This study addresses the problem of finding an anti-Berge equilibrium in a bimatrix game based on a global search algorithm. Finding the anti-Berge equilibrium reduces equivalently to a quadratic programming with an indefinite matrix and linear constraints, which belongs to a class of global optimization. To solve the problem numerically, we develop a modified parallel tangent algorithm. The proposed algorithm uses the one-dimensional nonlocal search procedure based on the Strongin and parabolas methods. The stop criterion of the algorithm is the sufficient condition for anti-Berge equilibrium. The proposed algorithm is implemented and numerically tested on a range of bimatrix games.
The quality, accessibility, and stability of the business environment directly affect the competitiveness and growth of enterprises. Therefore, this study aims to classify Mongolia’s business environment using machine learning techniques and identify each cluster’s specific characteristics and key influencing factors. The research utilizes data from 4,073 enterprises operating in Mongolia as of 2020, comprising 104 variables related to the business environment. Since the original dataset contained many missing values, imputation was necessary. Mode and K-Nearest Neighbors (KNN) imputation methods—commonly used in research—were tested. Their mean absolute errors (MAE) were evaluated against the original data, and KNN (MAE = 0.0119) demonstrated higher accuracy than Mode (MAE = 0.0438). As a result, the KNN-imputed dataset was used for further analysis. Five clustering algorithms were applied to group the data, and their performance was assessed using three metrics: Silhouette Score, Davies-Bouldin Index, and Calinski-Harabasz Score. Among these, the HDBSCAN model yielded the best results (Silhouette Score = 0.435, DB Index = 0.409), and after removing 1,051 enterprises classified as noise, the remaining 2,852 enterprises were grouped into three clusters. Additionally, Random Forest classification and SHAP analysis were used to identify the most influential features in cluster formation. The study showed that bank interest rates, the effectiveness of government policies, and household living standards had the most significant impact on cluster differentiation. By identifying these key drivers, this study provides scientifically grounded insights that support evidence-based decision-making for policymakers and researchers seeking to improve the business environment.
Critical node detection involves identifying key nodes whose removal significantly disrupts network connectivity or performance. The evaluation of vulnerability in networks is crucial because many real-world problems are represented as graphs with uncertainty, including social networks, communication infrastructures, epidemiological models, and transportation systems. In this paper, we consider a stochastic Critical Node Detection Problem (SCNDP) with edge uncertainty, aiming to minimize the expected pairwise connectivity (EPC) in the resulting residual network. We propose a heuristic method for the SCNDP and compare it with the existing algorithm. Experimental results performed on random graphs with different edge-probability configurations demonstrate the effectiveness of the heuristics.
The n-dimensional folded hypercube FQ_n is a well-known extension of the n-dimensional hypercube that can be constructed by adding an edge to every pair of vertices with complementary addresses. The existence of the Hamiltonian cycle provides advantages for implementing algorithms that require a ring structure. In addition, k (≥ 2) edge-disjoint Hamiltonian cycles also provide higher transmission efficiency for all-to-all communication algorithms and provide fault-tolerant routing for network transmission. This paper proves that there exist two edge-disjoint Hamiltonian cycles on FQ_n while n ≥ 3 ; and for k edge-disjoint Hamiltonian cycles on FQ_n , the maximum k is 2 while n ∈{3, 4} and 3 while n ∈{5, 6} .
Submodular optimization is an important topic in the field of combinatorial optimization. The k-submodular function is a generalization of the submodular function. Many machine learning problems, such as placing k types of sensors, can be formulated as maximizing a monotone k-submodular function. In real-world optimization problems where the constraints involve random components, chance constraints are often used to limit the probability of constraint violations due to random factors. In this paper, we investigate the problem of maximizing a monotone k-submodular function under the chance constraint. Specifically, each element e is assigned a random weight independently sampled from a uniform distribution. We study two distinct cases, in which the weights are drawn from a common uniform distribution or from element-specific uniform distributions, all with the same dispersion. For the first case, this paper designs a threshold-decreasing algorithm with an approximation ratio of 1/2-ϵ . For the second case, this paper first designs a deterministic algorithm, which uses the method of dividing the ground set into two parts, and then combines its output solution with the threshold-greedy algorithm to design an algorithm with an approximation ratio of 1/3-ϵ .
