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New graph algorithms via polyhedral techniques

  • Jakub Tarnawski

    Jakub Tarnawski is an algorithms researcher at Microsoft Research. He is broadly interested in theoretical computer science and combinatorial optimization, particularly in graph algorithms and approximation algorithms. He works on fundamental problems in these domains, such as the traveling salesman problem, submodular maximization, matching, and various variants of scheduling. He is a recipient of the Best Paper Award at STOC 2018 for his work on the traveling salesman problem, and of the Best Paper Award at FOCS 2017 for his work on matchings. He completed his PhD in Computer Science in 2019 at EPFL, Switzerland, advised by Ola Svensson, for which he has been awarded the ACM Dissertation Award Honorable Mention, the EPFL Doctorate Award, the EATCS Dissertation Award and the Chorafas Foundation Prize. He received his BSc and MSc degrees in Mathematics and Computer Science from the University of Wrocław, Poland.

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Published/Copyright: April 15, 2021

Abstract

This article gives a short overview of my dissertation, where new algorithms are given for two fundamental graph problems. We develop novel ways of using linear programming formulations, even exponential-sized ones, to extract structure from problem instances and to guide algorithms in making progress.

The first part of the dissertation addresses a benchmark problem in combinatorial optimization: the asymmetric traveling salesman problem (ATSP). It consists in finding the shortest tour that visits all vertices of a given edge-weighted directed graph. A ρ-approximation algorithm for ATSP is one that runs in polynomial time and always produces a tour at most ρ times longer than the shortest tour. Finding such an algorithm with constant ρ had been a long-standing open problem. Here we give such an algorithm.

The second part of the dissertation addresses the perfect matching problem. We have known since the 1980s that it has efficient parallel algorithms if the use of randomness is allowed. However, we do not know if randomness is necessary – that is, whether the matching problem is in the class NC. We show that it is in the class quasi-NC. That is, we give a deterministic parallel algorithm that runs in poly-logarithmic time on quasi-polynomially many processors.

ACM CCS:

Article note

The dissertation of Dr. Jakub Tarnawski has been awarded by the GI Dissertation Award 2019.


About the author

Jakub Tarnawski

Jakub Tarnawski is an algorithms researcher at Microsoft Research. He is broadly interested in theoretical computer science and combinatorial optimization, particularly in graph algorithms and approximation algorithms. He works on fundamental problems in these domains, such as the traveling salesman problem, submodular maximization, matching, and various variants of scheduling. He is a recipient of the Best Paper Award at STOC 2018 for his work on the traveling salesman problem, and of the Best Paper Award at FOCS 2017 for his work on matchings. He completed his PhD in Computer Science in 2019 at EPFL, Switzerland, advised by Ola Svensson, for which he has been awarded the ACM Dissertation Award Honorable Mention, the EPFL Doctorate Award, the EATCS Dissertation Award and the Chorafas Foundation Prize. He received his BSc and MSc degrees in Mathematics and Computer Science from the University of Wrocław, Poland.

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Received: 2021-03-10
Accepted: 2021-03-16
Published Online: 2021-04-15
Published in Print: 2021-07-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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