An Updated Experimental Evaluation of Graph Bipartization Methods

We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal (OCT)), motivated by the need for good algorithms for embedding problems into near-term quantum computing hardware. We assemble a preprocessing suite of fast input reduction routines from the OCT and Vertex Cover (VC) literature and compare algorithm implementations using Quadratic Unconstrained Binary Optimization problems from the quantum literature. We also generate a corpus of frustrated cluster loop graphs, which have previously been used to benchmark quantum annealing hardware. The diversity of these graphs leads to harder OCT instances than in existing benchmarks. In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We extend our results with a large corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data. In total, over 8,000 graph instances are evaluated, compared to the previous canonical corpus of 100 graphs. Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.

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A QUBO Formulation of Minimum Multicut Problem Instances in Trees for D-Wave Quantum Annealers

Scientific Reports ◽

10.1038/s41598-019-53585-5 ◽

2019 ◽

Vol 9 (1) ◽

Author(s):

William Cruz-Santos ◽

Salvador E. Venegas-Andraca ◽

Marco Lanzagorta

Keyword(s):

Optimization Problems ◽

Theoretical Computer Science ◽

Scaling Analysis ◽

Quantum Annealing ◽

Theoretical Computer ◽

Combinatorial Optimization Problems ◽

Hard Problems ◽

Problem Instances ◽

Annealing Algorithms ◽

D Wave

AbstractQuantum annealing algorithms were introduced to solve combinatorial optimization problems by taking advantage of quantum fluctuations to escape local minima in complex energy landscapes typical of NP − hard problems. In this work, we propose using quantum annealing for the theory of cuts, a field of paramount importance in theoretical computer science. We have proposed a method to formulate the Minimum Multicut Problem into the QUBO representation, and the technical difficulties faced when embedding and submitting a problem to the quantum annealer processor. We show two constructions of the quadratic unconstrained binary optimization functions for the Minimum Multicut Problem and we review several tradeoffs between the two mappings and provide numerical scaling analysis results from several classical approaches. Furthermore, we discuss some of the expected challenges and tradeoffs in the implementation of our mapping in the current generation of D-Wave machines.

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Grover Adaptive Search for Constrained Polynomial Binary Optimization

Quantum ◽

10.22331/q-2021-04-08-428 ◽

2021 ◽

Vol 5 ◽

pp. 428

Author(s):

Austin Gilliam ◽

Stefan Woerner ◽

Constantin Gonciulea

Keyword(s):

Optimization Problems ◽

Adaptive Search ◽

Binary Optimization ◽

Combinatorial Optimization Problems ◽

Quadratic Unconstrained Binary Optimization ◽

Speed Up ◽

Near Term ◽

Quantum Arithmetic ◽

Portfolio Optimization Problem ◽

Special Case

In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation and experimental results obtained on real quantum hardware. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems.

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Qubit-efficient encoding schemes for binary optimisation problems

Quantum ◽

10.22331/q-2021-05-04-454 ◽

2021 ◽

Vol 5 ◽

pp. 454

Author(s):

Benjamin Tan ◽

Marc-Antoine Lemonde ◽

Supanut Thanasilp ◽

Jirawat Tangpanitanon ◽

Dimitris G. Angelakis

Keyword(s):

Quantum State ◽

Optimization Problems ◽

State Of The Art ◽

Quantum Algorithms ◽

Approximate Solutions ◽

Problem Instance ◽

Quadratic Unconstrained Binary Optimization ◽

Max Cut Problem ◽

Multi Body

We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of nc classical variables can be implemented on O(log⁡nc) number of qubits. The underlying encoding scheme allows for a systematic increase in correlations among the classical variables captured by a variational quantum state by progressively increasing the number of qubits involved. We first examine the simplest limit where all correlations are neglected, i.e. when the quantum state can only describe statistically independent classical variables. We apply this minimal encoding to find approximate solutions of a general problem instance comprised of 64 classical variables using 7 qubits. Next, we show how two-body correlations between the classical variables can be incorporated in the variational quantum state and how it can improve the quality of the approximate solutions. We give an example by solving a 42-variable Max-Cut problem using only 8 qubits where we exploit the specific topology of the problem. We analyze whether these cases can be optimized efficiently given the limited resources available in state-of-the-art quantum platforms. Lastly, we present the general framework for extending the expressibility of the probability distribution to any multi-body correlations.

