Search Results (28)

In graph theory and network design, the minimum cut is a fundamental measure of system connectivity and communication capacity. While prior research has largely focused on computing the minimum cut for a fixed source–sink pair, practical scenarios such as data center communication often demand a different objective: identifying the source node whose minimum cut to a designated sink is maximized. This task, which we term the Global Maximum Minimum Cut with Fixed Sink (GMMC-FS) problem, captures the goal of locating a high-capacity source relative to a shared sink node that aggregates multiple servers. The problem is of significant engineering importance, yet it is computationally challenging as it involves a nested max–min optimization. In this paper, we present a recursive reduction (RR) algorithm for solving the GMMC-FS problem. The key idea is to iteratively select pivot nodes, compute their minimum cuts with respect to the sink, and prune dominated candidates whose cut values cannot exceed that of the pivot. By recursively applying this elimination process, RR dramatically reduces the number of max-flow computations required while preserving exact correctness. Compared with classical contraction-based and Gomory–Hu tree approaches that rely on global cut enumeration, the proposed RR framework offers a more direct and scalable mechanism for identifying the source that maximizes the minimum cut to a fixed sink. Its novelty lies in exploiting the structural properties of the sink side of suboptimal cuts, which leads to both theoretical efficiency and empirical robustness across large-scale networks. We provide a rigorous theoretical analysis establishing both correctness and complexity bounds, and we validate the approach through extensive experiments. Results demonstrate that RR consistently achieves optimal solutions while significantly outperforming baseline methods in runtime, particularly on large and dense networks. Full article

This follow-up scientific case study builds on prior research to explore the computational challenges of applying quantum algorithms to financial asset management, focusing specifically on solving the graph-cut problem for investment recommendation. Unlike our prior study, which focused on idealized QAOA performance, this work introduces a graph compression pipeline that enables QAOA deployment under real quantum hardware constraints. This study investigates quantum-accelerated spectral graph compression for financial asset recommendations, addressing scalability and regulatory constraints in portfolio management. We propose a hybrid framework combining the Quantum Approximate Optimization Algorithm (QAOA) with spectral graph theory to solve the Max-Cut problem for investor clustering. Our methodology leverages quantum simulators (cuQuantum and Cirq-GPU) to evaluate performance against classical brute-force enumeration, with graph compression techniques enabling deployment on resource-constrained quantum hardware. The results underscore that efficient graph compression is crucial for successful implementation. The framework bridges theoretical quantum advantage with practical financial use cases, though hardware limitations (qubit counts, coherence times) necessitate hybrid quantum-classical implementations. These findings advance the deployment of quantum algorithms in mission-critical financial systems, particularly for high-dimensional investor profiling under regulatory constraints. Full article

A novel algorithm was proposed for solving the max-cut problem, which seeks to identify the cut with the maximum weight in a given graph. Our technique is based on the bundle approach, applied to a newly formulated semidefinite relaxation. This research establishes the theoretical convergence of our approximation technique and presents the numerical results obtained on several large-scale graphs from the BiqMac library, specifically with 100, 250, and 500 nodes. The resulting performance was compared with that produced by two alternative semidefinite programming-based approximation methods, namely the BiqMac and BiqBin solvers, by comparing the CPU time and the number of function calls. The primary objective of this work was to enhance the scalability and computational efficiency in solving the max-cut problem, particularly for large-scale graph instances. Despite the development of numerous approximation algorithms, a persistent challenge lies in effectively handling problems with a large number of constraints. Our algorithm addresses this by integrating a novel semidefinite relaxation with a bundle-based optimization framework, achieving faster convergence and fewer function calls. These advancements mark a meaningful step forward in the efficient resolution of NP-hard combinatorial optimization problems. Full article

The Maximum Colored Cut problem aims to seek a bipartition of the vertex set of a graph, maximizing the number of colors in the crossing edges. It is a classical Max-Cut problem if the host graph is rainbow. Let $mcc\left(G\right)$ denote the maximum number of colors in a cut of an edge-colored graph G. Let $C_k$ be a cycle of length k; we say G is PC-$C_k$-free if G contains no properly colored $C_k$. We say G is a p-edge-colored graph if there exist p colors in G. In this paper, we first show that if G is a PC-$C_3$-free p-edge-colored complete 4-partite graph, then $mcc\left(G\right)=p$. Let $k\ge 3$ be an integer. Then, we show that if G is a PC-$C_4$-free p-edge-colored complete k-partite graph, then $mcc\left(G\right)\ge min\{p-1,15p/16\}$. Finally, for a p-edge-colored complete graph G, we prove that $mcc\left(G\right)\ge p-1$ if G is PC-$C_4$-free, and $mcc\left(G\right)\ge min\{p-6,7p/8\}$ if G is PC-$C_5$-free and $p\ge 7$. Full article

