An algorithm for Hierarchical Chinese postman problem using minimum spanning tree approach based on Kruskal's algorithm

A number of modeling and simulation algorithms using internal coordinates rely on hierarchical representations of molecular systems. Given the potentially complex topologies of molecular systems, though, automatically generating such hierarchical decompositions may be difficult. In this article, we present a fast general algorithm for the complete construction of a hierarchical representation of a molecular system. This two-step algorithm treats the input molecular system as a graph in which vertices represent atoms or pseudo-atoms, and edges represent covalent bonds. The first step contracts all cycles in the input graph. The second step builds an assembly tree from the reduced graph. We analyze the complexity of this algorithm and show that the first step is linear in the number of edges in the input graph, whereas the second one is linear in the number of edges in the graph without cycles, but dependent on the branching factor of the molecular graph. We demonstrate the performance of our algorithm on a set of specifically tailored difficult cases as well as on a large subset of molecular graphs extracted from the protein data bank. In particular, we experimentally show that both steps behave linearly in the number of edges in the input graph (the branching factor is fixed for the second step). Finally, we demonstrate an application of our hierarchy construction algorithm to adaptive torsion-angle molecular mechanics.

We study the task of estimating the number of edges in a graph, where the access to the graph is provided via an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n -vertex graph, using (i) polylog( n ) bipartite independent set queries or (ii) n 2/3 polylog( n ) independent set queries.

Extremal graphs for blow-ups of stars and paths

The formula for Turán number of spanning linear forests

Covering Non-uniform Hypergraphs

A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)|=k for all v, and a graph G is k-list-critical if G is not (k−1)-choosable, but every proper subgraph of G is (k−1)-choosable. The problem of determining the minimum number of edges in a k-critical graph with n vertices has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best known lower bound on the number of edges in a k-list-critical graph. In fact, our result on k-list-critical graphs is derived from a lower bound on the number of edges in a graph with Alon–Tarsi number at least k. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.

Constructing extremal triangle-free graphs using integer programming

This article describes an algorithm that solves a fully dynamic variant of the minimum spanning tree (MST) problem. The fully retroactive MST allows adding an edge to time $t$, or to obtain the current MST at time $t$. By using the square root technique and a data structure link-cut tree, it was possible to obtain an algorithm that runs each query in $O(\sqrt{m} \lg{|V(G)|})$ amortized, in which $|V(G)|$ is the number of nodes in graph $G$ and $m$ is the size of the timeline. We use a different approach to solve the MST problem instead of the standard algorithms, such as Prim or Kruskal, and this allows using the square root technique to improve the final complexity of the algorithm. Our empirical analysis shows that the proposed algorithm runs faster than re-executing the standard algorithms, and this difference only increases when the number of nodes in these graphs is larger.

In 1959 Erd\H os and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. We investigate a rainbow version of the theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$.

Minimum number of edges in graphs that are both P2- and Pi-connected

Graph neural networks (GNNs) can extract features by learning both the representation of each objects (i.e., graph nodes) as well as the relationship across different objects (i.e., the edges that connect nodes), achieving state-of-the-art performance on a wide range of graph-based tasks. Despite its strengths, utilizing these algorithms in a production environment faces several key challenges as the number of graph nodes and edges amount to several billions to hundreds of billions scale, requiring substantial storage space for training. Unfortunately, existing ML frameworks based on the in-memory processing model significantly hamper the productivity of algorithm developers as it mandates the overall working set to fit within DRAM capacity constraints. In this work, we first study state-of-the-art, large-scale GNN training algorithms. We then conduct a detailed characterization on utilizing capacity-optimized non-volatile memory solutions for storing memory-hungry GNN data, exploring the feasibility of SSDs for large-scale GNN training.

Graph neural networks (GNNs) can extract features by learning both the representation of each objects (i.e., graph nodes) and the relationship across different objects (i.e., the edges that connect nodes), achieving state-of-the-art performance in various graph-based tasks. Despite its strengths, utilizing these algorithms in a production environment faces several challenges as the number of graph nodes and edges amount to several billions to hundreds of billions scale, requiring substantial storage space for training. Unfortunately, state-of-the-art ML frameworks employ an in-memory processing model which significantly hampers the productivity of ML practitioners as it mandates the overall working set to fit within DRAM capacity. In this work, we first conduct a detailed characterization on a state-of-the-art, large-scale GNN training algorithm, GraphSAGE. Based on the characterization, we then explore the feasibility of utilizing capacity-optimized NVMe SSDs for storing memory-hungry GNN data, which enables large-scale GNN training beyond the limits of main memory size. Given the large performance gap between DRAM and SSD, however, blindly utilizing SSDs as a direct substitute for DRAM leads to significant performance loss. We therefore develop SmartSAGE, our software/hardware co-design based on an in-storage processing (ISP) architecture. Our work demonstrates that an ISP based large-scale GNN training system can achieve both high capacity storage and high performance, opening up opportunities for ML practitioners to train large GNN datasets without being hampered by the physical limitations of main memory size.

Frozen development in graph coloring

Improved solutions for the Chinese postman problem on mixed networks

Agent technology offers a large number of unique features which makes it a potential choice for implementing real world applications. Owing to their tremendous capabilities, it is regarded as a promising paradigm for future distributed computing. In this paper, we have proposed smart delivery: a cooperative multi-agent framework for optimised delivery service to transfer packets from source to destinations in randomly varying environment in the most efficient way with the help of mobile agents. We are considering packet moving problem in 2D environment consisting of packets and destinations. The task of each mobile agent is to move the packet from the source to the destination. Simulated environment is dynamically generated with varying graph size (i.e., number of nodes in graph), packet ratio and number of destinations. Packets and destinations are randomly placed in the graph and the graph is divided into societies to encourage an efficient and manageable environment for the agents to work in parallel and share expertise within and outside the society. The proposed framework uses decentralised approach and based on layered agent architecture and once initialised manages all the activities autonomously with the help of mobile agents.