[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86359-en":3,"doc-seo-86359-105":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},86359,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",8,"Research & Report","Efficiently Coloring the Intersection of a General Matroid and Combinatorial Matroids","This paper presents a polynomial-time algorithm for matroid intersection coloring when one matroid is general and the remaining matroids are partition matroids. Given a general matroid M1 and k−1 partition matroids M2,…,Mk, it colors the intersection M=∩i=1kMi using at most 1+∑i=1k(χ(Mi)−1) colors. It is the first polynomial-time O(k)-approximation for this setting, and it extends to standard combinatorial matroid types via reductions to partition matroids.","arXiv :2508 . 19473v2 [ cs .DS] 13 Jul 2026  \nEfficiently Coloring the Intersection of a General Matroid and Combinatorial Matroids  \nStephen Arndt 1 [0009-0008-2847-0721], Benjamin Moseley 1 [0000-0001-8162-017X], Kirk Pruhs2 [0000-0001-5680-1753], and Michael Zlatin3 [0000-0003-1773-1152]  \n1 Carnegie Mellon University, Pittsburgh, PA, USA  \n{sarndt,[moseleyb}@andrew.cmu.edu](moseleyb}@andrew.cmu.edu)[ ](moseleyb}@andrew.cmu.edu)2 University of Pittsburgh, Pittsburgh, PA, [USA](USA kirk@cs.pitt.edu)[ kirk@cs.pitt.edu](USA kirk@cs.pitt.edu)[ ](USA kirk@cs.pitt.edu)3 Pomona College, Claremont, CA, [USA](USA michael.zlatin@pomona.edu)[ michael.zlatin@pomona.edu](USA michael.zlatin@pomona.edu)  \n[Abstract](Abstract). This paper shows a polynomial-time algorithm that, given a general matroid M1 and k−1 partition matroids M2 , . . . , Mk , produces a coloring of the intersection M = ∩ki=1 Mi using at most 1 +Pki=1 (χ(Mi) − 1) colors. This is the first polynomial-time O (k)-approximation algorithm for matroid intersection coloring where one of the matroids may be a general matroid. Leveraging the fact that most of the standard combinatorial matroids reduce to partition matroids ata loss of a factor of two in the chromatic number, this algorithm also yields a polynomial-time O (k)-approximation algorithm for matroid intersection coloring in the case where each of the matroids M2 , . . . , Mk are one of these standard combinatorial types. Even when k = 2, the previous best-known approximation ratio was O(log n) via a reduction to Set Cover.  \n1 Introduction  \nSome of the most generally applicable tools in an algorithmist’s toolkit are efficient algorithms for various standard matroid-related optimization problems. In particular, a reasonable number of natural optimization problems can be viewed as set cover problems, in which the constraints can be partitioned into parts that induce a matroid [30,38] . Such problems can be viewed as matroid intersection coloring problems, defined as follows:  \n– The input is a collection of matroids M1 = (X, I1 ) ,..., Mk = (X, Ik) on common ground set X. The intersection M = ∩ki=1 Mi is defined to be the set system 􀀀X, ∩ki=1Ii 􀀁 .  \n– A feasible solution is a coloring of the elements of X such that, for every color, theset of elements assigned that color is independent in M , i.e. independent in each matroid Mi for i ∈ [k] .  \n– The objective is to minimize the number of colors used. For any set system M, we use χ (M) to denote the chromatic number of M , that is the minimum number of colors necessary for the existence of a feasible coloring.  \nThere is a significant literature within the combinatorics/mathematics community that establishes existential results related in some way to matroid intersection coloring  \n2 Stephen Arndt , Benjamin Moseley , Kirk Pruhs, and Michael Zlatin  \n(and/or packing) [3,8,2, 13,22, 19,32, 11] . However, most of the proofs of these results are nonconstructive, in that they are not readily adaptable to yield polynomial-time coloring algorithms. In particular, many of these results are established with topological fixed-point arguments that employ Sperner’s lemma [3,8,2] . We now highlight some of these existential results that are most relevant for this paper.  \nTheorem 1. [3] For k general matroids M1 ,..., Mk, it is the case that χ 􀀀∩ki=1 Mi 􀀁 ≤(2k − 1)maxki=1 χ (Mi).  \nFor the intersection of two general matroids (k = 2), this can be improved to the following.  \nTheorem 2. [8] For two general matroids M1 and M2 it is the case that χ(M1 ∩M2 ) ≤χ (M1 ) + χ(M2 ).  \nThe proofs of Theorems 1 and 2 both hinge on topological fixed-point arguments, deriving from an influential earlier paper by Aharoni and Berger [2] that shows that χ (M1 ∩ M2 ) ≤ 2max{χ(M1 ) ,χ (M2 )} . These theorems show that in some sense the chromatic number of the intersection of k matroids grows at most linearly with k. Further, there is a lower bound that rules out the possibility of many natural typ","cbCaildsyifrp4Uc","https://ap.wps.com/l/cbCaildsyifrp4Uc","pdf",4290100,3,1,26,"English","en",105,"# Introduction\n## Matroid intersection coloring definition\n## Prior existential results and bounds\n## Algorithmic landscape for polynomial-time coloring","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper studies matroid intersection coloring: assigning elements a minimum number of colors so that each color class is independent in every matroid in the intersection.\"},{\"question\":\"What is the main algorithmic result?\",\"answer\":\"It gives a polynomial-time algorithm that, for one general matroid and k−1 partition matroids, produces a coloring using at most 1+∑i=2k(χ(Mi)−1) colors.\"},{\"question\":\"How does the paper improve over previous approximation ratios?\",\"answer\":\"It provides the first polynomial-time O(k)-approximation algorithm in this setting; prior work for k=2 achieved only O(log n) using a reduction to Set Cover.\"}]",1784210840,66,{"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},"efficiently-coloring-the-intersection-of-a-general-matroid-and-combinatorial-matroids","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/efficiently-coloring-the-intersection-of-a-general-matroid-and-combinatorial-matroids/86359/",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 problem does the paper address?","Question",{"text":75,"@type":76},"The paper studies matroid intersection coloring: assigning elements a minimum number of colors so that each color class is independent in every matroid in the intersection.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main algorithmic result?",{"text":80,"@type":76},"It gives a polynomial-time algorithm that, for one general matroid and k−1 partition matroids, produces a coloring using at most 1+∑i=2k(χ(Mi)−1) colors.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper improve over previous approximation ratios?",{"text":84,"@type":76},"It provides the first polynomial-time O(k)-approximation algorithm in this setting; 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