Efficient Solutions for Large-Scale Max-Cut Problems: A Hybrid Local Search Heuristic Approach and Comparative Analysis with Quantum Annealing

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Efficient Solutions for Large-Scale Max-Cut Problems: A Hybrid Local Search Heuristic Approach and Comparative Analysis with Quantum Annealing

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23 June 2024

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Abstract

This study addresses the formidable challenge of solving large-scale Max-Cut problems (MCP). We introduce a rapid computational procedure utilizing a hybrid 1-flip/r-flip local search heuristic. This innovative strategy significantly reduces the computational time required for MCP problems while consistently generating solutions of exceptional quality. The paper presents substantial computational insights, showcasing the effectiveness of our approach on large-scale Max-Cut instances with varying densities. Our proposed heuristic is rigorously evaluated by comparing its performance against a quantum annealing solver, leveraging a multi-start Tabu Search framework. The results underscore the potency of this unique combination as an efficient and effective solution for large-scale QUBO problems. Notably, our hybrid heuristic consistently delivers high-quality solutions within the stringent CPU time limits of 600 seconds, demonstrating its efficacy across Max-Cut instances ranging from 10,000 to 40,000 variables. This research contributes to advancing the state-of-the-art in large-scale QUBO problem-solving, offering a powerful and time-efficient approach with broad applicability.

Keywords:

Subject: Business, Economics and Management - Other

1. Introduction

The Quadratic Unconstrained Binary Optimization (QUBO) problem is relevant across various applications, particularly as a unifying approach for many combinatorial optimization challenges [1]. The emergence of quantum computing, particularly quantum annealing solvers, has brought QUBO into the spotlight [2]. However, the complexities of modeling and analyzing very large-scale problems on conventional computing platforms often hinder effective solutions through mathematical programming methods.

While quantum computing platforms offer the potential to overcome hardware limitations, transforming real-world problems into suitable qubit architecture remains challenging. This study addresses the gap between quantum and traditional computing platforms, focusing on the very large-scale Max-Cut problem (MCP) – a critical research area with applications spanning statistical physics [3], communication infrastructure [3], machine learning [4], and computer chip design [5].

The computational landscape of the Max-Cut problem has sparked diverse solution strategies, ranging from approximation algorithms leveraging semidefinite programming to metaheuristic methods and exact approaches. While approximation algorithms like the seminal work of Goemans and Williamson [6] and subsequent contributions by Karish et al. offer theoretical performance guarantees [7], empirical evaluations often reveal their inferiority to other methodologies.

As exemplified by Krishnan and Mitchell's cut and price approach [8] and Rendl et al.'s branch and bound method [9], exact methods have emerged as viable alternatives, particularly for smaller problem instances. However, their applicability remains constrained to scenarios with relatively few nodes, typically in the hundreds.

Metaheuristic methods become imperative for more substantial Max-Cut instances characterized by thousands of nodes. Recent advancements in this domain include the rank-2 relaxation method, dubbed CirCut, by Burer et al. [10], Festa et al.'s hybrid randomized method [11], and Marti et al.'s scatter search method [12]. Computational evaluations consistently demonstrate the superiority of these techniques, with the scatter search method notably achieving outstanding performance across test problems featuring up to 3000 nodes. Additionally, Alidaee et al. proposed a diversification strategy based on sequential local improvement, demonstrating substantial enhancements in solutions for a significant portion of benchmark Max-Cut problems within reasonable CPU time constraints [13].

Further breakthroughs have been made to tackle even larger problem sizes. Kochenberger et al. introduced the DDT approach, delivering state-of-the-art solutions for test problems boasting up to 10,000 nodes [14].

