Graph Problems Research Articles

Solving Edges Deletion Problem of Complete Graphs

مع زيادة حجم الشبكة فان احتمالية الفشل في بعض مكوناتها (العقد او الحافات) يزداد، لذلك من الضروري ان تكون الشبكة قادرة على التواصل بين مكوناتها في حالة وجود خلل في بعض مكوناتها. ان احتمالية حدوث عطل في التوصيلات بين محطات الشبكة أكبر من احتمالية حدوث عطل في المحطات نفسها بسبب طبيعة عمل العديد من الشبكات. يمكن دراسة فعالية وموثوقية الشبكة باستخدام مجموعة من تقنيات نظرية البيان. ان التواصل في الشبكة هو ما يحدد مدى صلابتها (موثوقيتها) حيث ان فعالية الشبكة تتأثر عند حدوث تلف في أحد مكوناتها. يستخدم قطر الرسم البياني لقياس كفاءة عمل الشبكة من خلال قياس اقصى تأخير في استلام إشارة (رسالة) معينة او حدوث خلل في الاستلام. في هذا البحث ندرس مدى زيادة القطر للرسم البياني الناتج بعد حذف من الحافات من الرسم البياني . وايجاد أكبر قطر للرسم بياني الذي يحتوي من الرؤوس ونحصل عليه بعد حذف من الحافات من وتحديد بعد حساب

A heuristic method for solving the Steiner tree problem in graphs using network centralities

We propose a heuristic method of using network centralities for constructing small-weight Steiner trees in this paper. The Steiner tree problem in graphs is one of the practical NP-hard combinatorial optimization problems. Given a graph and a set of vertices called terminals in the graph, the objective of the Steiner tree problem in graphs is to find a minimum weight Steiner tree that is a tree containing all the terminals. Conventional construction methods make a Steiner tree based on the shortest paths between terminals. If these shortest paths are overlapped as much as possible, we can obtain a small-weight Steiner tree. Therefore, we proposed to use network centralities to distinguish which edges should be included to make a small-weight Steiner tree. Experimental results revealed that using the vertex or the edge betweenness centralities contributes to making small-weight Steiner trees.

Exploring Clique Transversal Problems for d-degenerate Graphs with Fixed d: From Polynomial-Time Solvability to Parameterized Complexity

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and {b}-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where d≥2, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable.

Energy landscapes for the quantum approximate optimization algorithm

Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary noisy intermediate-scale quantum (NISQ) era. The quantum approximate optimization algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of L quantum circuit layers within a corresponding Trotterized ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here we employ basin-hopping global optimization methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyze QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualizations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organization, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimization. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima. Published by the American Physical Society 2024

Quantum-Inspired Classical Algorithm for Graph Problems by Gaussian Boson Sampling

We present a quantum-inspired classical algorithm that can be used for graph-theoretical problems, such as finding the densest k subgraph and finding the maximum weight clique, which are proposed as applications of a Gaussian boson sampler. The main observation from Gaussian boson samplers is that a given graph’s adjacency matrix to be encoded in a Gaussian boson sampler is non-negative and that computing the output probability of Gaussian boson sampling restricted to a non-negative adjacency matrix is thought to be strictly easier than general cases. We first provide how to program a given graph problem into our efficient classical algorithm. We then numerically compare the performance of ideal and lossy Gaussian boson samplers, our quantum-inspired classical sampler, and the uniform sampler for finding the densest k subgraph and finding the maximum weight clique and show that the advantage from Gaussian boson samplers is not significant in general. We finally discuss the potential advantage of a Gaussian boson sampler over the proposed quantum-inspired classical sampler. Published by the American Physical Society 2024

The maximum 3-star packing problem in claw-free cubic graphs

The maximum 3-star packing problem in claw-free cubic graphs

A new temporal interpretation of cluster editing

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of P3; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.

On the Zarankiewicz Problem for Graphs with Bounded VC-Dimension

On the Zarankiewicz Problem for Graphs with Bounded VC-Dimension

Edges deletion problem of hypercube graphs for some n

Numerous graph-theoretic methods can be used to investigate the efficiency and reliability of networks. The reliability of a network is determined based on its connectivity. At the very least, the system must continue to function in the event of part of its component (node or link) failures. An ideal system remains reliable even when errors occur. Diameter is a measure of network efficiency used to study the effects of link failures in networks. In this paper, we look into the eliminating edges affect to the hypercube graph’s diameter (call it by [Formula: see text]) and calculate the maximum diameter that is denoted by [Formula: see text] after deleting [Formula: see text] edges from hypercube graphs [Formula: see text] for some n.

Ant colony optimization using two-dimensional pheromone for single-objective transport problems

One of the acclaimed algorithms that is used to solve combinatorial graph problems is ant colony optimization (ACO). In this article, we focus on a novel extended model of the pheromone that is responsible for storing collective knowledge. The presented two-dimensional pheromone is able to accommodate more information that is extracted from feasible solutions that can be used to improve the search of a solution space. The idea is positively evaluated on TSP and VRP problems, achieving better results as compared to the original algorithm. Since it is a universal concept, it can be applied to any single-objective problem that is solvable by ACO.

