Grid Graphs Research Articles

Scheduling to reduce close contacts: resolvable grid graph decomposition and packing

Scheduling to reduce close contacts: resolvable grid graph decomposition and packing

On the P3-Coloring of Bipartite Graphs

The advancement in coloring schemes of graphs is expanding over time to solve emerging problems. Recently, a new form of coloring, namely P3-coloring, was introduced. A simple graph is called a P3-colorable graph if its vertices can be colored so that all the vertices in each P3 path of the graph have different colors; this is called the P3-coloring of the graph. The minimum number of colors required to form a P3-coloring of a graph is called the P3-chromatic number of the graph. The aim of this article is to determine the P3-chromatic number of different well-known classes of bipartite graphs such as complete bipartite graphs, tree graphs, grid graphs, and some special types of bipartite graphs. Moreover, we have also presented some algorithms to produce a P3-coloring of these classes with a minimum number of colors required.

Variety of mutual-visibility problems in graphs

If X is a subset of vertices of a graph G, then vertices u and v are X-visible if there exists a shortest u,v-path P such that V(P)∩X⊆{u,v}. If each two vertices from X are X-visible, then X is a mutual-visibility set. The mutual-visibility number of G is the cardinality of a largest mutual-visibility set of G and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be X-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.

Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II: Numerical applications

In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain Ω⊂Rd, d≥1. When Ω=[0,1], such graphs include the standard Toeplitz graphs and, for Ω=[0,1]d, the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences, Finite Elements, and Isogeometric Analysis. Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.

A CLOSER LOOK AT A PATH DOMINATION NUMBER IN GRID GRAPHS

This article exposes the combinatorial formula that determines the pathdomination number in a grid graph and discusses some of its properties. Seven propertiesare derived regarding the path domination number of grid graphs. Furthermore, some additional properties as direct consequences of the derived main properties are alsodiscussed.

Insights into an analytical simulation of a natural convection flow controlled by Arrhenius kinetics in a micro-channel

The focus of this paper is the investigation of an Arrhenius-driven chemical reaction in an upstanding micro-channel over an imposed transverse magnetic field with fully developed constant free convection flow. Subject to suitable boundary conditions, the temperature and velocity equations are resolved in non-dimensional form employing the homotopy perturbation method (HPM). the fundamental flow behaviors of temperature, velocity, and volumetric flow are explored as a consequence of regulating characteristics such as fluid-wall interaction parameter, rarefaction parameter, chemical reaction parameters, wall-ambient temperature difference ratio, and Hartman number. The findings are carefully investigated and graphically represented in several mesh grid graphs. It was established that increasing the values of the rarefaction parameters and chemical reaction results in an upsurge in the fluid velocity and volume flow rate, respectively, whereas increasing the Hartman number results in observable flow retardation. Additionally, when the chemical reactant parameter is ignored, the numerical comparison is in excellent agreement with the previously published results.

BiLGAT: Bidirectional lattice graph attention network for chinese short text classification

BiLGAT: Bidirectional lattice graph attention network for chinese short text classification

AlphaSnake: Policy Iteration on a Nondeterministic NP-Hard Markov Decision Process (Student Abstract)

Reinforcement learning has been used to approach well-known NP-hard combinatorial problems in graph theory. Among these, Hamiltonian cycle problems are exceptionally difficult to analyze, even when restricted to individual instances of structurally complex graphs. In this paper, we use Monte Carlo Tree Search (MCTS), the search algorithm behind many state-of-the-art reinforcement learning algorithms such as AlphaZero, to create autonomous agents that learn to play the game of Snake, a game centered on properties of Hamiltonian cycles on grid graphs. The game of Snake can be formulated as a single-player discounted Markov Decision Process (MDP), where the agent must behave optimally in a stochastic environment. Determining the optimal policy for Snake, defined as the policy that maximizes the probability of winning -- or win rate -- with higher priority and minimizes the expected number of time steps to win with lower priority, is conjectured to be NP-hard. Performance-wise, compared to prior work in the Snake game, our algorithm is the first to achieve a win rate over 0.5 (a uniform random policy achieves a win rate < 2.57 x 10^{-15}), demonstrating the versatility of AlphaZero in tackling NP-hard problems.

On the size-Ramsey number of grids

Abstract We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$ , improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.

