Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer

Djidjev, Hristo Nikolov; Pelofske, Elijah Autumn Rain; Hahn, Georg

doi:10.1007/s11265-020-01550-1

Title: Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer

Abstract

NP-hard problems such as the maximum clique or minimum vertex cover problems, two of Karp’s 21 NP-hard problems, have several applications in computational chemistry, biochemistry and computer network security. Adiabatic quantum annealers can search for the optimum value of such NP-hard optimization problems, given the problem can be embedded on their hardware. However, this is often not possible due to certain limitations of the hardware connectivity structure of the annealer. This paper studies a general framework for a decomposition algorithm for NP-hard graph problems aiming to identify an optimal set of vertices. Our generic algorithm allows us to recursively divide an instance until the generated subproblems can be embedded on the quantum annealer hardware and subsequently solved. Furthermore, the framework is applied to the maximum clique and minimum vertex cover problems, and we propose several pruning and reduction techniques to speed up the recursive decomposition. The performance of both algorithms is assessed in a detailed simulation study.

Authors:

^[1];

^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Mon Jun 29 00:00:00 EDT 2020

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE Laboratory Directed Research and Development (LDRD) Program

OSTI Identifier:: 1822729

Report Number(s):: LA-UR-19-30809
Journal ID: ISSN 1939-8018

Grant/Contract Number:: 89233218CNA000001

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Signal Processing Systems

Additional Journal Information:: Journal Volume: 93; Journal Issue: 4; Journal ID: ISSN 1939-8018

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Decomposition algorithm; D-Wave; Maximum clique; Minimum vertex cover; NPhard; Optimization.

Citation Formats


                    Djidjev, Hristo Nikolov, Pelofske, Elijah Autumn Rain, and Hahn, Georg. Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer.  United States: N. p., 2020. 
Web.  doi:10.1007/s11265-020-01550-1.

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                    Djidjev, Hristo Nikolov, Pelofske, Elijah Autumn Rain, & Hahn, Georg. Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer.  United States.  https://doi.org/10.1007/s11265-020-01550-1

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                    Djidjev, Hristo Nikolov, Pelofske, Elijah Autumn Rain, and Hahn, Georg. Mon .  
"Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer".  United States.  https://doi.org/10.1007/s11265-020-01550-1.  https://www.osti.gov/servlets/purl/1822729.

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@article{osti_1822729,

  title        = {Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer},

  author       = {Djidjev, Hristo Nikolov and Pelofske, Elijah Autumn Rain and Hahn, Georg},

  abstractNote = {NP-hard problems such as the maximum clique or minimum vertex cover problems, two of Karp’s 21 NP-hard problems, have several applications in computational chemistry, biochemistry and computer network security. Adiabatic quantum annealers can search for the optimum value of such NP-hard optimization problems, given the problem can be embedded on their hardware. However, this is often not possible due to certain limitations of the hardware connectivity structure of the annealer. This paper studies a general framework for a decomposition algorithm for NP-hard graph problems aiming to identify an optimal set of vertices. Our generic algorithm allows us to recursively divide an instance until the generated subproblems can be embedded on the quantum annealer hardware and subsequently solved. Furthermore, the framework is applied to the maximum clique and minimum vertex cover problems, and we propose several pruning and reduction techniques to speed up the recursive decomposition. The performance of both algorithms is assessed in a detailed simulation study.},

  doi          = {10.1007/s11265-020-01550-1},

  journal      = {Journal of Signal Processing Systems},

  number       = 4,

  volume       = 93,

  place        = {United States},

  year         = {Mon Jun 29 00:00:00 EDT 2020},

  month        = {Mon Jun 29 00:00:00 EDT 2020}

}

Copy to clipboard

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Exact bounds on the order of the maximum clique of a graph
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Works referencing / citing this record:

Picking Efficient Portfolios from 3,171 US Common Stocks with New Quantum and Classical Solvers
preprint, January 2020

Cohen, Jeffrey; Alexander, Clark
arXiv
DOI: 10.48550/arxiv.2011.01308

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Title: Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer

Abstract

Citation Formats

Fast algorithms for determining (generalized) core groups in social networks journal, November 2010

Improved upper bounds for vertex cover journal, September 2010

Algorithm 457: finding all cliques of an undirected graph journal, September 1973

An improved fixed-parameter algorithm for vertex cover journal, February 1998

Algorithms for maximum independent sets journal, September 1986

Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning journal, February 2016

Vertex Cover: Further Observations and Further Improvements journal, November 2001

An exact algorithm for the maximum clique problem journal, November 1990

Solving some NP-complete problems using split decomposition journal, July 2008

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width journal, March 2000

Measure and conquer: a simple O(2 0.288 n ) independent set algorithm conference, January 2006