Due to its natural formulations, the max-cut problem is a popular testbed for Quantum Computing platforms and algorithms. Recent developments in quantum computing, especially quantum annealing solvers, put Max-Cut in the spotlight (15; 16; 17; 18; 19; 20; 21; 22; 23; 24). Alam et al. used the graph Max-Cut problem as a prototype while using the quantum approximate optimization algorithm (QAOA), where a quantum circuit and a classical optimizer operate in a closed loop, solving hard combinatorial optimization problems [15]. Basso et al. (2022) also applied QAOA to Max-Cut on large-girth regular graphs [16]. Galda et al. showed that the transferability of the parameters among QAOA instances and convergence of the optimal parameters around specific values can be predicted based on local graph properties [17]. Bianchi et al. proposed a pooling operator for graph neural networks to generate coarser graphs while maintaining the overall topology of the graph and used a spectral algorithm approximating the Max-Cut solution [18]. Shaydulin et al. utilized a multi-start optimization technique in a QAOA framework to improve the performance of quantum machines on various graph clustering problems [19] and introduced a QAOA Kit for the combination and standardization of previously known parameters for Max-Cut problems [20]. Shaydulin and Wild proposed a new method to accelerate the assessment of QAOA energy by using the problem symmetry. They implemented the proposed method on the Max-Cut problem using a graph auto-morphism solver and tensor network simulator [21]. Umasankar et al. used a Time-Multiplexed Opto-Electronic Oscillator-based Coherent Ising Machine on Max-Cut Problems [22]. Wurtz and Love defined the Spanning Tree QAOA for Max-Cut and utilized an “ansatz” whose structure results from an approximate classical solution and reaches the same performance level as the classical algorithm to outperform QAOA at low depth [23].

In this research, we introduce a local search strategy based on the r-flip strategy for QUBO [24]. The dynamic adjustment of the r value in the r-flip strategy strikes a balance between computational cost and solution quality. This approach effectively reduces computational time while reaching superior solutions [25]. Our proposed heuristic achieves results within a strict CPU time limit of 600 seconds, addressing Max-Cut instances ranging from 10,000 to 40,000 variables. Comparative evaluations against a quantum annealing solver based on multi-start Tabu Search [26] provide insights into the efficiency and effectiveness of our proposed approach.

By bridging the gap between quantum and conventional computing, our study contributes to the exploration of quantum computing's potential in solving complex optimization problems. It showcases a novel local search strategy tailored for QUBO with promising results in the context of the challenging Max-Cut problem.

2. QUBO and MCP Solution

Before we present the r-flip strategy to solve QUBO and MCP in this study, the notations used are given as follows:

n The number of variables

x An initial feasible solution

x* The best solution found by the algorithm

K The largest value of k for r-flip, k

\leq

π(i) The i-th element of x in the order π(1)⋯π(n)

S {i: x_i is tentatively chosen to receive a new value to produce a new solution x_i'} restricting consideration to |S| = r

D The set of candidates for an improving move

E (x_{i})

Derivative of f(x) concerning

x_{i}

o n e E (x) = (E (x_{1}), \dots, E (x_{n}))

The vector of derivatives

x(.) A vector representing the solution of x

E(.) A vector representing the value of the derivative

E (x_{i})

The general form of QUBO can be given as [27],

M a x f (x) = \sum_{i = 1}^{n} q_{i} x_{i} + \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \neq i}^{n} q_{i, j} x_{i} x_{j}, s . t . x_{i} \in \{0,1\}, i = 1, \dots, n

(1)

In (1),

\frac{1}{2} q_{i, j}

is the i,j-th entry of a given n by n symmetric matrix Q, and the linear term

q_{i}

can be the entry in the diagonal of the matrix. Since

{x_{i}}^{2} = x_{i}

, and Q may be written as an upper triangular matrix by doubling each entry of the upper triangle part of the matrix and letting

q_{i, i} = q_{i}

, then we can write (1) as (2).

M a x f (x) = \sum_{i = 1}^{n} \sum_{j \geq i}^{n} q_{i, j} x_{i} x_{j} = x^{T} Q x, s . t . x_i \in {0,1}, i = 1, \dots, n

(2)

Numerous combinatorial optimization problems lend themselves to reformulation as QUBO by employing a quadratic infeasibility penalty. However, determining the optimal penalty values poses a considerable challenge. In light of this, we focus our investigation on the very large-scale MCP as a prime candidate application, given its inherent and natural formulation within the QUBO framework.