THE DEVELOPMENT OF MATHEMATICS LEARNING ACHIEVEMENT ON THE TOPIC OF SOLVING QUADRATIC FUNCTION GRAPHS OF MATHAYOM 3 STUDENTS BY USING THE GEOGEBRA PROGRAM TOGETHER WITH POLYA'S PROBLEM SOLVING PROCESS

This research used GeoGebra program and organized learning management activities in accordance with Polya’s problem-solving process to develop mathematical learning achievement by solving problems in quadratic function graphs. With the help of GeoGebra program and step-by-step practice of problem-solving techniques using Polya’s problem-solving process, the aforementioned learning management system increases students' enthusiasm and interest in learning about graphs. Consequently, it encourages students to comprehend the knowledge of quadratic function graphs in mathematics efficiently, gain a more thorough comprehension of them, and develop their skills in solving problems. The target group used in the research is 14 Mathayom 3 students studying in the first semester of the academic year 2023 at Chumchonyodkaengsongkhro School, Namon District, Kalasin Province, which obtained purposive random sampling. The tools used in the research were lesson plans using GeoGebra with Polya’s problem-solving process, learning achievement tests, and a satisfaction questionnaire. The statistics used in the data analysis include mean (x̄), standard deviation (SD), percentage, and t-test statistics. The results of the research found that: 1) the results of the efficiency of the lesson plans using GeoGebra program with Polya’s problem solving process are equal to 73.75/74.64, which is higher than with the specified criteria, 2) The post-test scores’ learning achievement using GeoGebra with Polya's problem-solving process was higher than the pre-test scores using GeoGebra with Polya's problem-solving process, with statistical significance at the .05 level, and 3) the satisfaction of students with learning management using GeoGebra program with Polya’s problem-solving process is at a high level with an average of 4.31 and a standard deviation of 0.81.

Compiling Recurrences over Dense and Sparse Arrays

We present a framework for compiling recurrence equations into native code. In our framework, users specify a system of recurrences, the types of data structures that store inputs and outputs, and scheduling commands for optimization. Our compiler then lowers these specifications into native code that respects the dependencies in the recurrence equations. Our compiler can generate code over both sparse and dense data structures, and determines if the recurrence system is solvable with the provided scheduling primitives. We evaluate the performance and correctness of the generated code on several recurrences, from domains as diverse as dense and sparse matrix solvers, dynamic programming, graph problems, and sparse tensor algebra. We demonstrate that the generated code has competitive performance to hand-optimized implementations in libraries. However, these handwritten libraries target specific recurrences, specific data structures, and specific optimizations. Our system, on the other hand, automatically generates implementations from recurrences, data formats, and schedules, giving our system more generality than library approaches.

RFG-TVIU: robust factor graph for tightly coupled vision/IMU/UWB integration.

High precision navigation and positioning technology, as a fundamental function, is gradually occupying an indispensable position in the various fields. However, a single sensor cannot meet the navigation requirements in different scenarios. This paper proposes a "plug and play" Vision/IMU/UWB multi-sensor tightly-coupled system based on factor graph. The difference from traditional UWB-based tightly-coupled models is that the Vision/IMU/UWB tightly-coupled model in this study uses UWB base station coordinates as parameters for real-time estimation without pre-calibrating UWB base stations. Aiming at the dynamic change of sensor availability in multi-sensor integrated navigation system and the serious problem of traditional factor graph in the weight distribution of observation information, this study proposes an adaptive robust factor graph model. Based on redundant measurement information, we propose a novel adaptive estimation model for UWB ranging covariance, which does not rely on prior information of the system and can adaptively estimate real-time covariance changes of UWB ranging. The algorithm proposed in this study was extensively tested in real-world scenarios, and the results show that the proposed system is superior to the most advanced combination method in all cases. Compared with the visual-inertial odometer based on the factor graph (FG-VIO), the RMSE is improved by 62.83 and 64.26% in scene 1 and 82.15, 70.32, and 75.29% in scene 2 (non-line-of-sight environment).

OptFlowCam: A 3D‐Image‐Flow‐Based Metric in Camera Space for Camera Paths in Scenes with Extreme Scale Variations

AbstractInterpolation between camera positions is a standard problem in computer graphics and can be considered the foundation of camera path planning. As the basis for a new interpolation method, we introduce a new Riemannian metric in camera space, which measures the 3D image flow under a small movement of the camera. Building on this, we define a linear interpolation between two cameras as shortest geodesic in camera space, for which we provide a closed‐form solution after a mild simplification of the metric. Furthermore, we propose a geodesic Catmull‐Rom interpolant for keyframe camera animation. We compare our approach with several standard camera interpolation methods and obtain consistently better camera paths especially for cameras with extremely varying scales.

