research-article

The Simplex Algorithm Is NP-Mighty

Authors:

Martin SkutellaAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 15, Issue 1

Article No.: 5, Pages 1 - 19

https://doi.org/10.1145/3280847

Published: 16 November 2018 Publication History

Abstract

We show that the Simplex Method, the Network Simplex Method—both with Dantzig’s original pivot rule—and the Successive Shortest Path Algorithm are NP-mighty. That is, each of these algorithms can be used to solve, with polynomial overhead, any problem in NP implicitly during the algorithm’s execution. This result casts a more favorable light on these algorithms’ exponential worst-case running times. Furthermore, as a consequence of our approach, we obtain several novel hardness results. For example, for a given input to the Simplex Algorithm, deciding whether a given variable ever enters the basis during the algorithm’s execution and determining the number of iterations needed are both NP-hard problems. Finally, we close a long-standing open problem in the area of network flows over time by showing that earliest arrival flows are NP-hard to obtain.

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Cited By

Skutella M(2024)An Introduction to Transshipments Over TimeSurveys in Combinatorics 202410.1017/9781009490559.009(239-270)Online publication date: 23-May-2024
https://doi.org/10.1017/9781009490559.009
Cardinal JSteiner R(2023)Inapproximability of shortest paths on perfect matching polytopesMathematical Programming10.1007/s10107-023-02025-4Online publication date: 21-Oct-2023
https://doi.org/10.1007/s10107-023-02025-4
Cardinal JSteiner R(2023)Inapproximability of Shortest Paths on Perfect Matching PolytopesInteger Programming and Combinatorial Optimization10.1007/978-3-031-32726-1_6(72-86)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-32726-1_6
Show More Cited By

Index Terms

The Simplex Algorithm Is NP-Mighty
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial optimization
2. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes
  2. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Network flows
    2. Mathematical optimization
      1. Discrete optimization

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 15, Issue 1

January 2019

366 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3281277

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2018 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 16 November 2018

Accepted: 01 August 2018

Revised: 01 May 2018

Received: 01 September 2017

Published in TALG Volume 15, Issue 1

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Alexander von Humboldt-Foundation
DFG Priority Programme 1736 “Algorithms for Big Data”

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Cited By

Skutella M(2024)An Introduction to Transshipments Over TimeSurveys in Combinatorics 202410.1017/9781009490559.009(239-270)Online publication date: 23-May-2024
https://doi.org/10.1017/9781009490559.009
Cardinal JSteiner R(2023)Inapproximability of shortest paths on perfect matching polytopesMathematical Programming10.1007/s10107-023-02025-4Online publication date: 21-Oct-2023
https://doi.org/10.1007/s10107-023-02025-4
Cardinal JSteiner R(2023)Inapproximability of Shortest Paths on Perfect Matching PolytopesInteger Programming and Combinatorial Optimization10.1007/978-3-031-32726-1_6(72-86)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-32726-1_6
Gavrus CIvan NOancea G(2022)Machining Parameters Optimization Based on Objective Function LinearizationMathematics10.3390/math1005080310:5(803)Online publication date: 3-Mar-2022
https://doi.org/10.3390/math10050803
Kara GÖzturan C(2021)Parallel network simplex algorithm for the minimum cost flow problemConcurrency and Computation: Practice and Experience10.1002/cpe.665934:4Online publication date: 20-Oct-2021
https://doi.org/10.1002/cpe.6659
Gautam B(2020)Energy MinimizationHomology Molecular Modeling - Perspectives and Applications [Working Title]10.5772/intechopen.94809Online publication date: 18-Nov-2020
https://doi.org/10.5772/intechopen.94809

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