Helsinki Algorithms Seminar: Cornelius Brand, Saarland University "Extensor-coding: A unified algebraic approach to the longest path problem"

Weekly meeting of researchers in the Helsinki area interested in the art of algorithms and algorithm design.
Department of Computer Science, image: Matti Ahlgren

Extensor-coding: A unified algebraic approach to the longest path problem

Cornelius Brand
Saarland University

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 4^k*(n+m)*poly(k)/ε². 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 2^k*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.

Helsinki Algorithms Seminar is a weekly meeting of researchers in the Helsinki area interested in the art of algorithms and algorithm design, broadly interpreted to cover both theoretical ideas and algorithm engineering on concrete computing platforms. In most cases we have a presentation prepared for each meeting to communicate an idea, a recent result, work-in-progress, or demo, but this should not be at the expense of discussion and simply having fun with algorithms.

Our affiliations are with Aalto University and the University of Helsinki, and accordingly our activities alternate between the Otaniemi Campus of Aalto University and the Kumpula Campus of University of Helsinki, catalyzed by the Helsinki Institute for Information Technology HIIT, under the Algorithmic Data Analysis (ADA) programme.

For the season programme, please see the seminar webpage.

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