Efficient 2-Distance Coloring Method for Maximum and Average Sparse Graphs in Resource Allocation Optimization of Microgrids

In microgrids management, optimizing resource allocation and enhancing efficiency are critical challenges. This paper introduces the 2-distance coloring technique as a novel approach to address these issues by focusing on the coloring problem of sparse graphs. Through rigorous theoretical derivation, we establish the minimum number of colors required for 2-distance coloring in sparse graph scenarios. Specifically, we demonstrate that for a graph G with an average maximum degree less than 2+17/20 and a maximum degree of 6, it can be list 2-distance colored using no more than 11 colors. This means that, given any set of 11 colors for each vertex, a valid 2-distance coloring can be achieved, ensuring no two adjacent vertices share the same color. This finding is pivotal, as it sets a precedent for the number of resources needed to efficiently manage and allocate resources in microgrids systems through 2-distance coloring. The application of this technique in microgrids promises to revolutionize resource distribution, reducing overlap and maximizing efficiency across the network.

Sufficient control of complex networks

In this paper, we propose to study sufficient control of complex networks, which is to control a sufficiently large portion of the network, where only the quantity of controllable nodes matters. To the best of our knowledge, this is the first time that such a problem is investigated. We prove that the sufficient controllability problem can be converted into a minimum-cost flow problem, for which an algorithm with polynomial complexity can be devised. Further, we study the problem of minimum-cost sufficient control, which is to drive a sufficiently large subset of the network nodes to any predefined state with the minimum cost using a given number of controllers. The problem is NP-hard. We propose an “extended L0-norm-constraint-based Projected Gradient Method” (eLPGM) algorithm, which achieves suboptimal solutions for the problems at small or medium sizes. To tackle the large-scale problems, we propose to convert the control problem into a graph problem and devise an efficient low-complexity “Evenly Divided Control Paths” (EDCP) algorithm to tackle the graph problem. Simulation results on both synthetic and real-life networks are provided, demonstrating the satisfactory performance of the proposed methods.

Application of graphical problems in chemistry education

Application of graphical problems in chemistry education

Solving optimization problems with local light-shift encoding on Rydberg quantum annealers

We provide a non-unit-disk framework to solve combinatorial optimization problems such as maximum cut and maximum independent set on a Rydberg quantum annealer. Our setup consists of a many-body interacting Rydberg system where locally controllable light shifts are applied to individual qubits in order to map the graph problem onto the Ising spin model. Exploiting the flexibility that optical tweezers offer in terms of spatial arrangement, our numerical simulations implement the local-detuning protocol while globally driving the Rydberg annealer to the desired many-body ground state, which is also the solution to the optimization problem. Using optimal control methods, these solutions are obtained for prototype graphs with varying sizes at timescales well within the system lifetime and with approximation ratios close to one. The nonblockade approach facilitates the encoding of graph problems with specific topologies that can be realized in two-dimensional Rydberg configurations and is applicable to both unweighted as well as weighted graphs. A comparative analysis with fast simulated annealing is provided which highlights the advantages of our scheme in terms of system size, hardness of the graph, and the number of iterations required to converge to the solution. Published by the American Physical Society 2024

Bipartite (P6,C6)-Free Graphs: Recognition and Optimization Problems

The canonical decomposition of a bipartite graph is a new decomposition method that involves three operators: parallel, series, and K⨁ S. The class of weak-bisplit graphs is the class of totally decomposable graphs with respect to these operators, and the class of bicographs is the class of totally decomposable graphs with respect to parallel and series operators. We prove in this paper that the class of bipartite (P6,C6)-free graphs is the class of bipartite graphs that are totally decomposable with respect to parallel and K⨁S operators. We present a linear time recognition algorithm for (P6,C6)-free graphs that is symmetrical to the linear recognition algorithms of weak-bisplit graphs and star1,2,3-free bipartite graphs. As a result of this algorithm, we present efficient solutions in this class of graphs for two optimization graph problems: the maximum balanced biclique problem and the maximum independent set problem.

Effect of constraint relaxation on the minimum vertex cover problem in random graphs.

A statistical-mechanical study of the effect of constraint relaxation on the minimum vertex cover problem in Erdős-Rényi random graphs is presented. Using a penalty-method formulation for constraint relaxation, typical properties of solutions, including infeasible solutions that violate the constraints, are analyzed by means of the replica method and cavity method. The problem involves a competition between reducing the number of vertices to be covered and satisfying the edge constraints. The analysis under the replica-symmetric (RS) ansatz clarifies that the competition leads to degeneracies in the vertex and edge states, which determine the quantitative properties of the system, such as the cover and penalty ratios. A precise analysis of these effects improves the accuracy of RS approximation for the minimum cover ratio in the replica symmetry-breaking (RSB) region. Furthermore, the analysis based on the RS cavity method indicates that the RS/RSB boundary of the ground states with respect to the mean degree of the graphs is expanded, and the critical temperature is lowered by constraint relaxation.