TransPath: Learning Heuristics for Grid-Based Pathfinding via Transformers

Heuristic search algorithms, e.g. A*, are the commonly used tools for pathfinding on grids, i.e. graphs of regular structure that are widely employed to represent environments in robotics, video games, etc. Instance-independent heuristics for grid graphs, e.g. Manhattan distance, do not take the obstacles into account, and thus the search led by such heuristics performs poorly in obstacle-rich environments. To this end, we suggest learning the instance-dependent heuristic proxies that are supposed to notably increase the efficiency of the search. The first heuristic proxy we suggest to learn is the correction factor, i.e. the ratio between the instance-independent cost-to-go estimate and the perfect one (computed offline at the training phase). Unlike learning the absolute values of the cost-to-go heuristic function, which was known before, learning the correction factor utilizes the knowledge of the instance-independent heuristic. The second heuristic proxy is the path probability, which indicates how likely the grid cell is lying on the shortest path. This heuristic can be employed in the Focal Search framework as the secondary heuristic, allowing us to preserve the guarantees on the bounded sub-optimality of the solution. We learn both suggested heuristics in a supervised fashion with the state-of-the-art neural networks containing attention blocks (transformers). We conduct a thorough empirical evaluation on a comprehensive dataset of planning tasks, showing that the suggested techniques i) reduce the computational effort of the A* up to a factor of 4x while producing the solutions, whose costs exceed those of the optimal solutions by less than 0.3% on average; ii) outperform the competitors, which include the conventional techniques from the heuristic search, i.e. weighted A*, as well as the state-of-the-art learnable planners. The project web-page is: https://airi-institute.github.io/TransPath/.

Optimal planning of Microgrids using portfolio optimization with considering uncertainty

In this paper, the candidate points for distribution and substation transformers, low and medium voltage feeders, and the Distributed Energy Resources (DERs) are considered as a construction stock portfolio. The main goal of this research is to present a design method with the availability of smart grid infrastructure, which a main network is divided into several Microgrids (MGs). The design is done from the point of view of long-term operation (30 years), and all technical and economic constraints are taken into account for the optimal design of the MG. The most optimal portfolio is selected using the Imperialistic Competitive Optimization (ICA) algorithm and constituted the network design. Moreover, the uncertainty of load values is considered and the costs deviance as well as the expected value are obtained. The simulation is performed on a green field network with known geographical characteristics, and the load information is obtained using the regional load forecasting strategy. Using random values generated for the decision variable of substations, some candidate locations of DTs are selected. The low voltage feeders are connected to the not-selected substations. In the next step, the loads are considered to the selected substations based on the load allocation algorithm. Next, the assigned load points are classified into various groups by the K-means clustering algorithm. With applying the Prim method, the minimum spanning tree, with the grid graph, is calculated. The resilience of the designed grid is examined in a robust manner and in the presence of MGs. In order to study the resilience, the considered MG is divided into 4 zones based on the wind speed. The studied network is divided into 4 states as main network without dividing to several MGs, with 20 % more robustness, with 20 % less robustness, and with several MGs. The simulations are implemented with MATLAB in 11 scenarios that 3 scenarios include the controllable DERs and the remained scenarios are rendered for various uncertainty factors of loads related to the first scenario. The results show that the total cost of MG and operation cost of DER units in the third scenario are decreased about 16 % and 18 %, respectively. The results also show that the network with 20 % more robustness has better results and lower outage of loads in comparison with other states. Moreover, the connected MGs outperformed in comparison with the wide network.

Constructive–destructive heuristics for the Safe Set Problem

The Weighted Safe Set Problem aims to determine in an undirected graph a subset of vertices such that the weights of the connected components they induce exceed the weights of the adjacent components induced by the complementary subset. We tackle the problem with various approaches based on randomised constructive and destructive procedures, comparing them with each other and with the only other existing heuristic approach, that is a randomised destructive heuristic. The experiments concern all the available benchmark instances (random graphs up to 60 vertices), but also new benchmark instances with larger size and different topologies, namely random, small-world, regular, planar, grid and real-world graphs up to 300 vertices. Dense random graphs, in fact, appear to become progressively easier to solve as their size grows. On the other instances, the best performance is obtained by combining a randomised constructive procedure, a deterministic destructive procedure, and delayed termination conditions that allow a deeper exploration of the feasible region.

Parallel Strong Connectivity Based on Faster Reachability

Computing strongly connected components (SCC) is among the most fundamental problems in graph analytics. Given the large size of today's real-world graphs, parallel SCC implementation is increasingly important. SCC is challenging in the parallel setting and is particularly hard on large-diameter graphs. Many existing parallel SCC implementations can be even slower than Tarjan's sequential algorithm on large-diameter graphs. To tackle this challenge, we propose an efficient parallel SCC implementation using a new parallel reachability approach. Our solution is based on a novel idea referred to as vertical granularity control (VGC). It breaks the synchronization barriers to increase parallelism and hide scheduling overhead. To use VGC in our SCC algorithm, we also design an efficient data structure called the parallel hash bag. It uses parallel dynamic resizing to avoid redundant work in maintaining frontiers (vertices processed in a round). We implement the parallel SCC algorithm by Blelloch et al. (J. ACM, 2020) using our new parallel reachability approach. We compare our implementation to the state-of-the-art systems, including GBBS, iSpan, Multi-step, and our highly optimized Tarjan's (sequential) algorithm, on 18 graphs, including social, web, k-NN, and lattice graphs. On a machine with 96 cores, our implementation is the fastest on 16 out of 18 graphs. On average (geometric means) over all graphs, our SCC is 6.0× faster than the best previous parallel code (GBBS), 12.8× faster than Tarjan's sequential algorithms, and 2.7× faster than the best existing implementation on each graph. We believe that our techniques are of independent interest. We also apply our parallel hash bag and VGC scheme to other graph problems, including connectivity and least-element lists (LE-lists). Our implementations improve the performance of the state-of-the-art parallel implementations for these two problems.