2.1. Max-Cut Problems

The max-cut problem with many real-world implementations is an NP-hard problem, meaning that exact solution algorithms are insufficient for large-scale problems since obtaining a solution is very time-consuming [28]. Following the advances in quantum computers, max-cut problems have received much interest recently [29]. Two formulations, linear and nonlinear, exist for the MCP [27]. Initially, we present the natural QUBO model for MCP.

M a x \sum_{i = 1}^{n} \sum_{j = 1, i < j}^{n} q_{i j} (x_{i} + x_{j} - 2 * x_{i} x_{j})

(3)

s.t.

x_{i} \in \{0,1\} \forall i = 1, \dots, n

The QUBO model (3) can be transformed into a linear model using classical linearization techniques [30].

M a x \sum_{i = 1}^{n} \sum_{j = 1, i < j}^{n} q_{i j} (x_{i} + x_{j} - 2 * y_{i j})

(4)

s . t . y_{i j} \leq x_{i}, \forall i, j = 1, \dots, n a n d i < j

(5)

y_{i j} \leq x_{j}, \forall i, j = 1, \dots, n a n d i < j

(6)

y_{i j} \geq x_{i} + x_{j} - 1, \forall i, j = 1, \dots, n a n d i < j

(7)

x_{i} \in \{0,1\}, \forall i = 1, \dots, n

y_{i j} \in [0, 1], \forall j = 1, \dots, n a n d i < j

The linear model for the MCP undergoes a substantial increase in size on very large-scale instances, both in terms of the number of variables and constraints. In this research, we present solutions for the QUBO model solely to conduct a comparative analysis with a quantum annealing solver.

2.2. Solution Representation

Here Q is a symmetric matrix. Given a solution

x = (x_{1}, \dots, x_{n})

define

Δ_{i} (x) = q (i, i) + \sum_{k \neq i} 2 x_{k} q_{i k}, i \in N

(8)

Concerning 1-flip neighborhood, x is locally optimal if and only if it satisfies

Δ_{i} (x) \leq 0, \forall x_{i} = 0, a n d Δ_{i} (x) > 0, \forall x_{i} = 1

(9)

Then the objective function will satisfy

∆ f = \sum_{i \in S} (x'_{i} - x_{i}) E (x_{i}) + \sum_{i, j \in S, i < j} (x'_{i} - x_{i}) (x'_{j} - x_{j}) q_{i, j}

(10)

and

\forall j \in S, E (x_{j}) \leftarrow E (x_{j}) + \sum_{i \in S \ {j}} (x_{i}^{'} - x_{i}) q_{i, j}

(11)

The rationale behind the r-flip move is rooted in the notion that a larger value of moves (here, a larger value of r) offers the opportunity to explore a more extensive and diverse solution space. Moreover, when improvement within a specific neighborhood (e.g., a 1-flip move) is exhausted, and a solution x attains local optimality concerning the 1-flip move, employing another neighborhood move (such as a 2-flip move) becomes viable for further exploration of the solution space. This strategic approach forms the basis for various meta-heuristics. Upon obtaining a locally optimal solution x concerning a 1-flip move, the elements of x are ordered based on the ascending absolute value of derivatives, as follows:

| E (x_{π (1)}) | \leq \dots \leq | E (x_{π (n)}) |

(12)

Here,

π (i)

means the i-th element of x in the order

(π (1), \dots, π (n))

. Now, one at a time in the given order, check the summation in (13) for k=1,2,…,n. For m<n, define (m, n) to be the number of combinations of m elements out of n, and let

φ = \underset{i, j \in N}{Max} \{|q_{i, j}|\}

, and for the r-flip move let

M = φ * (2, r)

. Let K be the largest value of k where the inequality (13) holds, and define the set D(n) (14).