The max-cut problem is a well-known topic in combinatorial optimization, with a wide range of practical applications. Given its NP-hard nature, heuristic approaches—such as genetic algorithms, tabu search, and harmony search—have been extensively employed. Recent research has demonstrated that harmony search can outperform genetic algorithms by effectively avoiding redundant searches, a strategy similar to tabu search. In this study, we propose a modified genetic algorithm that integrates tabu search to enhance solution quality. By preventing repeated exploration of previously visited solutions, the proposed method significantly improves the efficiency of traditional genetic algorithms and achieves performance levels comparable to harmony search. The experimental results confirm that the proposed algorithm outperforms standard genetic algorithms on the max-cut problem. This work demonstrates the effectiveness of combining tabu search with genetic algorithms and offers valuable insights into the enhancement of heuristic optimization techniques. The novelty of our approach lies in integrating solution-level tabu constraints directly into the genetic algorithm’s population dynamics, enabling redundancy prevention without additional memory overhead, a strategy not previously explored in the proposed hybrids. Full article

This study explores the application of quantum computing in asset management, focusing on the use of the Quantum Approximate Optimization Algorithm (QAOA) to solve specific classes of financial asset recommendation problems. While quantum computing holds promise for combinatorial optimization tasks, its application to portfolio management faces significant challenges in scalability for practical implementations. In this work, we model the problem using a graph representation where nodes represent investors, and edges reflect significant similarities in asset choices. We test the proposed method using quantum simulators, including cuQuantum, Cirq-GPU, and Cirq with IonQ, and compare the performance of quantum optimization against classical brute-force methods. Our results suggest that quantum algorithms may offer computational advantages for certain use cases, though classical heuristics also provide competitive performance for smaller datasets. This study contributes to the ongoing investigation into the potential of quantum computing for real-time financial decision-making, providing insights into both its applicability and limitations in asset management for larger and more complex investor datasets. Full article

This paper introduces an improved Grover Adaptive Search (GAS) algorithm. The GAS algorithm has been prove to achieve quadratic acceleration in the Constrained Polynomial Binary Optimization (CPBO) problem. Nevertheless, the acceleration effect of the GAS algorithm can be decreased by the poor threshold selection. This article uses the Quantum Approximate Optimization Algorithm (QAOA) to improve the initial threshold selection, thereby accelerating the convergence speed of the original GAS algorithm. The acceleration effect of the improved GAS algorithm is presented by the Max-Cut problem and the CPBO problem. Full article

The bipartite polarization problem is an optimization problem where the goal is to find the highest polarized bipartition on a weighted and labeled graph that represents a debate developed through some social network, where nodes represent user’s opinions and edges agreement or disagreement between users. This problem can be seen as a generalization of the maxcut problem, and in previous work, approximate solutions and exact solutions have been obtained for real instances obtained from Reddit discussions, showing that such real instances seem to be very easy to solve. In this paper, we further investigate the complexity of this problem by introducing an instance generation model where a single parameter controls the polarization of the instances in such a way that this correlates with the average complexity to solve those instances. The average complexity results we obtain are consistent with our hypothesis: the higher the polarization of the instance, the easier is to find the corresponding polarized bipartition. In view of the experimental results, it is computationally feasible to implement transparent mechanisms to monitor polarization on online discussions and to inform about solutions for creating healthier social media environments. Full article

We explore strong Nash equilibria in the max k-cut game on an undirected and unweighted graph with a set of k colors. Here, the vertices represent players, and the edges denote their relationships. Each player, v, selects a color as its strategy, and its payoff (or utility) is determined by the number of neighbors of v who have chosen a different color. Limited findings exist on the existence of strong equilibria in max k-cut games. In this paper, we make advancements in understanding the characteristics of strong equilibria. Specifically, our primary result demonstrates that optimal solutions are seven-robust equilibria. This implies that for a coalition of vertices to deviate and shift the system to a different configuration, i.e., a different coloring, a number of coalition vertices greater than seven is necessary. Then, we establish some properties of the minimal subsets concerning a robust deviation, revealing that each vertex within these subsets will deviate toward the color of one of its neighbors. Full article

The Constraint Satisfaction Problem (CSP) is a significant research area in artificial intelligence, and includes a large number of symmetric or asymmetric structures. A backtracking search combined with constraint propagation is considered to be the best CSP-solving algorithm, and the consistency algorithm is the main algorithm used in the process of constraint propagation, which is the key factor in constraint-solving efficiency. Max-restricted path consistency (maxRPC) is a well-known and efficient consistency algorithm, whereas the lmaxRPC3${}^{rm}$ algorithm is a classic lightweight algorithm for maxRPC. In this paper, we leverage the properties of symmetry to devise an improved pruning strategy aimed at efficiently diminishing the problem’s search space, thus enhancing the overall solving efficiency. Firstly, we propose the maxRPC3${}^{sim}$ algorithm, which abandons the two complex data structures used by lmaxRPC3${}^{rm}$. We can render the algorithm to be more concise and competitive compared to the original algorithm while ensuring that it maintains the same average performance. Secondly, inspired by the RCP3 algorithm, we propose the maxRPC3${}^{simR}$ algorithm, which uses the idea of residual support to cut down the redundant operation of the lmaxRPC3${}^{rm}$ algorithm. Finally, combining the domain/weighted degree (dom/wdeg) heuristic with the activity-based search (ABS) heuristic, a new variable ordering heuristic, ADW, is proposed. Our heuristic prioritizes the selection of variables with symmetry for pruning, further enhancing the algorithm’s pruning capabilities. Experiments were conducted on both random and structural problems separately. The results indicate that our two algorithms generally outperform other algorithms in terms of performance on both problem classes. Moreover, the new heuristic algorithm demonstrates enhanced robustness across different problem types when compared to various existing algorithms. Full article