The SIS model is one of the fundamental frameworks used to describe infectious diseases in which recovered individuals do not acquire permanent immunity. However, traditional numerical schemes such as the Euler and Runge–Kutta methods often fail to preserve key qualitative properties of the model—namely, the positivity of solutions, conservation of total population, and dynamic consistency—especially when large time steps are used. In this study, we propose a novel Nonstandard Finite Difference (NSFD) scheme for the SIS model. The scheme guarantees the positivity of solutions, conservation of total population, and elementary stability, while exhibiting weak dependence on both the step size and parameter variations. Theoretical analysis and numerical experiments demonstrate that the proposed scheme provides more stable, robust, and dynamically consistent solutions than the standard Euler, second-order Runge–Kutta (RK2), and Mickens-type NSFD methods. The results indicate that the proposed NSFD approach offers an accurate and reliable numerical tool for simulating SIS and other structurally similar epidemic models.
In this study, we introduce a so-called Inverse Malfatti’s problem which is to find a triangle including three non-overlapping circles with the minimum area. This problem will be an inverse version of Malfatti’s problem, which dates back to 200 years ago, proposed by the Italian mathematician Malfatti. The Malfatti’s problem for the first time has been examined from the point of view of global optimization in [1]. The Inverse Malfatti’s problem has also been formulated as the non-convex optimization problem of minimizing a convex function over a non-convex set. Some computational results are provided
Urine bubble analysis serves as a crucial diagnostic indicator for Heyi syndrome (a key Mongolian medical syndrome) and others. However, current diagnostic methods primarily depend on manual visual examination, which lacks robust automated image analysis methods. To bridge this gap, this paper propose a urine bubble image segmentation approach combining backbone replacement and loss function analysis to facilitate Mongolian medicine diagnosis. Methodologically, we adopt the Vision Mamba U-Net (VMUNet) as the base framework, and explore a backbone replacement strategy to enhance segmentation performance. Specifically, we substitute the original VMUNet backbone with four mainstream segmentation architectures—U ^2 -Net, U-Net, PSPNet, and DeepLabV3+—and systematically evaluate their performance on the urine bubble segmentation task. For each architecture, we adopt representative loss functions based on their prior relevance to medical image segmentation, including Binary Cross-Entropy (BCE), Dice loss, Tversky loss, Focal Tversky loss, and Boundary loss. This results in a comprehensive set of architecture–loss combinations, enabling a rigorous comparative analysis. Experimental results demonstrate that integrating the DeepLabV3+ architecture with the BCE–Dice loss function yields the best performance in edge restoration, small bubble detection, and background suppression. In particular, it achieves an Intersection over Union (IoU) of 72.88
This study constructs Chisel Verification Assertions (CVA), a new assertion language and formal verification tool. CVA heavily relies on the ChiselTest framework and integrates the powerful expressive power of Propositional Projection Temporal Logic (PPTL). A temporal operator system that adapts to Chisel characteristics is designed independently in CVA, achieving a leapfrog improvement in the expressiveness of the Chisel assertion language. This operator system, following hardware validation logic, can accurately describe complex temporal and functional properties. Several typical cases of Chisel program verification are selected to validate CVA. The results show that CVA can effectively capture logic and temporal errors in design, verifying its technical effectiveness and practicality in hardware verification scenarios.
Finding all solutions to a system of non-linear equations is a longstanding and challenging problem with applications across science and engineering. The recent emergence of large language models raises the question of whether these models can assist, or even autonomously solve, such problems. This paper reviews classical and global optimization approaches for finding all roots to systems of nonlinear equations and empirically evaluates nine contemporary large language models on a suite of small-sized representative systems. The solution quality is analyzed, common failure modes are investigated, and the impact of prompt design is considered, offering guidelines for practitioners and directions for future research.