\sum_{i = 1}^{k} | E (x_{π (i)}) | < M, for k = 1,2, 3, \dots

(13)

D (n) = {x_{π (1)}, \dots, x_{π (K)}}

(14)

To implement an r-flip local search, the main decision is to choose a set S with |S|=r at each step and change

x_{i}

1 - x_{i}

for all

i \in S

if it improves the objective function. This search process is inherently computationally demanding. The strategies mentioned above, including Lin and Kernighan's approach and very large-scale neighborhood search, are inherently tailored to specific problem domains. For instance, these strategies have predominantly been found to be applicable to graph theoretic problems. However, this paper introduces a novel implementation of the r-flip strategy coupled with a sequential diversification technique. This approach enables the exploration of diverse sets S, each comprising r elements, while strategically avoiding non-improving sets. The incorporation of sequential diversification fundamentally reduces the computational time required for QUBO.

2.3. Implementation Details

The solution implementation is detailed in Section 2.2 through Algorithm 1 and Algorithm 2, with a comprehensive comparison of very large-scale MCP instances presented in the subsequent section. For additional insights and theoretical underpinnings, we draw upon the work and proofs by Alidaee and Wang [31], particularly for the novel r-flip results in general QUBO problems. It's worth noting that in Algorithm 2, the sets S and M dynamically evolve as the search progresses.

In a 1-flip search, we only look at one variable at a time to possibly change its value. However, in r-flip search the process is similar to 1-flip search except that up to r variables may be considered for a change of values. The idea behind the r-flip search is that the larger value of moves (here, a larger value of r) may provide an opportunity to explore a larger and, thus, more diverse solution space. In this process, when a neighborhood search, for example, k-flip search, is exhausted, i.e., x is locally optimal concerning k-flip move, another neighborhood move, for example, (k+1)-flip search, may be implemented to explore the solution space. Such a strategy is the basis for several meta-heuristics.

An alternative way to implement this algorithm is that k=r throughout the process. Algorithm 2 is simple but inefficient; it requires some mechanism to choose the set S and a process mechanism to implement elements of the k-flip move. Below, we give several results that allow an efficient implementation of the r-flip search process.

Lemma 1. Given a locally optimal solution x concerning a 1-flip search, we have

(15)

Proof. Directly follows from the local optimality condition (6), which states

(16)

Lemma 2. Let Preprints 110137 i005

, and |S|=r, given any solution x, and M as defined earlier, we have

(17)

Proof. For each pair of elements, i, j in S, the left-hand-side (LHS) in (17) is either q_ij or −q_ij. Using the definition of M, the summation in LHS is at most equal to M.

Theorem 3: Given a locally optimal solution x of f(x) concerning 1-flip search, a subset Preprints 110137 i005

, with |S|=r, is an improving r-flip move if and only if we have

(18)

Proof. Using (10), a subset S with r elements is an improving r-flop move if and only if we have (19)

(19)

Since x is a locally optimal solution concerning a 1-flip search, it follows from Lemma 1

that inequality (19) is equivalent to (20); which completes the proof.

(20)

Proposition 2: Let x be any locally optimal solution of f(x) concerning a 1-flip search. If a subset Preprints 110137 i005

, with |S|=r, is an improving r-flip move, then we must have Preprints 110137 i010

Proof: The result follows from Theorem 3 and Lemma 2.

A consequence of Theorem 3 is as follows. Given a locally optimal solution x concerning a 1-flip search, if no subset of S with |S|=r satisfies (18), then x is also a locally optimal solution concerning an r-flip search. If there is no subset S of any size that (18) is satisfied, then x is also a locally optimal solution concerning an r-flip search for all

r \leq n

. Similar statements are also true regarding Proposition 2. The result of Proposition 2 is significant in the implementation of an r-flip search. It illustrates that, after having a 1-flip search implemented, if an r-flip search is next served as a locally optimal solution, only those elements with the sum of the absolute value of derivatives less than M are eligible for consideration. Furthermore, when deciding about the elements of an r-flip search, we can easily check to see if any element

x_{i}

by itself or with a combination of other elements is eligible to be a member of an improving r-flip move.

We adopted the following notation for computational results in the next section:

BFS Best found solution among 10 runs within the CPU time limit.

APD Average percentage difference: the measurement for the performance of individual algorithms across runs on the same instance.