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A novel approach for solving travelling thief problem using enhanced simulated annealing

PeerJ Computer Science ◽

10.7717/peerj-cs.377 ◽

2021 ◽

Vol 7 ◽

pp. e377

Author(s):

Hamid Ali ◽

Muhammad Zaid Rafique ◽

Muhammad Shahzad Sarfraz ◽

Muhammad Sheraz Arshad Malik ◽

Mohammed A. Alqahtani ◽

...

Keyword(s):

Simulated Annealing ◽

Optimization Problem ◽

Optimization Problems ◽

State Of The Art ◽

Minimum Cost ◽

Medium Size ◽

Time Interval ◽

Maximum Profit ◽

Novel Approach ◽

Hard Problems

Real-world optimization problems are getting more and more complex due to the involvement of inter dependencies. These complex problems need more advanced optimizing techniques. The Traveling Thief Problem (TTP) is an optimization problem that combines two well-known NP-Hard problems including the 0/1 knapsack problem and traveling salesman problem. TTP contains a person known as a thief who plans a tour to collect multiple items to fill his knapsack to gain maximum profit while incurring minimum cost in a standard time interval of 600 s. This paper proposed an efficient technique to solve the TTP problem by rearranging the steps of the knapsack. Initially, the picking strategy starts randomly and then a traversal plan is generated through the Lin-Kernighan heuristic. This traversal is then improved by eliminating the insignificant cities which contribute towards profit adversely by applying the modified simulated annealing technique. The proposed technique on different instances shows promising results as compared to other state-of-the-art algorithms. This technique has outperformed on a small and medium-size instance and competitive results have been obtained in the context of relatively larger instances.

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Charged particle tracking with quantum annealing optimization

Quantum Machine Intelligence ◽

10.1007/s42484-021-00054-w ◽

2021 ◽

Vol 3 (2) ◽

Author(s):

Alexander Zlokapa ◽

Abhishek Anand ◽

Jean-Roch Vlimant ◽

Javier M. Duarte ◽

Joshua Job ◽

...

Keyword(s):

Combinatorial Optimization ◽

Optimization Problems ◽

Hadron Collider ◽

Hopfield Network ◽

Quantum Annealing ◽

Track Reconstruction ◽

Combinatorial Optimization Problems ◽

Charged Particle Tracking ◽

Near Term ◽

Quantum Speedup

AbstractAt the High Luminosity Large Hadron Collider (HL-LHC), traditional track reconstruction techniques that are critical for physics analysis will need to be upgraded to scale with track density. Quantum annealing has shown promise in its ability to solve combinatorial optimization problems amidst an ongoing effort to establish evidence of a quantum speedup. As a step towards exploiting such potential speedup, we investigate a track reconstruction approach by adapting the existing geometric Denby-Peterson (Hopfield) network method to the quantum annealing framework for HL-LHC conditions. We develop additional techniques to embed the problem onto existing and near-term quantum annealing hardware. Results using simulated annealing and quantum annealing with the D-Wave 2X system on the TrackML open dataset are presented, demonstrating the successful application of a quantum annealing algorithm to the track reconstruction challenge. We find that combinatorial optimization problems can effectively reconstruct tracks, suggesting possible applications for fast hardware-specific implementations at the HL-LHC while leaving open the possibility of a quantum speedup for tracking.

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Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

Algorithms ◽

10.3390/a12040077 ◽

2019 ◽

Vol 12 (4) ◽

pp. 77 ◽

Cited By ~ 5

Author(s):

Tomas Vyskocil ◽

Hristo Djidjev

Keyword(s):

Optimization Problems ◽

Energy State ◽

Minimum Energy ◽

Combinatorial Design ◽

Mixed Integer ◽

Processing Unit ◽

Alternative Approach ◽

Quadratic Unconstrained Binary Optimization ◽

Hard Problems ◽

Binary Quadratic Optimization

Quantum annealers such as D-Wave machines are designed to propose solutions for quadratic unconstrained binary optimization (QUBO) problems by mapping them onto the quantum processing unit, which tries to find a solution by measuring the parameters of a minimum-energy state of the quantum system. While many NP-hard problems can be easily formulated as binary quadratic optimization problems, such formulations almost always contain one or more constraints, which are not allowed in a QUBO. Embedding such constraints as quadratic penalties is the standard approach for addressing this issue, but it has drawbacks such as the introduction of large coefficients and using too many additional qubits. In this paper, we propose an alternative approach for implementing constraints based on a combinatorial design and solving mixed-integer linear programming (MILP) problems in order to find better embeddings of constraints of the type ∑ x i = k for binary variables x i. Our approach is scalable to any number of variables and uses a linear number of ancillary variables for a fixed k.