In this paper, we investigate the twin bridges problem in polygons, whose counterpart on trees has been studied recently. Given two disjoint simple polygons P and Q, each with at most n vertices, the problem is to find two vertices p_1 and p_2 in P (and q_1 and q_2 in Q) to build two bridges p_1q_1 and p_2q_2 such that the constrained diameter of the resulting geometric structure (e.g., a polygon with a hole when the two bridges do not intersect), i.e., the maximum of the shortest distance between two vertices in P, Q, or just in one of them, through at least one of the bridges, is minimized. The main results are summarized as follows: (1) If P and Q are arbitrary polygons, the problem can be solved in O(n^6log n) time. (2) These results hold even if the bridges between p_i and q_i , i∈{1,2} , are geodesic. (3) We show the general problem is NP-hard: given m disjoint polygons P_1,...,P_m , add two bridges between P_i and P_i+1 , i∈ [m-1] , such that the constrained diameter of the resulting geometric structure (ideally, a polygon with multiple holes, but could be more complex if some of the bridges intersect) is minimized.
This paper addresses the interactions in the economic system with one importer and several exporters of the homogeneous product on the example of the international coal market. The importer sets prices and additional tariffs for each supplier country for a number of coal resources types. The suppliers determine the quantity of the supplied product, in consideration of their costs and supply capabilities. To optimize such a system, we propose a hierarchical (bilevel) model where the upper-level (the importer) aims to minimize the costs of each product type, taking into account the satisfaction of its needs. And the goal of each exporter is to maximize its profit, given the established prices and tariffs, as well as its own production costs and supply capacity. To develop numerical methods for the considered problem, we begin by reformulating the bilevel problem as a single-level nonconvex optimization problem, substituting the lower level with its optimality conditions. Next, we apply the Exact Penalization Theory along with the Global Search Theory (GST) to tackle the transformed problem. GST involves expressing nonconvex functions as DC representations (that is, as the difference of two convex functions) and constructing both local and global search methods tailored to the distinct features of the examined problem.
This paper proposes a family of m-step derivative-free iterative methods for solving nonlinear systems. It is shown that these methods achieve a local convergence order of 2m while maintaining low computational cost due to their simple structure. Accelerated variants with memory are also developed and demonstrate improved performance. Numerical experiments confirm the theoretical results and show that the proposed methods are computationally efficient when compared with existing approaches in terms of CPU time.
In this work, we studied the operational effectiveness of the Call Center of the Public Transport Department (PTD) of Ulaanbaatar. Our focus was on its responsiveness to incoming calls, review of the workload based on the actual data, and the integration of gender and social inclusion issues. This research enables the PTD to make evidence-based decisions regarding human resources while also addressing critical gender and social inclusion concerns. Enhancing the performance and efficiency of the PTD is crucial for improving public transportation services, a significant contribution to alleviating the pressing issue of traffic congestion in Ulaanbaatar. We conducted statistical and simulation analyses using queueing theory on historical data from March 22 to September 20, 2023. Our findings indicate that the call center receives 12 calls on average, spending 12.4 min each hour. However, 31
The tessellation cover number on graphs is an -hard optimization problem important to model staggered walks in quantum computation. A tessellation is a partition of the vertices of a graph into vertex disjoint cliques. The tessellation cover problem aims to determine the minimum number t of tessellations that covers all edges of a given graph G. We denote by t-tessellability the decision version where it is asked if G admits a tessellation cover number of size t. Tessellations on graphs also have an interesting value for graph theory, since the tessellation cover number parameter T(G) is related to several others in the literature such as chromatic number, chromatic index, and maximum clique. In this study, we develop QUBO formulations for the following -hard problems: the determination of the maximum induced star on G, which yields a lower bound on T(G); the determination of the chromatic number of the clique graph of G and the chromatic index of G, which yield upper bounds on T(G); the determination of T(G), by transforming t-tessellability to integer programming and then to QUBO. In order to demonstrate the order-based formulation presented, we implement and analyze these formulations.