RPD Relative percentage deviation: the measurement of the relative performance across algorithms/methods on the same instance

3. Computational Experiments

In this investigation, we conducted extensive computational experiments to assess the efficacy of the proposed r-flip strategy across varying problem sizes, densities, and r values. The implementation of Algorithm 2 is executed using the GNU C++ compiler and undergoes 10 runs on the AMD Ryzen 5 3500U processor equipped with 16GB RAM. Simultaneously, the Quantum Annealing Solver (QAS), employing a multi-start tabu search, is implemented via a Python client on the same processor.

For the hybrid 1-flip/r-flip strategy in Algorithm 2 applied to large-scale MCP instances, we set a CPU time limit of 600 seconds. To comprehensively evaluate the r-flip strategy's performance on very large-scale instances, we generate a diverse set of planar graph instances tailored for the MCP. This dataset comprises 120 instances, ranging from Q matrices sized 10,000 by 10,000 to 40,000 by 40,000. Among these instances, three distinct weight types with discrete values are introduced: Type 'a' with values of 1 and -1, Type 'b' with random discrete values between -10 and 10, and Type 'c' with random discrete values between -1,000 and 1,000. The instance data, ranging in size from 10,000 to 40,000, is conveniently accessible in the supplementary materials provided.

4. Numerical Results

Table 1, Table 2 and Table 3 present the outcomes obtained from both the hybrid 1-flip/r-flip Algorithm 2 and QAS across the 120 instances. It's crucial to note that the results from the QAS are derived using the sampler approach (sampler=10). In reporting the QAS results, we focus on Relative Percentage Deviation (RPD) as the quality of solutions hinges on the parameter of sufficient reads (num_reads) and the collection of samples for potential solutions.

Concurrently, for Algorithm 2, we provide the Average Percentage Deviation (APD) value to illustrate the consistent performance over 10 runs. This dual reporting approach allows for a comprehensive evaluation of the comparative performance between the hybrid 1-flip/r-flip Algorithm 2 and the Quantum Annealing Solver across the diverse set of instances.

The r-flip strategy holds the potential for integration into various other local search heuristics as an enhancement procedure. A judicious implementation of the hybrid 1-flip/r-flip strategy not only diminishes computing time but also enhances solution quality. The outcomes reported here represent the best solutions obtained from 10 independent runs for each instance using the hybrid 1-flip/r-flip strategy. Given this study's constraints on computing resources, the solution deviation across these 10 runs is calculated within the specified short CPU time limits. Figure 1, Figure 2 and Figure 3 illustrate the performance of Algorithm 2 vs. QAS for 10, 20, and 40 thousand variables.

5. Conclusions

In this paper, we explore the application of r-flip search within the context of a very large-scale Max-Cut problem, treating it as a case of QUBO. Specifically, our investigation centers on assessing the local optimality of the r-flip search when a 1-flip search has already reached its local optimum. Our computational findings demonstrate a significant reduction in the pool of candidates for potential r-flip implementation, showcasing the efficiency of this novel strategy in solving very large-scale Max-Cut instances within a strict 600-second timeframe.

In our approach, we employed a 1-flip strategy to achieve local optimality. Yet, it's well recognized that numerous problems, particularly those in constrained optimization, can reap substantial benefits from r-flip strategies as local procedures. Comparisons with alternative methodologies documented in the literature underscore the remarkable performance of our approach, surpassing leading competitors by a significant margin. Notably, our method consistently discovered previously undiscovered optimal solutions for the majority of the problem instances tackled. What distinguishes our achievement is that our solution framework isn't tailored specifically for the Max-Cut problem but rather accommodates a broader class of problems. Our findings strongly indicate the potential for significant advancements in r-flip solution techniques, hinting at their capacity to tackle even larger-scale Max-Cut challenges with further refinement.

These results present notable advantages, particularly in scenarios where variable neighborhood strategies are employed for large-scale problems or sparse matrices. Furthermore, we benchmark our outcomes against those obtained from a quantum annealing solver based on a multi-start tabu search. Notably, the quantum annealing solver has also proven effective in addressing the challenges posed by very large-scale MCP. This comparative analysis provides insights into the strengths and applicability of both the r-flip strategy and quantum annealing solver in tackling complex optimization problems.

