[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86282-en":3,"doc-seo-86282-105":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},86282,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Any Proof of Polynomial Hirsch Must Be Completely Incoherent","The paper studies the polynomial Hirsch conjecture for polytopes, motivated by the simplex method in linear programming, where one traces a directed path along edges while the linear objective increases. Building on coherent monotone path results of Billera–Sturmfels and their topology characterization with Kapranov, the work asks whether polynomial-length coherent monotone paths always exist for every linear-induced orientation. It disproves this by constructing polytopes and linear functions forcing all coherent monotone paths to be exponentially long, and applies the method to strengthen lower bounds for the shadow simplex method, geometric transversals, and parametric linear optimization.","arXiv :2607 . 11628v1 [math .CO] 13 Jul 2026  \nANY PROOF OF POLYNOMIAL HIRSCH MUST BE COMPLETELY  \nINCOHERENT  \nALEXANDER E. BLACK AND LEI XUE  \nAbstract . In 1992, Billera and Sturmfels introduced coherent monotone paths on polytopes as part of their description of the fiber polytope construction, and later in 1994 showed with Kapranov that these coherent monotone paths capture the topology of the space of all monotone paths, paths from a minimum to a maximum, in the directed graph of a polytope with orientation induced by a linear function. Those results motivate the following analog of the polynomial Hirsch conjecture:  \nDoes there always exist a coherent monotone path of polynomial length on a polytope for any choice of orientation induced by a linear function? We show this is not the case by exhibiting a family of polytopes and corresponding linear functions for which every coherent monotone path is exponentially long. As applications, we strengthen longstanding results pertaining to lower bounds for the shadow simplex method, geometric transversals in discrete geometry, and parametric linear optimization.  \n1. Introduction  \nThe polynomial Hirsch conjecture is among the most notorious and well studied problems in polyhedral theory. It asks for a polynomial bound in terms of the number of facets n and dimension d of a polytope on the length of a shortest path between any pair of vertices in its one-skeleton, its set of vertices and edges. This problem is so well studied that to survey all partial results is well beyond the scope of this paper, and we defer to the survey of Kim and Santos [40] and the comprehensive references to the simplex method literature found in [9] . Famously, Hirsch originally conjectured abound of n − d, and this was disproven by Santos in [49] . Despite this progress, the best known lower bounds remain linear on the order of about 1.05(n − d) . The best known upper bounds are on the order of nlog(d) originally due to Kalai and Kleitman in [37] .  \nThe core motivation for this problem comes from the simplex method for linear programming. Namely, the simplex method solves a linear program max(c⊺x) such that Ax ≤ b by tracing a path in the one-skeleton of the polytope such that at each step the linear objective function increases. It is a major open problem in linear optimization whether there is a version of the simplex method that runs in polynomial time, and if the polynomial Hirsch conjecture is false, the answer to that question is unconditionally no.  \nIn an effort to make progress on this very difficult problem, several variants and abstractions of the polynomial Hirsch conjecture have appeared in the literature. Open variants include the polynomial monotone Hirsch conjecture motivated by the simplex method’s requirement that the objective function increases at each step, the strict monotone Hirsch conjecture by Ziegler in [54], and the abstract Hirsch conjecture by Kalai for the polymath3 project motivated by the connected layer families introduced in [29] . A positive resolution to any of those would imply the polynomial Hirsch  \nDepartment of Mathematics, Bowdoin College  \nDepartment of Mathematics, Colby College  \nE-mail addresses: [a.black@bowdoin.edu](a.black@bowdoin.edu) , [leixue@colby.edu](leixue@colby.edu).  \n2020 Mathematics Subject Classification. 90C05, 52B12, 90C31, 52A35 .  \nKey words and phrases. Simplex Method, Parametric Optimization, Lower Bounds.  \n2  \nconjecture. Other variants have been resolved including the circuit diameter conjecture of [18] recently proven in its polynomial version in [44] and the continuous variant for interior point methods from [25] that turned out to be false by breakthrough work in [2] . The analogous question for simplicial complexes is known to have a negative answer even for pseudomanifolds [22 , 50] and a positive resolution for flag normal simplicial complexes [1] . It is open for simplicial manifolds with a notable case being that of simpli","cbCaiaQeQCqRFExT","https://ap.wps.com/l/cbCaiaQeQCqRFExT","pdf",491332,2,1,21,"English","en",105,"# Introduction\n## Polynomial Hirsch conjecture and background\n## Connection to the simplex method\n## Variants and related conjectures\n## Monotone paths, Baues complex, and coherent paths","[{\"question\":\"What question does the paper ask about the polynomial Hirsch conjecture?\",\"answer\":\"It asks whether, for any orientation of a polytope induced by a linear function, there always exists a coherent monotone path whose length is polynomial in the polytope parameters.\"},{\"question\":\"What is the paper’s main negative result?\",\"answer\":\"It shows there are families of polytopes and corresponding linear functions for which every coherent monotone path is exponentially long.\"},{\"question\":\"Which areas are mentioned as applications of these results?\",\"answer\":\"The paper applies the approach to strengthen lower bounds for the shadow simplex method, results about geometric transversals in discrete geometry, and lower bounds for parametric linear optimization.\"}]",1784210024,53,{"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},"any-proof-of-polynomial-hirsch-must-be-completely-incoherent","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/any-proof-of-polynomial-hirsch-must-be-completely-incoherent/86282/",4,{"url":51,"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","2026-07-26","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 question does the paper ask about the polynomial Hirsch conjecture?","Question",{"text":75,"@type":76},"It asks whether, for any orientation of a polytope induced by a linear function, there always exists a coherent monotone path whose length is polynomial in the polytope parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the paper’s main negative result?",{"text":80,"@type":76},"It shows there are families of polytopes and corresponding linear functions for which every coherent monotone path is exponentially long.",{"name":82,"@type":73,"acceptedAnswer":83},"Which areas are mentioned as applications of these results?",{"text":84,"@type":76},"The paper applies the approach to strengthen lower bounds for the shadow simplex method, results about geometric transversals in discrete geometry, and lower bounds for parametric linear 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