Algebraic Algorithms for Finding Patterns in Graphs (Invited Talk)

Author Thore Husfeldt

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Author Details

Thore Husfeldt
  • IT University of Copenhagen, Denmark
  • Lund University, Sweden


Based on joint work with Cornelius Brand and Holger Dell.

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Thore Husfeldt. Algebraic Algorithms for Finding Patterns in Graphs (Invited Talk). In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Graph algorithms
  • paths
  • exterior algebra
  • wedge product
  • color-coding
  • parameterized complexity


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  1. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. URL:
  2. Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, page 151–164, New York, NY, USA, 2018. Association for Computing Machinery. Google Scholar
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