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The text introduces key concepts such as efficiency, complexity, feasible solution sets, and problem size growth, and motivates why efficient algorithms may not exist. It surveys major problem classes including minimum spanning trees, shortest paths, maximum flow, and dynamic programming, with related formulations and algorithmic approaches.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/combinatorial-optimisation-msm-3m02b-preliminary-version/565003/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/combinatorial-optimisation-msm-3m02b-preliminary-version/565003.png","ImageObject",442,249,{"name":88,"@type":89},"CatatanPagi","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-21","2026-10-12",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What is combinatorial optimisation and why is it challenging?","Question",{"text":108,"@type":109},"Combinatorial optimisation is an area of optimisation in applied mathematics and computer science, closely related to operations research, algorithm theory, and computational complexity. It addresses problems whose feasible sets are discrete, but the number of feasible solutions can grow very rapidly, making efficient algorithms difficult.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What does the document mean by an efficient algorithm?",{"text":113,"@type":109},"The document’s main aim is to construct efficient algorithms for combinatorial optimisation problems. It explicitly discusses the idea of efficiency in the context of computational complexity and how runtime scales with problem size.",{"name":115,"@type":106,"acceptedAnswer":116},"Which major algorithmic problem areas are covered?",{"text":117,"@type":109},"The contents outline minimum spanning trees, shortest path problems, maximum flow problems, and dynamic programming. 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Gerdts  \nChapter 1  \nNotation  \nNumbers: R is the set of reel numbers, Z is the set of integers, N is the set of positive integers, N0 is the set of non-negative integers.  \nVectors and matrices: A vector x 2 Rn is a column vector with components x 1 ; x2 ; : : : ; xn:  \n0 B  \nx = B  \nB @  \nx 1  \nx2  \n.  \n.  \n.  \nxn  \n1  \nC  \nC :  \nC  \nA  \nA 2 Rm􀀂n is a matrix with m rows and n columns and entries aij , i = 1; : : : ; m, j = 1; : : : ; n:  \n0 B  \nA = (aij)i=1;:::;m = Bj=1;:::;n B @  \na 11 a 12  \na21 a22  \n. .  \n. .  \n. .  \nam1 am2  \n􀀁 􀀁 􀀁 a 1n  \n􀀁 􀀁 􀀁 a2n  \n.  \n.  \n.  \n.  \n.  \n.  \n.  \n.. amn  \n1  \nC  \nC :  \nC  \nA  \nA> denotes the transposed matrix of the matrix A.  \nk 􀀁 k = k 􀀁 k2 denotes the Euclidean norm in Rn , i.e. for x = (x1 ; : : : ; xn)> it holds kxk =px>x =pP x2i .  \nChapter 2  \nIntroduction  \nCombinatorial optimisation is a branch of optimisation in applied mathematics and computer scienc","cbCaimhjDnZp0sEb","https://ap.wps.com/l/cbCaimhjDnZp0sEb","pdf",575407,87,"English","# Notation\n# Introduction\n## Efficiency and Complexity\n# Minimum Spanning Trees\n# Shortest Path Problems\n# Maximum Flow Problems\n# Dynamic Programming\n## What is a Dynamic Optimisation Problem?\n# Software\n# Lecture plan","[{\"question\":\"What is combinatorial optimisation and why is it challenging?\",\"answer\":\"Combinatorial optimisation is an area of optimisation in applied mathematics and computer science, closely related to operations research, algorithm theory, and computational complexity. It addresses problems whose feasible sets are discrete, but the number of feasible solutions can grow very rapidly, making efficient algorithms difficult.\"},{\"question\":\"What does the document mean by an efficient algorithm?\",\"answer\":\"The document’s main aim is to construct efficient algorithms for combinatorial optimisation problems. It explicitly discusses the idea of efficiency in the context of computational complexity and how runtime scales with problem size.\"},{\"question\":\"Which major algorithmic problem areas are covered?\",\"answer\":\"The contents outline minimum spanning trees, shortest path problems, maximum flow problems, and dynamic programming. For each area, the document points to formulations and specific algorithms such as greedy methods, shortest path algorithms, and dynamic programming principles.\"}]","Combinatorial Optimisation - MSM 3M02b - Preliminary Version | PDF",1791845664]