearing in a very wide range of real-world applications. In this work, we develop a new exact algorithm which e􀀋ectively solves the BCSP by reformulating it as a k-clique problem on the underlying microstructure graph representation. Our new algorithm exploits the cutting-edge branching scheme of the state-ofthe-art maximum clique algorithms combined with two 􀀌ltering phases in which the domains of the variables are reduced. Our 􀀌ltering phases are based on coloring techniques and on heuristically solving an associated boolean satis􀀌ability (SAT) problem. In addition, the algorithm initialization phase performs a reordering of the microstructure graph vertices which produces an often easier reformulation to solve. We carry out an extensive computational campaign on a benchmark of almost 2000 instances which encompasses numerous real and synthetic problems from the literature. The performance of the new algorithm is compared against two SAT-based solvers and three general purpose CP solvers. Our tests reveal that the new algorithm signi􀀌cantly outperforms all the others in several classes of BCSP instances.\u003Cbr>Keywords: Binary Constraint Satisfaction Problems, Constraint Programming, Exact Algorithm, Computational Experiments. |\n| --- |\n|  |\n\n1 Corresponding author  \nPreprint submitted to . . .  \nDecember 27, 2020  \n1. Introduction  \nA constraint satisfaction problem (CSP), in its general form, asks for an assignment of values to the variables of the problem respecting a set of constraints. It is a central topic in Constraint Programming (CP) with many practical applications due to its broad representational scope: location of facilities, scheduling, car sequencing, vehicle routing and many others, see, e.g., [9] and part II of [54] for a survey.  \nFormally, we are given a set X = fx1 ; x2 ; : : : ; xkg of k variables, each of which is associated to a 􀀌nite set of values D (xi ), called the domain of the variable. Without loss of generality (w.l.o.g.), we assume D (xi ) is a subset of Z. In addition, we are given a set of q constraints C = fc1 ; c2 ; : : : ; cqg, and each constraint cj is a pair (X(cj); R (cj )) where X(cj) 􀀒 X is the subset of variables involved in the constraint. This set of variables is called the scope of the constraint and R (cj ) is a set of tuples, called the relation of the constraint. Each tuple R (cj) 2 Z jX(cj)j is composed of jX(cj)j values allowed by the constraint for the variables in its scope. In other words, each tuple represents a feasible combination of variable values. The CSP calls for 􀀌nding a feasible assignment of values to each one of the variables which satis􀀌es all the constraints. If no solution exists, the problem is said to be inconsistent or unsatis􀀌able. Determining whether a CSP has a solution is N P-complete, since the problem admits the boolean satis􀀌ability (SAT) problem as a special case, the archetypal NP-complete decision problem. The state-of-the-art exact algorithms to solve CSPs are general purpose solvers based on CP and SAT techniques. A review of these algorithms is provided in Section ","cbCail3XwMs7NUZB","https://ap.wps.com/l/cbCail3XwMs7NUZB","pdf",625709,38,"English","# Abstract\n# 1. Introduction","[{\"question\":\"What is the binary constraint satisfaction problem (BCSP) and why is it important?\",\"answer\":\"BCSP asks for an assignment to variables that satisfies constraints involving only pairs of variables. It is a fundamental problem in constraint programming and appears in many real-world applications.\"},{\"question\":\"How does the proposed algorithm solve BCSP exactly?\",\"answer\":\"It reformulates the BCSP as a k-clique problem on the microstructure graph, then applies an advanced branching scheme for maximum clique along with two filtering phases that reduce variable domains.\"},{\"question\":\"What techniques drive the filtering phases and how is their effectiveness assessed?\",\"answer\":\"The filtering phases are based on coloring methods and on heuristically solving an associated Boolean SAT problem. Effectiveness is evaluated through a large benchmark campaign of nearly 2000 instances, comparing against SAT-based and general-purpose CP solvers.\"}]","A new branch-and-filter exact algorithm for binary constraint satisfaction problems | PDF",1791090000,96]