Finding Maximum Cliques on a Quantum Annealer conference, May 2017

Improvement on vertex cover for low-degree graphs journal, January 2000

Pseudo-Boolean optimization journal, November 2002

Solving large minimum vertex cover problems on a quantum annealer conference, April 2019

BigBrain: An Ultrahigh-Resolution 3D Human Brain Model journal, June 2013

Decomposition by clique separators journal, July 1985

Optimizing Binary MRFs via Extended Roof Duality conference, June 2007

Reducing Binary Quadratic Forms for More Scalable Quantum Annealing conference, November 2017

On the computational complexity of Ising spin glass models journal, October 1982

Fast maximum clique algorithms for large graphs conference, April 2014

Open problems around exact algorithms journal, February 2008

Solving Large Maximum Clique Problems on a Quantum Annealer book, January 2019

Minor-embedding in adiabatic quantum computation: I. The parameter setting problem journal, September 2008

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width journal, March 2000

Exact algorithms for maximum independent set journal, August 2017

Improved upper bounds for vertex cover journal, September 2010

Exact bounds on the order of the maximum clique of a graph journal, May 2003

On efficient fixed-parameter algorithms for weighted vertex cover journal, July 2003

On extended P4-reducible and extended P4-sparse graphs journal, June 1997

On the computational complexity of Ising spin glass models journal, October 1982

BigBrain: An Ultrahigh-Resolution 3D Human Brain Model journal, June 2013

Parallel Maximum Clique Algorithms with Applications to Network Analysis journal, January 2015

Measure and conquer: a simple O(2 0.288 n ) independent set algorithm conference, January 2006

Algorithm 457: finding all cliques of an undirected graph journal, September 1973

An improved fixed parameter tractable algorithm for vertex cover text, January 1999

Solving large Maximum Clique problems on a quantum annealer text, January 2019

Picking Efficient Portfolios from 3,171 US Common Stocks with New Quantum and Classical Solvers preprint, January 2020

Fast algorithms for determining (generalized) core groups in social networks
journal, November 2010

Improved upper bounds for vertex cover
journal, September 2010

Algorithm 457: finding all cliques of an undirected graph
journal, September 1973

An improved fixed-parameter algorithm for vertex cover
journal, February 1998

Algorithms for maximum independent sets
journal, September 1986

Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning
journal, February 2016

Vertex Cover: Further Observations and Further Improvements
journal, November 2001

An exact algorithm for the maximum clique problem
journal, November 1990

Solving some NP-complete problems using split decomposition
journal, July 2008

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width
journal, March 2000

Measure and conquer: a simple O(2 ^{0.288 n} ) independent set algorithm
conference, January 2006

Finding Maximum Cliques on a Quantum Annealer
conference, May 2017

Improvement on vertex cover for low-degree graphs
journal, January 2000

Pseudo-Boolean optimization
journal, November 2002

Solving large minimum vertex cover problems on a quantum annealer
conference, April 2019

BigBrain: An Ultrahigh-Resolution 3D Human Brain Model
journal, June 2013

Decomposition by clique separators
journal, July 1985

Optimizing Binary MRFs via Extended Roof Duality
conference, June 2007

Reducing Binary Quadratic Forms for More Scalable Quantum Annealing
conference, November 2017

On the computational complexity of Ising spin glass models
journal, October 1982

Fast maximum clique algorithms for large graphs
conference, April 2014

Open problems around exact algorithms
journal, February 2008

Solving Large Maximum Clique Problems on a Quantum Annealer
book, January 2019

Minor-embedding in adiabatic quantum computation: I. The parameter setting problem
journal, September 2008

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width
journal, March 2000

Exact algorithms for maximum independent set
journal, August 2017

Improved upper bounds for vertex cover
journal, September 2010

Exact bounds on the order of the maximum clique of a graph
journal, May 2003

On efficient fixed-parameter algorithms for weighted vertex cover
journal, July 2003

On extended P4-reducible and extended P4-sparse graphs
journal, June 1997

On the computational complexity of Ising spin glass models
journal, October 1982

BigBrain: An Ultrahigh-Resolution 3D Human Brain Model
journal, June 2013

Parallel Maximum Clique Algorithms with Applications to Network Analysis
journal, January 2015

Measure and conquer: a simple O(2 ^{0.288 n} ) independent set algorithm
conference, January 2006

Algorithm 457: finding all cliques of an undirected graph
journal, September 1973

An improved fixed parameter tractable algorithm for vertex cover
text, January 1999

Solving large Maximum Clique problems on a quantum annealer
text, January 2019

Picking Efficient Portfolios from 3,171 US Common Stocks with New Quantum and Classical Solvers
preprint, January 2020