Problem In Graph Theory Research Articles

Inverse problems of graph theory: Graphs without triangles

Inverse problems of graph theory: Graphs without triangles

Research on the Optimal Strategy of "Crossing the Desert" Game Based on Dijkstra Algorithm

This article mainly analyzes the problem of "crossing the desert" game, and transforms the problem of cross-desert game into the shortest path problem in graph theory. Taking the consumable materials and consumable funds as the weights, using Dijkstra's algorithm, path planning, dynamic planning and recursive methods, the model is established in consideration of the weather, and the optimal strategy is given to maximize the remaining funds when the destination is reached.

Robust ENF Estimation Based on Harmonic Enhancement and Maximum Weight Clique

We present a framework for robust electric network frequency (ENF) extraction from real-world audio recordings, featuring multi-tone ENF harmonic enhancement and graph-based optimal harmonic selection. Specifically, We first extend the recently developed single-tone ENF signal enhancement method to the multi-tone scenario and propose a harmonic robust filtering algorithm (HRFA). It can respectively enhance each harmonic component without cross-component interference, thus further alleviating the effects of unwanted noise and audio content on the much weaker ENF signal. In addition, considering the fact that some harmonic components could be severely corrupted even after enhancement, disturbing rather than facilitating ENF estimation, we propose a graph-based harmonic selection algorithm (GHSA), which finds the optimal combination of harmonic components for more accurate ENF estimation. Noticeably, the harmonic selection problem is equivalently formulated as a maximum weight clique (MWC) problem in graph theory, and the Bron-Kerbosch algorithm (BKA) is adopted in the GHSA. With the enhanced and optimally selected harmonic components, both the existing maximum likelihood estimator (MLE) and weighted MLE (WMLE) are incorporated to yield the final ENF estimation results. The proposed framework is extensively evaluated using both synthetic signals and our ENF-WHU dataset consisting of $130$ real-world audio recordings, demonstrating substantially improved capability of extracting the ENF from realistically noisy observations over the existing single- and multi-tone competitors. This work further improves the applicability of the ENF as a forensic criterion in real-world situations.

A parallel biological computing algorithm to solve the vertex coloring problem with polynomial time complexity

The vertex coloring problem is a well-known combinatorial problem that requires each vertex to be assigned a corresponding color so that the colors on adjacent vertices are different, and the total number of colors used is minimized. It is a famous NP-hard problem in graph theory. As of now, there is no effective algorithm to solve it. As a kind of intelligent computing algorithm, DNA computing has the advantages of high parallelism and high storage density, so it is widely used in solving classical combinatorial optimization problems. In this paper, we propose a new DNA algorithm that uses DNA molecular operations to solve the vertex coloring problem. For a simple n-vertex graph and k different kinds of color, we appropriately use DNA strands to indicate edges and vertices. Through basic biochemical reaction operations, the solution to the problem is obtained in the O (kn2) time complexity. Our proposed DNA algorithm is a new attempt and application for solving Nondeterministic Polynomial (NP) problem, and it provides clear evidence for the ability of DNA calculations to perform such difficult computational problems in the future.

Simple heuristics for efficient parallel tensor contraction and quantum circuit simulation

Tensor networks are the main building blocks in a wide variety of computational sciences, ranging from many-body theory and quantum computing to probability and machine learning. Here we propose a parallel algorithm for the contraction of tensor networks using probabilistic graphical models. Our approach is based on the heuristic solution of the $\mu$-treewidth deletion problem in graph theory. We apply the resulting algorithm to the simulation of random quantum circuits and discuss the extensions for general tensor network contractions.

