[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86309-en":3,"doc-seo-86309-105":30,"detail-sidebar-cat-0-en-105":83},{"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},86309,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Improving Upper Bounds for the Maximum Clique Problem using Reduction Rules","The work investigates the interaction between reduction rules and upper-bound functions for the Maximum Clique Problem (MCP). It shows how MCP upper-bound functions can strengthen classical core and truss reductions by replacing local size conditions with upper-bound tests, yielding (k,ωu)-core, (k,ωu)-truss, and a general (k,d,ωu)-truss. For each notion, clique preservation, correctness of the peeling algorithm, and running-time bounds are proved. A framework for improving upper-bound values is developed and validated on benchmark graphs.","arXiv :2607 . 11726v1 [math .CO] 13 Jul 2026  \nImproving Upper Bounds for the Maximum Clique Problem  \nusing Reduction Rules  \nAljaˇz Krpan ∗† Janez Povh ‡  \nAbstract  \nWe study the interaction between reduction rules and upper-bound functions for the Maximum Clique Problem (MCP) . We show how MCP upper-bound functions can strengthen classical core and truss reductions by replacing local size conditions with upper-bound tests. This leads to the (k,ωu )-core, the (k,ωu )-truss, and the more general (k, d,ωu )-truss, where the parameter d controls the trade-off between stronger reductions and additional computational cost. For each of these notions, we prove clique-preservation properties, correctness of the corresponding peeling algorithm, and running-time bounds. Based on these reductions, we introduce a general framework for improving upper-bound values for MCP. We give two concrete instantiations of the framework: one that uses only the combined truss and core reductions, and one that combines the truss and core reductions with repeated applications of structions. Computational experiments on 73 benchmark graphs show that the proposed reductions can substantially improve several standard upper-bound functions and that combining multiple reduction methods can be beneficial in practice. In particular, the combination of structions, truss and core reductions with a DSatur-based bound often reached SDP-level upper-bound values faster than direct SDP computation; on the tested graphs with edge density below 0 .7, it did so in every case. Using the truss and core reduction with the Lov´asz theta upper-bound function, we also improve the previously best certified integer upper-bound values for three difficult DIMACS instances whose exact clique numbers are not known. In particular, we improve upper-bound values for graph C500 .9 from 83 to 73, for graph C1000 .9 from 122 to 115, and for graph C2000 .9 from 177 to 168 .  \nKeywords: Maximum Clique Problem; upper bounds; reductions; core decomposition; truss decomposition; structions; DIMACS benchmarks.  \nMSC 2020: Primary 05C85; Secondary 05C69, 90C27, 68R10, 68W40 .  \n1 Introduction  \n1.1 Motivation  \nThe Maximum Clique Problem (MCP) is among the most extensively studied problems in graph theory and combinatorial optimization. Given a graph G, the MCP asks to determine the clique number ω(G), that is, the size of a largest subset of pairwise adjacent vertices. In the constructive variant, one also asks to output such a clique. Since MCP is NP-hard [25], it has received sustained attention in both theoretical and applied research. Practical instances arise in numerous areas, including bioinformatics (e.g., protein–protein interaction prediction) [55], social network analysis (e.g., community detection) [17], and structural biology or drug design (e.g., local structure alignment) [28] .  \n∗ Faculty of mathematics and physics, University of Ljubljana, Slovenia [krpan.aljaz@gmail.com](krpan.aljaz@gmail.com)[ ](krpan.aljaz@gmail.com)†Rudolfovo, Science and Technology Centre Novo mesto, Slovenia  \n‡Rudolfovo, Science and Technology Centre Novo mesto, [Slovenia](Slovenia janez.povh@rudolfovo.eu)[ janez.povh@rudolfovo.eu](Slovenia janez.povh@rudolfovo.eu)  \nSince MCP is NP-hard, practical algorithms often rely on techniques that reduce the search space. Two important tools for this purpose are reduction rules and upper-bound functions. Reduction rules simplify the input graph while preserving the clique sizes relevant to the problem, and are widely used in preprocessing, exact solvers, and local-search algorithms [12, 42, 52] . On the other hand, upper-bound functions certify that a graph cannot contain a clique larger than a certain value, without necessarily finding a maximum clique. A common example of this is pruning in branch-and-bound algorithms.  \nIn this paper, we study the interaction between reductions and upper-bound functions. On one hand, upper-bound functions can make reductio","cbCaidK2GHG7HFVe","https://ap.wps.com/l/cbCaidK2GHG7HFVe","pdf",767676,4,1,40,"English","en",105,"# Introduction\n## Motivation\n## Our contribution","[{\"question\":\"How are reductions used to improve upper-bound values for the MCP?\",\"answer\":\"When a reduction preserves all cliques of a target size, computing an upper bound on the reduced graph certifies the absence of such cliques in the original graph and can yield tighter or easier-to-compute bounds.\"}]",1784210381,101,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"improving-upper-bounds-for-the-maximum-clique-problem-using-reduction-rules","",{"@graph":36,"@context":77},[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/improving-upper-bounds-for-the-maximum-clique-problem-using-reduction-rules/86309/",{"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],{"name":72,"@type":73,"acceptedAnswer":74},"How are reductions used to improve upper-bound values for the MCP?","Question",{"text":75,"@type":76},"When a reduction preserves all cliques of a target size, computing an upper bound on the reduced graph certifies the absence of such cliques in the original graph and can yield tighter or easier-to-compute bounds.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,111,114,119,122,126],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":22,"slug":110},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]