Cornelius Brand, Holger Dell, Thore Husfeldt

Research output: Conference Article in Proceeding or Book/Report chapterArticle in proceedingsResearchpeer-review


We devise an algorithm that approximately computes the number of paths of length k in a given directed graph with n vertices up to a multiplicative error of 1 ± ε. Our algorithm runs in time ε−2 4k(n+m) poly(k). The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic 2k·poly(n) time algorithm to find a k-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect k-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.
Original languageEnglish
Title of host publicationProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing : STOC 2018
Number of pages164
PublisherAssociation for Computing Machinery
Publication date2018
ISBN (Electronic)978-1-4503-5559-9
Publication statusPublished - 2018


  • Graph Algorithms
  • Path Counting
  • Exterior Algebra
  • Subgraph Isomorphism
  • Multivariate Polynomials


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