References

Kochenberger, G.A., Glover, F., Alidaee, B., and Rego, C.: ‘A unified modeling and solution framework for combinatorial optimization problems’, OR Spectrum, 2004, 26, (2), pp. 237-250.
Rosenberg, G., Vazifeh, M., Woods, B., and Haber, E.: ‘Building an iterative heuristic solver for a quantum annealer’, Computational Optimization and Applications, 2016, 65, (3), pp. 845-869.
Barahona, F., Grötschel, M., Jünger, M., and Reinelt, G.: ‘An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design’, Operations Research, 1988, 36, (3), pp. 493-513.
Kel’manov, A.V., and Pyatkin, A.V.: ‘On complexity of some problems of cluster analysis of vector sequences’, Journal of Applied and Industrial Mathematics, 2013, 7, (3), pp. 363-369.
Liers, F., Nieberg, T., and Pardella, G. ‘Via Minimization in VLSI Chip Design - Application of a Planar Max-Cut Algorithm’. Working Paper. 2011.
Goemans, M., Williamson, D.: Improved approximation algorithms for Max-Cut and satisfiability problems using Semidefinite programming. J. ACM 42, 1115–1145 (1995).
Karish, S., Rendl, F., Clausen, J.: Solving graph bisection problems with semidefinite programming. SIAM J. Comput. 12, 177–191 (2000).
Krishnan, K., Mitchell, J.: A semidefinite programming based polyhedral cut and price approach for the Max-Cut problem. Comput. Optim. Appl. 33, 51–71 (2006).
Rendl, F., Rinaldi, G., Wiegele, A.: Solving Max-Cut to optimality by intersecting semidefinite and poly-hedral relaxations. Math. Program. 121, 307–335 (2008).
Burer, S., Monteiro, D., Zang, Y.: Rank-two relaxation heuristic for Max-Cut and other binary quadratic programs. SIAM J. Optim. 12, 503–521 (2001).
Festa, P., Pardalos, P., Resende, M., Ribeiro, C.: Randomized heuristics for the Max-Cut problem. Optim. Methods Softw. 7, 1033–1058 (2002).
Marti, R., Duarte, A., Laguna, M.: Advanced scatter search for the Max-Cut problem. INFORMS J. Com- put. 21, 26–38 (2009).
Alidaee, Bahram, Hugh Sloan, and Haibo Wang. "Simple and fast novel diversification approach for the UBQP based on sequential improvement local search." Computers & Industrial Engineering 111 (2017): 164-175.
Kochenberger, Gary A., Jin-Kao Hao, Zhipeng Lü, Haibo Wang, and Fred Glover. "Solving large scale max cut problems via tabu search." Journal of Heuristics 19 (2013): 565-571.
Alam M, Ash-Saki A, Ghosh S, editors. Accelerating Quantum Approximate Optimization Algorithm using Machine Learning. 2020 Design, Automation & Test in Europe Conference & Exhibition (DATE); 2020 9-13 March 2020.
Basso J, Farhi E, Marwaha K, et al. The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model. 2022. p. 7:1-7:21.
Galda A, Liu X, Lykov D, et al., editors. Transferability of optimal QAOA parameters between random graphs. 2021 IEEE International Conference on Quantum Computing and Engineering (QCE); 2021 17-22 Oct. 2021.
Bianchi FM, Grattarola D, Livi L, et al. Hierarchical Representation Learning in Graph Neural Networks With Node Decimation Pooling. IEEE Transactions on Neural Networks and Learning Systems. 2022;33(5):2195-2207. [CrossRef]
Shaydulin R, Safro I, Larson J, editors. Multistart Methods for Quantum Approximate optimization. 2019 IEEE High Performance Extreme Computing Conference (HPEC); 2019 24-26 Sept. 2019.
Shaydulin R, Marwaha K, Wurtz J, et al., editors. QAOAKit: A Toolkit for Reproducible Study, Application, and Verification of the QAOA. 2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS); 2021 15-15 Nov. 2021.
Shaydulin R, Wild SM. Exploiting Symmetry Reduces the Cost of Training QAOA. IEEE Transactions on Quantum Engineering. 2021;2:1-9. [CrossRef]
Umasankar G, Shah PS, Chandrachoodan N, et al., editors. Benchmarking the Poor Man's Ising Machine. 2021 IEEE Photonics Society Summer Topicals Meeting Series (SUM); 2021 19-21 July 2021.
Wurtz J, Love PJ. Classically Optimal Variational Quantum Algorithms. IEEE Transactions on Quantum Engineering. 2021;2:1-7. [CrossRef]
Alidaee, B., Kochenberger, G., and Wang, H.: ‘Theorems Supporting r-flip Search for Pseudo-Boolean Optimization’, Int. J. Appl. Metaheuristic Comput., 2010, 1, (1), pp. 93-109.
Alidaee, B.; Wang, H.; Sua, L.S. An Efficient Closed-Form Formula for Evaluating r-Flip Moves in Quadratic Unconstrained Binary Optimization. Algorithms 2023, 16, 557. [Google Scholar] [CrossRef]
Palubeckis, G.: ‘Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem’, Annals of Operations Research, 2004, 131, (1), pp. 259-282.
Garey, M.R., and Johnson, D.S.: ‘Computers and Intractability; A Guide to the Theory of NP-Completeness’ (W. H. Freeman & Co., 1990. 1990).
Gu, S.; Yang, Y. A Deep Learning Algorithm for the Max-Cut Problem Based on Pointer Network Structure with Supervised Learning and Reinforcement Learning Strategies. Mathematics 2020, 8, 298. [Google Scholar] [CrossRef]
Rehfeldt, D., Koch, T. & Shinano, Y. Faster exact solution of sparse MaxCut and QUBO problems. Math. Prog. Comp. 15, 445–470 (2023). [CrossRef]
Glover, F., and Woolsey, E.: ‘Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program’, Operations Research, 1974, 22, (1), pp. 180-182.
Alidaee, B., and Wang, H.: ‘A note on heuristic approach based on UBQP formulation of the maximum diversity problem’, Journal of the Operational Research Society, 2017, 68, (1), pp. 102-110.

