We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

/pdf/a-note-on-two-problems-in-connexion-with-graphs-4btl91o8g9.pdf

A note on two problems in connexion with graphs

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

/pdf/community-detection-in-graphs-34rogm3r6p.pdf

Community detection in graphs

(1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.

https://link.springer.com/content/pdf/10.1057%2Fjors.1977.45.pdf

Graph theory with applications

Many tasks in computer vision involve assigning a label (such as disparity) to every pixel. A common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist, e.g., at object boundaries. These tasks are naturally stated in terms of energy minimization. The authors consider a wide class of energies with various smoothness constraints. Global minimization of these energy functions is NP-hard even in the simplest discontinuity-preserving case. Therefore, our focus is on efficient approximation algorithms. We present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves, namely expansion moves and swap moves. These moves can simultaneously change the labels of arbitrarily large sets of pixels. In contrast, many standard algorithms (including simulated annealing) use small moves where only one pixel changes its label at a time. Our expansion algorithm finds a labeling within a known factor of the global minimum, while our swap algorithm handles more general energy functions. Both of these algorithms allow important cases of discontinuity preserving energies. We experimentally demonstrate the effectiveness of our approach for image restoration, stereo and motion. On real data with ground truth, we achieve 98 percent accuracy.

/pdf/fast-approximate-energy-minimization-via-graph-cuts-3usqtddnkx.pdf

Fast approximate energy minimization via graph cuts

Flows in Networks.

Introduction. The problem discussed in this paper was formulated by T. Harris as follows: “Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.”

http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/FordFulkerson1956.pdf

Maximal Flow Through a Network

: The labeling algorithm for the solution of maximal network flow problems and its application to various problems of the transportation type are discussed.

Network flow theory

A network, in which two integers tij the traversal time and cij the capacity are associated with each arc PiPj, is considered with respect to the following question. What is the maximal amount of goods that can be transported from one node to another in a given number T of time periods, and how does one ship in order to achieve this maximum? A computationally efficient algorithm for solving this dynamic linear-programming problem is presented. The algorithm has the following features a The only arithmetic operations required are addition and subtraction b In solving for a given time period T, optimal solutions for all lesser time periods are a by-product c The constructed optimal solution for a given T is presented as a relatively small number of activities chain-flows which are repeated over and over until the end of the T periods. Hence, in particular, hold-overs at intermediate nodes are not required d Arcs which serve as bottlenecks for the flow are singled out, as well as the time periods in which they act as such e In solving the problem for successive values of T, stabilization on a set of chain-flows seec above eventually occurs, and an a priori bound on when stabilization occurs can be established. The fact that there exist solutions to this problem which have the simple form described in c is remarkable, since other dynamic linear-programming problems that have been studied do not enjoy this property.

Constructing Maximal Dynamic Flows from Static Flows

(This article originally appeared in Management Science, October 1958, Volume 5, Number 1, pp 97-101, published by The Institute of Management Sciences)

A simplex computation for an arc-chain formulation of the maximal multi-commodity network flow problem is proposed Since the number of variables in this formulation is too large to be dealt with explicitly, the computation treats non-basic variables implicitly by replacing the usual method of determining a vector to enter the basis with several applications of a combinatorial algorithm for finding a shortest chain joining a pair of points in a network

L. R. Ford

Papers

Flows in Networks.

Maximal Flow Through a Network

Network flow theory

Constructing Maximal Dynamic Flows from Static Flows

A Suggested Computation for Maximal Multi-Commodity Network Flows