I will give a gentle introduction to algebraic graph algorithms by showing how to determine if a given graph contains a simple path of length k. This is a famous problem admitting a beautiful and widely-known algorithm, namely the colour-coding method of Alon, Yuster and Zwick (1995). Starting from this entirely combinatorial approach, I will carefully develop an algebraic perspective on the same problem. First, I will explain how the colour-coding algorithm can be understood as the evaluation of a well-known expression (sometimes called the "walk-sum" of the graph) in a commutative algebra called the zeon algebra. From there, I will introduce the exterior algebra and present the algebraic framework recently developed with Brand and Dell (2018). The presentation is aimed at a combinatorially-minded audience largely innocent of abstract algebra.
@InProceedings{husfeldt:LIPIcs.CPM.2020.1, author = {Husfeldt, Thore}, title = {{Algebraic Algorithms for Finding Patterns in Graphs}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {1:1--1:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-149-8}, ISSN = {1868-8969}, year = {2020}, volume = {161}, editor = {G{\o}rtz, Inge Li and Weimann, Oren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.1}, URN = {urn:nbn:de:0030-drops-121261}, doi = {10.4230/LIPIcs.CPM.2020.1}, annote = {Keywords: paths, exterior algebra, wedge product, color-coding, parameterized complexity} }
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