The Problem of Graphs Local Optimization and its Applications

Abstract

Introduction. We study the problem of constructing a locally optimal directed graph when each edge is assigned a fixed numerical weight. Due to the broad interpretation of the numerical weights of edges, the problem of finding the least-cost paths has many applications and is one of the most common and studied problems in the applied theory of algorithms. In various modifications, it can be found in algorithms for constructing optimal routes for flying and other moving objects, in pattern recognition, optimization of communication networks and integrated circuits. The property of local optimality is defined through a dynamic process that at every round assigns to each vertex the minimal value depending on its current value and sums of values stored in neighborhood vertices with weights of incoming edges. The graph is locally optimal when any vertex cannot allow further minimizations. Purpose. The purpose of the article is to investigate the problem of constructing a locally-optimal graph, establish properties of such graphs, and derive new applications. Methods. We consider the problem of finding the shortest paths in a broader context as a general computational problem of local graph optimization. This model of computations on a graph is studied. Each local vertex, regardless of the other vertices, performs the same iterative procedure of minimizing some non-negative numerical value assigned to that vertex. The operation of such minimization depends on the current value obtained in this vertex when applying the previous minimization and sums of the current numerical data located in the vertices from which the input edges lead to the selected local vertex and weights of the seedges. After several stages of such an operation, the graph always comes into a stable state: further minimization of vertices does not change the values assigned to the vertices. In this state, we call the graph locally optimal. Results. The conditions under which the graph reaches the state of local optimization are investigated. With different choices of interpretation of the function that determines the influence of neighboring vertices on the vertex minimization value, we obtain a variety of applications. For example, the gradient descent method from the local optimization state makes it easy to obtain the minimum weight (cost) paths from the given starting vertex to all vertices. If the graph is interpreted as a semantic network in the space of natural language words, it is shown how the problem of finding the minimum path among words or phrases is interpreted as solving a known problem in computational linguistics: find the word most associated with a given input set words or sentences (images). Finally, by calculating the weights of the paths in randomized (uniformly distributed) graphs, a fast-authentication protocol is proposed for the two coalition entities acting in a malicious environment. Conclusions. The local optimality is a stable constant state of the current graph marking reachable by using the dynamic minimization process. Local optimality provides a common method for solving many problems related to search engine optimization. It is shown how from solving the general problem of constructing a locally-optimal graph it follows the problem of finding the paths of least weights (cost). New applications include solving the problem of associative search in computational linguistics (creative thinking) and fast mutual authentication in coalition groups.