TOTAL EDGE IRREGULAR LABELING FOR TRIANGULAR GRID GRAPHS AND RELATED GRAPHS

Let be a graph with and are the set of its vertices and edges, respectively. Total edge irregular -labeling on is a map from to satisfies for any two distinct edges have distinct weights. The minimum for which the satisfies the labeling is spoken as its strength of total edge irregular labeling, represented by . In this paper, we discuss the tes of triangular grid graphs, its spanning subgraphs, and Sierpiński gasket graphs.

An Algorithm for the Numbers of Homomorphisms from Paths to Rectangular Grid Graphs

Let G and H be graphs. A mapping f from the vertices of G to the vertices of H is known as a homomorphism from G to H if, for every pair of adjacent vertices x and y in G, the vertices f(x) and f(y) are adjacent in H. A rectangular grid graph is the Cartesian product of two path graphs. In this paper, we provide a formula to determine the number of homomorphisms from paths to rectangular grid graphs. This formula gives the solution to the problem concerning the number of walks in the rectangular grid graphs.

Reconfiguration of Hamiltonian Cycles in Rectangular Grid Graphs

We study reconfiguration of simple Hamiltonian cycles in a rectangular grid graph [Formula: see text], where the Hamiltonian cycle in each step of the reconfiguration connects every internal vertex of [Formula: see text] to a boundary vertex by a single straight line segment. We introduce two operations, flip and transpose, which are local to the grid. We show that any simple cycle of [Formula: see text] can be reconfigured to any other simple cycle of [Formula: see text] using [Formula: see text] flip and transpose operations. Our result proves that the simple Hamiltonian cycle graph [Formula: see text] is connected with respect to those two operations and has diameter [Formula: see text].

Distributed Multi-Target Search and Surveillance Mission Planning for Unmanned Aerial Vehicles in Uncertain Environments

In this paper, a distributed, autonomous, cooperative mission-planning (DACMP) approach was proposed to focus on the problem of the real-time cooperative searching and surveillance of multiple unmanned aerial vehicles (multi-UAVs) with threats in uncertain and highly dynamic environments. To deal with this problem, a time-varying probabilistic grid graph was designed to represent the perception of a target based on its a priori dynamics. A heuristic search strategy based on pyramidal maps was also proposed. Using map information at different scales makes it easier to integrate local and global information, thereby improving the search capability of UAVs, which has not been previously considered. Moreover, we proposed an adaptive distributed task assignment method for cooperative search and surveillance tasks by considering the UAV motion environment as a potential field and modeling the effects of uncertain maps and targets on candidate solutions through potential field values. The results highlight the advantages of search task execution efficiency. In addition, simulations of different scenarios show that the proposed approach can provide a feasible solution for multiple UAVs in different situations and is flexible and stable in time-sensitive environments.

Faster goal-oriented shortest path search for bulk and incremental detailed routing

We develop new algorithmic techniques for VLSI detailed routing. First, we improve the goal-oriented version of Dijkstra’s algorithm to find shortest paths in huge incomplete grid graphs with edge costs depending on the direction and the layer, and possibly on rectangular regions. We devise estimates of the distance to the targets that offer better trade-offs between running time and quality than previously known methods, leading to an overall speed-up. Second, we combine the advantages of the two classical detailed routing approaches—global shortest path search and track assignment with local corrections—by treating input wires (such as the output of track assignment) as reservations that can be used at a discount by the respective net. We show how to implement this new approach efficiently.

Hypergraph Ramsey numbers of cliques versus stars

AbstractLet denote the complete 3‐uniform hypergraph on vertices and the 3‐uniform hypergraph on vertices consisting of all edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off‐diagonal Ramsey number exhibits an unusual intermediate growth rate, namely, for some positive constants and . The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum such that any 2‐edge‐coloring of the Cartesian product contains either a red rectangle or a blue ?

Risk‐averse optimization and resilient network flows

AbstractWe propose an approach to constructing metrics of network resilience, where resilience is understood as the network's amenability to restoring its optimal or near‐optimal operations subsequent to unforeseen (stochastic) disruptions of its topology or operational parameters, and illustrated it on the examples of the resilient maximum network flow problem and the resilient minimum cost network problem. Specifically, the network flows in these problems are designed for resilience against unpredictable losses of network carrying capacity, and the mechanism of attaining a degree of resilience is through preallocation of resources toward (at least partial) restoration of the capacities of the arcs. The obtained formulations of resilient network flow problems possess a number of useful properties, for example, similarly to the standard network flow problems, the network flow is integral if the arc capacities, costs, and so forth, are integral. It is also shown that the proposed formulations of resilient network flow problems can be viewed as “network measures of risk”, similar in properties and behavior to convex measures of risk. Efficient decomposition algorithms have been proposed for both the resilient maximum network flow problem and the resilient minimum cost network flow problem, and a study of the network flow resilience as a function of network's structure has been conducted on networks with three types of topology: that of uniform random graphs, scale‐free graphs, and grid graphs.