Figure 1. MCP with 10,000 variables

Figure 2. MCP with 20,000 variables

Figure 3. MCP with 40,000 variables

Table 1. Computational Results of MCP with 10,000 variables.

Instance ID	Type a				Type b			Type c
	Algorithm2			QAS	Algorithm2		QAS	Algorithm2		QAS
	BFS		APD	RPD	BFS	APD	RPD	BFS	APD	RPD
MC10000_1		8156	1.87	8.96	50228	1.94	1.35	4672506	2.03	1.79
MC10000_2		8067	2.00	2.45	50884	1.89	1.2	4572770	2.02	1.43
MC10000_3		8129	1.87	2.4	51390	1.87	7.62	4675380	1.96	1.33
MC10000_4		8253	1.93	2	51680	1.88	1.37	4574974	2.03	1.4
MC10000_5		8218	1.84	2.19	51078	1.9	3.5	4664286	1.98	1.32
MC10000_6		8369	1.75	2.17	50710	1.87	2.76	4622312	1.94	1.5
MC10000_7		8142	1.86	3.13	50322	1.87	5.76	4639744	1.99	1.02
MC10000_8		8218	1.81	2.29	50334	1.96	1.53	4728400	1.92	1
MC10000_9		8190	1.85	4.19	50588	1.98	1.74	4699270	2.02	1.27
MC10000_10		8074	1.84	2.29	50790	1.55	3.34	4749898	1.92	1.33
MC10000_11		8292	1.94	2.45	51800	1.61	3.32	4638566	1.99	1.25
MC10000_12		8011	1.93	4.12	50338	1.91	1.7	4698042	1.95	1.62
MC10000_13		8233	1.89	9.15	50718	1.87	1.64	4585190	1.97	1.25
MC10000_14		8276	1.91	15.08	51338	1.87	1.69	4681668	2.01	1.48
MC10000_15		8224	1.83	7.5	50604	1.94	2.06	4680394	1.93	1.43

Table 2. Computational Result of MCP with 20,000 variables.