In this paper, we employ PCA and t-SNE analyses to gain deeper insights into the behavior of entangled and non-entangled mixing operators within the Quantum Approximate Optimization Algorithm (QAOA) at various depths. We utilize a dataset containing optimized parameters generated for max-cut problems with cyclic and complete configurations. This dataset encompasses the resulting $RZ$, $RX$, and $RY$ parameters for QAOA models at different depths ($1L$, $2L$, and $3L$) with or without an entanglement stage within the mixing operator. Our findings reveal distinct behaviors when processing the different parameters with PCA and t-SNE. Specifically, most of the entangled QAOA models demonstrate an enhanced capacity to preserve information in the mapping, along with a greater level of correlated information detectable by PCA and t-SNE. Analyzing the overall mapping results, a clear differentiation emerges between entangled and non-entangled models. This distinction is quantified numerically through explained variance in PCA and Kullback–Leibler divergence (post-optimization) in t-SNE. These disparities are also visually evident in the mapping data produced by both methods, with certain entangled QAOA models displaying clustering effects in both visualization techniques. Full article

Quantum computing is a promising technology that may provide breakthrough solutions to today’s difficult problems such as breaking encryption and solving large-scale combinatorial optimization problems. An algorithm referred to as Quantum Approximate Optimization Algorithm (QAOA) has been recently proposed to approximately solve hard problems using a protocol know as bang–bang. The technique is based on unitary evolution using a Hamiltonian encoding of the objective function of the combinatorial optimization problem. The QAOA was explored in the context of several optimization problems such as the Max-Cut problem and the Traveling Salesman Problem (TSP). Due to the relatively small node-size solution capability of the available quantum computers and simulators, we developed a hybrid approach where sub-graphs of a TSP tour can be executed on a quantum computer, and the results from the quantum optimization are combined for a further optimization of the whole tour. Since the local optimization of a sub-graph is prone to becoming trapped in a local minimum, we overcame this problem by using a parallel Ant Colony Optimization (ACO) algorithm with periodic pheromone exchange between colonies. Our method exceeds existing approaches which have attempted partitioning a graph for small problems (less than 48 nodes) with sub-optimal results. We obtained optimum results for benchmarks with less than 150 nodes and results usually within 1% of the known optimal solution for benchmarks with around 2000 nodes. Full article

We investigate a class of challenging general semidefinite programming problems with extra nonconvex constraints such as matrix rank constraints. This problem has extensive applications, including combinatorial graph problems, such as MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints. A customized approach based on the alternating direction method of multipliers (ADMM) is proposed to solve the general large-scale nonconvex semidefinite programming efficiently. We propose two reformulations: one using vector variables and constraints, and the other further reformulating the Burer–Monteiro form. Both formulations admit simple subproblems and can lead to significant improvement in scalability. Despite the nonconvex constraint, we prove that the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this matrix form, when the matrix factors are wide enough, the local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation. Full article

The linear Diophantine fuzzy set notion is the main foundation of the interactive method of tackling nonlinear fractional programming problems that is presented in this research. When the decision maker (DM) defines the degree $\alpha$ of $\alpha$ level sets, the max-min problem is solved in this interactive technique using Zimmermann’s min operator method. By using the updating technique of degree $\alpha$, we can solve DM from the set of $\alpha$-cut optimal solutions based on the membership function and non-membership function. Fuzzy numbers based on $\alpha$-cut analysis bestowing the degree $\alpha$ given by DM can first be used to classify fuzzy Diophantine inside the coefficients. After this, a crisp multi-objective non-linear fractional programming problem (MONLFPP) is created from a Diophantine fuzzy nonlinear programming problem (DFNLFPP). Additionally, the MONLFPP can be reduced to a single-objective nonlinear programming problem (NLPP) using the idea of fuzzy mathematical programming, which can then be solved using any suitable NLPP algorithm. The suggested approach is demonstrated using a numerical example. Full article

The quantum approximate optimization algorithm (QAOA) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area of research, as bad initialization might deteriorate the quality of the results, especially at great circuit depths. We first discuss the patterns of optimal parameters in QAOA in two directions: the angle index and the circuit depth. Then, we discuss the symmetries and periodicity of the expectation that is used to determine the bounds of the search space. Based on the patterns in optimal parameters and the bounds restriction, we propose a strategy that predicts the new initial parameters by taking the difference between the previous optimal parameters. Unlike most other strategies, the strategy we propose does not require multiple trials to ensure success. It only requires one prediction when progressing to the next depth. We compare this strategy with our previously proposed strategy and the layerwise strategy for solving the Max-cut problem in terms of the approximation ratio and the optimization cost. We also address the non-optimality in previous parameters, which is seldom discussed in other works despite its importance in explaining the behavior of variational quantum algorithms. Full article