Cloud computing environments impose challenges in incorporating the maximum degree of parallelism constraints into Parallel Batch Machine Scheduling (PBMS). Unlike traditional PBMS that considers only fixed job widths, this paper studies generalized PBMS with malleable jobs that allow job width to be changed during the job execution, provided it does not exceed its maximum degree of parallelism. We propose a fast O(n log n) approximation algorithm by extending the state-of-the-art PBMS algorithm by setting each job’s width to its maximum degree of parallelism, where n is the number of jobs. Due to the unique nature of malleable jobs, previous ratio proofs are inapplicable. We develop new proof techniques and establish that our algorithm achieves a ratio of (4-2/Bm) , where B and m denote machine capacities and numbers, respectively. Furthermore, by exploiting the relationship between maximum job demand and average processing capacity per processor, we refine the algorithm and achieve an improved ratio of (4-4/Bm) while preserving O(n log n) runtime complexity. In addition, we fine-tune our algorithm to achieve a ratio of (3-2/Bm) when jobs are with identical release times.
Sweep coverage with Unmanned Aerial Vehicles (UAVs) can be used in various applications, such as disaster management, data collection, and regulated surveillance. However, in real-world scenarios, targets are often obscured by obstacles or other targets, resulting in a limited visible region. Furthermore, in some scenarios, such as surveillance and disaster management, UAVs have to cover the targets within a certain time since the information of targets is time-sensitive. In this paper, we address the Time-Sensitive Target Coverage Under Visibility Constraints with UAVs (TCVCU) problem, where the targets are time-sensitive and the visible region of each target is a series of line segments. We propose VLSS algorithm to determine the minimum number of required UAVs for the TCVCU problem. Experiment results show that the algorithm can reduce the required UAVs by about 15
For the aircraft routing and crew pairing problems, a sequential approach is usually used to solve they. When solving the crew pairing problem, the impact of aircraft routing problem is often neglected so that these two problems are independent. This approach reduces the complexity of the solution process, but it may obtain a suboptimal solution. In this paper, we consider an integrated aircraft routing and crew pairing problem. We propose an integrated model that integrates the aircraft routing and crew pairing problems. We propose a solution algorithm based on a heuristic alternating Lagrangian decomposition to address coupling constraint of the integrated model. The solution algorithm iterates between the first Lagrangian subproblem about aircraft routing and the second Lagrangian subproblem about crew pairing. These two Lagrangian subproblems are solved by a branch-and-price algorithm. In the branch-and-price algorithm, we present a heuristic branching strategy. The computational experiments are conducted on several real-world data sets.
Shortest Common Supersequence (SCS) is a well-known problem in string algorithms and related applications. For multiple input sequences the problem is NP-hard and for a fixed number of d input sequences it can be solved in O(n^d) time, where n is the maximum length of the input sequences. In this paper, we consider designing parameterized algorithms or proving the non-existence of these algorithms for SCS and several of its variants.
This paper investigates the Bi-objective Spanning Tree Problem (BSTP), an extension of the fundamental Minimum Spanning Tree problem with diverse applications. In this problem, each edge of a given graph is associated with two distinct weights, and the objective is to find a spanning tree that simultaneously minimizes both total weights. As a specific case of bi-objective optimization, we may be interested in enumerating all the Pareto-optimal solutions of the BSTP. However, as the BSTP has been shown to be NP-hard, all the existing approaches to enumerate efficient solutions cannot achieve a polynomial CPU time. In this paper, we propose a novel approach for finding preferred efficient solutions where the worst of the ratios of the two objective values to their maximum possible values, respectively, is minimized. This approach is based on the Kalai-Smorodinsky (KS) solution, a well-known concept in cooperative game theory. More precisely, we consider an extension of the KS solution concept to the non-convex case that can be found by optimizing convex combinations of two objectives. We first characterize the properties of such KS solution(s) for the BSTP. Next, we present a weakly polynomial-time algorithm for finding the KS solution(s) for the BSTP. Finally, we showcase the computational results in some instances and discuss the results. Beyond the particular case of the BSTP, this paper offers an efficient and explainable approach for solving bi-objective combinatorial optimization problems.