Instance ID	Type a				Type b			Type c
	Algorithm2			QAS	Algorithm2		QAS	Algorithm2		QAS
	BFS		APD	RPD	BFS	APD	RPD	BFS	APD	RPD
MC20000_1		16476	1.60	7.03	102580	2.46	3.4	9392660	0.40	2.21
MC20000_2		16368	2.12	6.77	100878	2.52	8.11	9407570	0.53	2.29
MC20000_3		16384	1.65	8.57	102006	2.55	4.99	9217500	2.45	6.18
MC20000_4		16547	2.07	8.22	101174	2.49	7.68	9359374	2.43	4.24
MC20000_5		16365	2.11	6.37	100236	2.59	9.84	9261876	2.45	4.43
MC20000_6		16253	1.69	10.66	101064	2.57	8.51	9268934	2.44	4.46
MC20000_7		16177	2.15	7.08	102820	2.55	8.95	9228882	2.45	4.43
MC20000_8		16492	2.14	7.43	101914	2.47	8.49	9236988	2.50	4.81
MC20000_9		16597	1.62	6.98	103390	2.51	14.7	9345570	2.42	5.02
MC20000_10		16586	1.38	7.65	102896	2.43	4.33	9126342	2.48	4.53
MC20000_11		16307	1.40	6.89	101566	2.49	3.93	9313278	2.48	5.33
MC20000_12		16433	2.10	6.66	102376	2.47	3.83	9439436	2.42	4.23
MC20000_13		16621	1.37	8.36	100238	2.50	4.11	9304690	2.50	4.76
MC20000_14		16407	2.10	8.26	100524	2.57	4.58	9390564	2.41	4.09
MC20000_15		16254	2.13	7.57	102768	2.52	4.03	9215550	2.47	6.36

Table 3. Computational Result of MCP with 40,000 variables.

Instance ID	Type a				Type b			Type c
	Algorithm2			QAS	Algorithm2		QAS	Algorithm2		QAS
	BFS		APD	RPD	BFS	APD	RPD	BFS	APD	RPD
MC40000_1		33036	4.52	11.02	201520	3.46	8.13	18412388	3.23	9.29
MC40000_2		32780	4.60	11.14	202278	2.71	8.19	18533702	3.28	9.47
MC40000_3		32971	4.62	10.87	204676	2.70	8.16	18443578	3.33	9.43
MC40000_4		32797	4.63	10.52	203482	2.69	8.18	18478330	3.28	9.41
MC40000_5		32676	4.68	10.73	204004	2.69	8.27	18351592	3.25	9.59
MC40000_6		32770	4.67	10.94	204288	2.71	8.22	18645472	3.25	10.33
MC40000_7		32833	4.67	10.89	202726	2.74	8.79	18511826	3.29	8.98
MC40000_8		32805	4.62	10.67	201954	3.45	9.1	18520832	3.23	10.18
MC40000_9		32374	4.75	10.43	201384	2.73	8.67	18574756	3.29	9.72
MC40000_10		32922	4.64	10.55	203348	2.68	8.57	18478474	3.31	9.31
MC40000_11		32733	4.64	10.83	201328	2.73	8.25	18516990	3.24	8.94
MC40000_12		33111	4.58	10.34	202800	3.42	8.78	18557510	3.24	9.6
MC40000_13		32863	4.62	11.1	201674	2.71	8.66	18613850	3.21	10.18
MC40000_14		32917	4.63	9.91	204160	2.70	8.27	18448596	3.25	9.76
MC40000_15		33079	4.63	10.75	204466	2.70	7.82	18510938	3.26	9.17

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Efficient Solutions for Large-Scale Max-Cut Problems: A Hybrid Local Search Heuristic Approach and Comparative Analysis with Quantum Annealing

Abstract

1. Introduction

2. QUBO and MCP Solution

2.1. Max-Cut Problems

2.2. Solution Representation

2.3. Implementation Details

3. Computational Experiments

4. Numerical Results

5. Conclusions

References

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