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Gumbel-softmax-based Optimization: A Simple General Framework for Optimization Problems on Graphs

10.21203/rs.3.rs-22822/v1 ◽

2020 ◽

Author(s):

Yaoxin Li ◽

Jing Liu ◽

Guozheng Lin ◽

Yueyuan Hou ◽

Muyun Mou ◽

...

Keyword(s):

Objective Function ◽

Statistical Physics ◽

Automatic Differentiation ◽

Optimization Problems ◽

Vertex Cover ◽

Independent Set ◽

Maximum Independent Set ◽

Gradient Descent Algorithm ◽

Hard Problems ◽

Traditional Approaches

Abstract In computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA) and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on three representative optimization problems on graph including modularity optimization from network science, Sherrington-Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph. High-quality solutions can be obtained with much less time consuming compared to traditional approaches.

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Nanolaser-based emulators of spin Hamiltonians

Nanophotonics ◽

10.1515/nanoph-2020-0230 ◽

2020 ◽

Vol 9 (13) ◽

pp. 4193-4198 ◽

Cited By ~ 2

Author(s):

Midya Parto ◽

William E. Hayenga ◽

Alireza Marandi ◽

Demetrios N. Christodoulides ◽

Mercedeh Khajavikhan

Keyword(s):

Large Scale ◽

Optimization Problems ◽

Exchange Interactions ◽

Np Hard ◽

Spin Models ◽

Geometric Frustration ◽

Spin Hamiltonians ◽

Hard Problems ◽

On Chip ◽

Classical Spin Models

AbstractFinding the solution to a large category of optimization problems, known as the NP-hard class, requires an exponentially increasing solution time using conventional computers. Lately, there has been intense efforts to develop alternative computational methods capable of addressing such tasks. In this regard, spin Hamiltonians, which originally arose in describing exchange interactions in magnetic materials, have recently been pursued as a powerful computational tool. Along these lines, it has been shown that solving NP-hard problems can be effectively mapped into finding the ground state of certain types of classical spin models. Here, we show that arrays of metallic nanolasers provide an ultra-compact, on-chip platform capable of implementing spin models, including the classical Ising and XY Hamiltonians. Various regimes of behavior including ferromagnetic, antiferromagnetic, as well as geometric frustration are observed in these structures. Our work paves the way towards nanoscale spin-emulators that enable efficient modeling of large-scale complex networks.

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Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

Computational Social Networks ◽

10.1186/s40649-021-00086-z ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Yaoxin Li ◽

Jing Liu ◽

Guozheng Lin ◽

Yueyuan Hou ◽

Muyun Mou ◽

...

Keyword(s):

Objective Function ◽

Statistical Physics ◽

Automatic Differentiation ◽

Optimization Problems ◽

Vertex Cover ◽

Independent Set ◽

Computational Social Science ◽

Maximization Problem ◽

Maximum Independent Set ◽

Gradient Descent Algorithm

AbstractIn computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

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Investigation of Improved Cooperative Coevolution for Large-Scale Global Optimization Problems

Algorithms ◽

10.3390/a14050146 ◽

2021 ◽

Vol 14 (5) ◽

pp. 146

Author(s):

Aleksei Vakhnin ◽

Evgenii Sopov

Keyword(s):

Global Optimization ◽

Evolutionary Algorithms ◽

Numerical Experiments ◽

Large Scale ◽

Optimization Problems ◽

State Of The Art ◽

Fixed Number ◽

High Dimensional ◽

Cooperative Coevolution ◽

Large Scale Problems

Modern real-valued optimization problems are complex and high-dimensional, and they are known as “large-scale global optimization (LSGO)” problems. Classic evolutionary algorithms (EAs) perform poorly on this class of problems because of the curse of dimensionality. Cooperative Coevolution (CC) is a high-performed framework for performing the decomposition of large-scale problems into smaller and easier subproblems by grouping objective variables. The efficiency of CC strongly depends on the size of groups and the grouping approach. In this study, an improved CC (iCC) approach for solving LSGO problems has been proposed and investigated. iCC changes the number of variables in subcomponents dynamically during the optimization process. The SHADE algorithm is used as a subcomponent optimizer. We have investigated the performance of iCC-SHADE and CC-SHADE on fifteen problems from the LSGO CEC’13 benchmark set provided by the IEEE Congress of Evolutionary Computation. The results of numerical experiments have shown that iCC-SHADE outperforms, on average, CC-SHADE with a fixed number of subcomponents. Also, we have compared iCC-SHADE with some state-of-the-art LSGO metaheuristics. The experimental results have shown that the proposed algorithm is competitive with other efficient metaheuristics.

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