Radio Map-Based 3D Path Planning for Cellular-Connected UAV

In this paper, we study the three-dimensional (3D) path planning for a cellular-connected unmanned aerial vehicle (UAV) to minimize its flying distance from given initial to final locations, while ensuring a target link quality in terms of the expected signal-to-interference-plus-noise ratio (SINR) at the UAV receiver with each of its associated ground base stations (GBSs) during the flight. To exploit the location-dependent and spatially varying channel as well as interference over the 3D space, we propose a new radio map based path planning framework for the UAV. Specifically, we consider the channel gain map of each GBS that provides its large-scale channel gains with uniformly sampled locations on a 3D grid, which are due to static and large-size obstacles (e.g., buildings) and thus assumed to be time-invariant. Based on the channel gain maps of GBSs as well as their loading factors, we then construct an SINR map that depicts the expected SINR levels over the sampled 3D locations. By leveraging the obtained SINR map, we proceed to derive the optimal UAV path by solving an equivalent shortest path problem (SPP) in graph theory. We further propose a grid quantization approach where the grid points in the SINR map are more coarsely sampled by exploiting the spatial channel/interference correlation over neighboring grids. Then, we solve an approximate SPP over the reduced-size SINR map (graph) with reduced complexity. Numerical results show that the proposed solution can effectively minimize the flying distance/time of the UAV subject to its communication quality constraint, and a flexible trade-off between performance and complexity can be achieved by adjusting the grid quantization ratio in the SINR map. Moreover, the proposed solution significantly outperforms various benchmark schemes without fully exploiting the channel/interference spatial distribution in the network.

Alternative Branching Strategies in the Branch and Bound Algorithm by Using a k-clique covering vertex set for Maximum Clique Problems.

The Maximum clique problem (MCP) is graph theory problem that demand complete subgraf with maximum cardinality (maximum clique) in arbitrary graph. Solving MCP usually use Branch and Bound (BnB) algorithm, in this paper we will show how n + 1 color classes (where n is the difference between upper and lower bound) selected to form k-clique covering vertex set which later used for branching strategy can guarenteed finnding maximum clique.

Alternative Branching Strategies in the Branch and Bound Algorithm by Using a K-Clique Covering Vertex Set for Maximum Clique Problems

The maximum clique problem (MCP) is graph theory problem that demand complete subgraph with maximum cardinality (maximum clique) in arbitrary graph. Solving MCP usually use Branch and Bound (BnB) algorithm. In this paper, we will show how n + 1 color classes (where n is the difference between upper and lower bound) selected to form k-clique covering vertex set which later used for branching strategy can guarantee finding maximum clique.

A clustering detector with graph theory for blind detection of spatial modulation systems

This paper considers the blind detection of spatial modulation systems multiple-input multiple-output systems. In this system, we propose a clustering detection framework with graph theory that conducts signal detection without the training of channel state information. In detail, firstly, by dynamically controlling the size of each cluster, we transform the original optimization problem of the traditional K-means clustering detector into a new optimization problem. In addition, the cluster assignment subproblem of the iterative clustering algorithm for solving the new optimization problem makes it equivalent to a minimum cost flow linear network optimization problem of graph theory, which can be addressed by the breadth-first algorithm. Moreover, a novel clustering detector with the breadth-first algorithm is presented correspondingly. Numerical results show that the proposed detector is efficient in avoiding the undesired local optima and can closely approach the performance of the optimal detector.

The partition dimension of a subdivision of a homogeneous firecracker

Finding the partition dimension of a graph is one of the interesting (and uncompletely solved) problems of graph theory. For instance, the values of the partition dimensions for most kind of trees are still unknown. Although for several classes of trees such as paths, stars, caterpillars, homogeneous firecrackers and others, we do know their partition dimensions. In this paper, we determine the partition dimension of a subdivision of a particular tree, namely homogeneous firecrackers. Let G be any graph. For any positive integer k and e \in E(G), a subdivision of a graph G , denoted by S ( G ( e;k )), is the graph obtained from G by replacing an edge $e$ with a ( k +1)-path. We show that the partition dimension of S ( G ( e;k )) is equal to the partition dimension of G if G is a homogeneous firecracker. We show that the partition dimension of S ( G ( e;k )) is equal to the partition dimension of G if G is a homogeneous firecracker.

