[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84439-en":3,"doc-seo-84439-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84439,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Graph Optimization Foundation Model: Tokenizing Graph via a Language-Model Paradigm","Graph optimization underpins vehicle routing, community detection, and other recurring decision problems, yet dominant approaches solve each instance from scratch and fail to reuse persistent structural knowledge in the underlying network. This omission causes knowledge loss across decision cycles, leading to computational redundancy, limited scalability, and weak real-world feasibility. GOFM introduces a task-agnostic structural prior by tokenizing graph structure via topology-aware random walks and Transformer masked reconstruction, enabling feasible lightweight constrained decoding. Experiments on multiple routing families and up to 1,945-node networks show competitive results with reduced computation, and reuse is validated beyond routing through community detection, influence maximization, and an Amazon Last Mile case study.","arXiv :2509 .24256v2 [ cs .LG] 13 Jul 2026  \nGRAPH OPTIMIZATION FOUNDATION MODEL: TOKENIZING GRAPH VIA A LANGUAGE-MODEL PARADIGM  \nA PREPRINT  \nYunhao Liang 1 Pujun Zhang 1 ∗ Yuan Qu 1 ∗ Jingyuan Yang2 Shaochong Lin 1 Max Z.J. Shen 1 ,3 ,4  \n1Faculty of Engineering, The University of Hong Kong, Hong Kong SAR, China  \n2 Costello College of Business, George Mason University, VA, USA  \n3Faculty of Business and Economics, The University of Hong Kong, Hong Kong, China  \n4 College of Engineering, University of California, Berkeley, CA, USA  \nABSTRACT  \nGraph optimization is foundational to modern operations, routing vehicles through road  \nnetworks, detecting communities in social networks, and so on. Yet, prevailing paradigms,  \nranging from classical heuristics to recent neural solvers, treat each task instance in iso  \nlation. This solve-from-scratch paradigm overlooks the persistent topological knowledge  \nembedded within the underlying network, leading to significant knowledge loss across  \nrecurring decision cycles and multiple optimization problems, which usually results in  \ncomputational redundancy, limits in scalability, and real-world infeasibility. Therefore,  \nwe introduce the Graph Optimization Foundation Model (GOFM), which shifts the focus  \nfrom task-specific solvers to the pretraining of a task-agnostic structural prior. GOFM  \ninternalizes network topology and distance geometry by encoding structure-aware random  \nwalks into a Transformer through progressive masked reconstruction. This process tokenizes  \na representation that captures the fundamental properties of the graph. At inference, this  \nlearned prior facilitates diverse optimization tasks via lightweight constrained decoding,  \nensuring strict feasibility without retraining. Evaluations across five routing families and  \nnetworks of up to 1,945 nodes demonstrate that GOFM achieves competitive performance  \ncompared to specialized solvers while significantly reducing computational time. Beyond  \nrouting, a real-world Amazon Last Mile case study and experiments on community detection  \nand influence maximization confirm the model’s capacity for representation reuse. For  \npractitioners, GOFM offers a path toward treating persistent networks as reusable decision  \nassets, reducing repeated solver engineering while enabling fast, feasible decision support at  \nscale.  \nKeywords: Graph Optimization Foundation Model; Pretrain–Transfer Learning; Self-Supervised Graph Representation  \n1 Introduction  \nIn Greek mythology, Sisyphus is condemned to roll a boulder up a hill for eternity. Each time he nearsthe summit, the boulder rolls back, and he must begin again from nothing. The punishment is not the effort itself but the absence of memory: no ascent teaches him anything that makes the next one easier. Network optimization today suffers from a strikingly similar curse. Every time a new routing, partitioning,  \n∗ Correspondence to: Pujun Zhang (pjzhang@hku.hk), Yuan Qu (yuanqu@hku.hk)  \nor targeting problem arrives on the same graph, the solver starts from the bottom of the hill again and again. The connection patterns it reconstructed yesterday, the bottlenecks it discovered last week, and the distance relationships it computed an hour ago, all the knowledge is not carried forward while the graph remains the same. If this persistent structural knowledge could be distilled from the graph, must every downstream optimization task still begin from the bottom of the hill?  \nThe answer may lie in the pretrain-transfer paradigm that has recently redefined artificial intelligence (Bommasani et al. 2021, Brown et al. 2020) . Large language models (LLMs) learn structured regularities from large-scale data through self-supervision. They then transfer that knowledge across tasks with minimal adaptation (Devlin et al. 2019, Achiam et al. 2023, Yang et al. 2025) . The model sequentially learns which completions are plausible in a given context, and the resulting prior, a form ","cbCaiaeP8mX5Txsy","https://ap.wps.com/l/cbCaiaeP8mX5Txsy","pdf",2609742,1,35,"English","en",105,"# Abstract\n# Introduction\n## Problem of solve-from-scratch in persistent networks\n## Motivation from pretrain–transfer learning in LLMs\n## Gap between OR combinatorial solvers and LLM-centric approaches","[{\"question\":\"What issue does GOFM address in graph optimization?\",\"answer\":\"GOFM targets the solve-from-scratch paradigm that discards previously learned topological knowledge even when the underlying graph remains persistent across many optimization queries.\"},{\"question\":\"How does GOFM build the reusable prior?\",\"answer\":\"GOFM internalizes network topology and distance geometry by encoding structure-aware random walks into a Transformer using progressive masked reconstruction, producing a tokenized representation of graph properties.\"},{\"question\":\"How are optimization tasks handled at inference without retraining?\",\"answer\":\"At inference, GOFM supports diverse tasks through lightweight constrained decoding using the learned prior, aiming to maintain strict feasibility without task-specific retraining.\"}]",1784195634,88,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"graph-optimization-foundation-model-tokenizing-graph-via-a-language-model-paradigm","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/graph-optimization-foundation-model-tokenizing-graph-via-a-language-model-paradigm/84439/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 issue does GOFM address in graph optimization?","Question",{"text":75,"@type":76},"GOFM targets the solve-from-scratch paradigm that discards previously learned topological knowledge even when the underlying graph remains persistent across many optimization queries.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does GOFM build the reusable prior?",{"text":80,"@type":76},"GOFM internalizes network topology and distance geometry by encoding structure-aware random walks into a Transformer using progressive masked reconstruction, producing a tokenized representation of graph properties.",{"name":82,"@type":73,"acceptedAnswer":83},"How are optimization tasks handled at inference without retraining?",{"text":84,"@type":76},"At inference, GOFM supports diverse tasks through lightweight constrained decoding using the learned prior, aiming to maintain strict feasibility without task-specific 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