research-article

An Exact Method for the Minimum Feedback Arc Set Problem

Authors:

Hermann Schichl,

Arnold Neumaier,

Tobias AchterbergAuthors Info & Claims

Journal of Experimental Algorithmics (JEA), Volume 26

Article No.: 1.4, Pages 1 - 28

https://doi.org/10.1145/3446429

Published: 23 April 2021 Publication History

Abstract

A feedback arc set of a directed graph G is a subset of its arcs containing at least one arc of every cycle in G. Finding a feedback arc set of minimum cardinality is an NP-hard problem called the minimum feedback arc set problem. Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in G can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have Ω (2ⁿ) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range.

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Index Terms

An Exact Method for the Minimum Feedback Arc Set Problem
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial optimization
    2. Graph theory
      1. Graph algorithms
      2. Paths and connectivity problems
  2. Mathematical analysis
    1. Mathematical optimization
      1. Discrete optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
    2. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming

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cover image ACM Journal of Experimental Algorithmics

ACM Journal of Experimental Algorithmics Volume 26, Issue

December 2021

479 pages

ISSN:1084-6654

EISSN:1084-6654

DOI:10.1145/3446425

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Publication History

Published: 23 April 2021

Accepted: 01 December 2020

Revised: 01 October 2020

Received: 01 March 2019

Published in JEA Volume 26

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Cavallaro CCutello VPavone M(2024)Efficient heuristics to compute minimal and stable feedback arc setsJournal of Combinatorial Optimization10.1007/s10878-024-01209-848:4Online publication date: 15-Oct-2024
https://dl.acm.org/doi/10.1007/s10878-024-01209-8
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