Cooperative Beam Routing for Multi-IRS Aided Communication

Intelligent reflecting surface (IRS) has been deemed as a transformative technology to achieve smart and reconfigurable environment for wireless communication. This letter studies a new IRS-aided communication system, where multiple IRSs assist in the communication between a multi-antenna base station (BS) and a remote single-antenna user by multi-hop signal reflection. Specifically, by exploiting the line-of-sight (LoS) link between nearby IRSs, a multi-hop cascaded LoS link between the BS and user is established where a set of IRSs are selected to successively reflect the BS's signal, so that the received signal power at the user is maximized. To tackle this new problem, we first present the closed-form solutions for the optimal active and cooperative passive beamforming at the BS and selected IRSs, respectively, for a given beam route. Then, we derive the end-to-end channel power, which unveils a fundamental trade-off in the optimal beam routing design between maximizing the multiplicative passive beamforming gain and minimizing the multi-reflection path loss. To reconcile this trade-off, we recast the IRS selection and beam routing problem as an equivalent shortest simple-path problem in graph theory and solve it optimally. Numerical results show significant performance gains of the proposed algorithm over benchmark schemes and also draw useful insights into the optimal beam routing design.

Minimum Cut Acceleration by Exploiting Tree-Cut Injection for Upper Bound Estimation

As a basic problem in graph theory, the minimum cut (min-cut) problem has important applications in communication related researches. There are injections from the set of all traversal trees to the set of all cuts. However, this kind of injection is completely ignored in existing methods, which may lead to the lose of acceleration opportunity. Accordingly, we propose Traversal Tree-Cut Injection based Min-Cut Acceleration algorithm (TCMA). By collecting enough cuts from different traversal trees in an undirected graph, the upper bound of min-cut value between any node pair can be calculated as the minimal value of all related cuts, which can match the accurate min-cut value with high probability. We compare TCMA with other methods using random and real-world graphs. The results show that different from other methods, TCMA is general enough to not rely on any special topology, and can achieve both high acceleration ratio and precision in various graphs, thus can be used as an effective supplement to existing methods.

Metric Learning in Graph Matching Problems

One of the important and computationally complex problems in graph theory is graph matching, i.e., finding a correspondence between the vertices of a pair of graphs. Modern rapidly developing methods are largely based on machine learning. We propose a new approach based on deep learning of a graph neuron network combining convolutional and Siamese techniques. This involves learning the metrics between graph vertices and then using these metrics for matching. Experiments on sets of real data demonstrate the usefulness of the proposed approach.

MATHEMATICAL THINKING PROCESS IN GRAPH THEORY MATERIAL IN TERMS OF STUDENT LEARNING STYLES

This research is motivated by the lack of understanding students have in working on graph theory material problems due to differences in students' thinking processes, and student learning styles when studying discrete mathematics, which requires different thinking with Boolean algebra material. Each student has a different way of thinking skills and learning style. This study uses descriptive qualitative methods so that this study has the objective to find out how the students' mathematical thinking processes in working on graph theory material problems and what obstacles are experienced by students in terms of their learning styles. The focus of research in this study, namely to analyze mathematical thinking processes on graph theory material in terms of student learning styles. The instruments in this study were researchers and also graph theory test questions that were used to view students' thought processes. The research subjects were 5 semester students, as many as 3 people with visual, auditory and kinesnetik learning styles. Then given the problem of graph theory as much as 2 questions, one question in the form of text and another in the form of images. The results show that students with visual learning styles tend to be included in students who have a conceptual thinking process, students with auditory learning styles tend to be included in students who have semi-conceptual thinking processes, while students with kinesthetic learning styles, including students who have computational thinking process . Barriers experienced by students include the lack of understanding of the basic concepts of graph theory and the trajectories and trajectories of Euler and Hamilton.

Vehicle Sequencing at Transshipment Terminals with Handover Relations

Operational planning at transshipment nodes is a wide and challenging field of research that covers a vast number of distinct relevant applications, spanning from seaport container terminals to rail terminals to cross-docks. In this work, we study the feasibility version of a fundamental synchronization problem that assigns incoming vehicles to docking resources subject to handover relations. We carry out a comprehensive analysis of computational complexity of various problem variants and establish structural connections to famous decision problems in graph theory. We further propose an exact solution algorithm for finding feasible dock assignments, if vehicles can visit the node only once and evaluate its performance in a comprehensive computational study.

The maximum number of triangles in a graph of given maximum degree

Maximizing or minimizing the number of copies of a fixed graph in a large host graph is one of the most classical topics in extremal graph theory. Indeed, one of the most famous problems in extremal graph theory, the Erdős-Rademacher problem, which can be traced back to the 1940s, asks to determine the minimum number of triangles in a graph with a given number of vertices and edges. It was conjectured that the mnimum is attained by complete multipartite graphs with all parts but one of the same size whilst the remaining part may be smaller. The problem was widely open in the regime of four or more parts until Razborov resolved the problem asymptotically in 2008 as one of the first applications of his newly developed flag algebra method. This catalyzed a line of research on the structure of extremal graphs and extensions. In particular, Reiher asymptotically solved in 2016 the conjecture of Lovász and Simonovits from the 1970s that the same graphs are also minimizers for cliques of arbitrary size. This paper deals with a problem concerning the opposite direction: _What is the maximum number of triangles in a graph with a given number $n$ of vertices and a given maximum degree $D$?_ Gan, Loh and Sudakov in 2015 conjectured that the graph maximizing the number of triangles is always a union of disjoint cliques of size $D+1$ and another clique that may be smaller, and showed that if such a graph maximizes the number of triangles, it also maximizes the number of cliques of any size $r\ge 4$. The author presents a remarkably simple and elegant argument that proves the conjecture exactly for all $n$ and $D$.

How to transform graph states using single-qubit operations: computational complexity and algorithms

Graph states are ubiquitous in quantum information with diverse applications ranging from quantum network protocols to measurement based quantum computing. Here we consider the question whether one graph (source) state can be transformed into another graph (target) state, using a specific set of quantum operations (LC + LPM + CC): single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. This question is of interest for effective routing or state preparation decisions in a quantum network or distributed quantum processor and also in the design of quantum repeater schemes and quantum error-correction codes. We first show that deciding whether a graph state |G⟩ can be transformed into another graph state |G′⟩ using LC + LPM + CC is -complete, which was previously not known. We also show that the problem remains NP-complete even if |G′⟩ is restricted to be the GHZ-state. However, we also provide efficient algorithms for two situations of practical interest. Our results make use of the insight that deciding whether a graph state |G⟩ can be transformed to another graph state |G′⟩ is equivalent to a known decision problem in graph theory, namely the problem of deciding whether a graph G′ is a vertex-minor of a graph G. The computational complexity of the vertex-minor problem was prior to this paper an open question in graph theory. We prove that the vertex-minor problem is -complete by relating it to a new decision problem on 4-regular graphs which we call the semi-ordered Eulerian tour problem.

An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops.

Graph enumeration with given constraints is an interesting problem considered to be one of the fundamental problems in graph theory, with many applications in natural sciences and engineering such as bio-informatics and computational chemistry. For any two integers and , we propose a method to count all non-isomorphic trees with n vertices, self-loops, and no multi-edges based on dynamic programming. To achieve this goal, we count the number of non-isomorphic rooted trees with n vertices, self-loops and no multi-edges, in time and space, since every tree can be uniquely viewed as a rooted tree by either regarding its unicentroid as the root, or in the case of bicentroid, by introducing a virtual vertex on the bicentroid and assuming the virtual vertex to be the root. By this result, we get a lower bound and an upper bound on the number of tree-like polymer topologies of chemical compounds with any “cycle rank”.

Elusive extremal graphs

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that `every extremal graph theory problem has a finitely forcible optimum'. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints.

Set Partition Modulation

In this paper, a novel modulation scheme called set partition modulation (SPM) is proposed. In this scheme, set partitioning and ordered subsets in the set partitions are used to form codewords. We define different SPM variants and depict a practical model for using SPM with orthogonal frequency division multiplexing (OFDM). For the OFDM-SPM schemes, different constellations are used to distinguish between different subsets in a set partition. To achieve good distance properties as well as better error performance for the OFDM-SPM codewords, we define a codebook selection problem and formulate such a problem as a clique problem in graph theory. In this regard, we propose a fast and efficient codebook selection algorithm. We analyze error and achievable rate performance of the proposed schemes and provide asymptotic results for the performance. It is shown that the proposed SPM variants are general schemes, which encompass multi-mode OFDM with index modulation (MM-OFDM-IM) and dual-mode OFDM with index modulation (DM-OFDM-IM) as special cases. It is also shown that OFDM-SPM schemes are capable of exhibiting better error performance and improved achievable rate than conventional OFDM, OFDM-IM, DM-OFDM-IM, and MM-OFDM-IM.