This paper develops for the first time an analytical approach inspired from the basic differential multiplier method (BDMM) in a framework concept involving coupled nonlinear ordinary differential equations (ODEs) for finding shortest path (SP) in dynamically reconfigurable graphs. The application of BDMM to the shortest-path problem (SPP) results into a set of coupled ODEs which do essentially describe/constitute a Recurrent Neural Network (RNN) model. This model is characterized by three fundamental parameters describing (or expressing) the following: (a) the graph topology (through the 'incidence matrix'), (b) the edge weights (dynamic external weights setting capability), and (c) the dynamic re-configurability through external input(s) of the source-destination nodes pair. The coupled ODEs constitute a universal modeling framework for SP detection. The three fundamental parameters of this framework are valid for all types of graphs (directed, undirected, connected, non-connected, complete- and non-complete) regardless of their topology, magnitude, size, etc. An overall thorough methodology and an illustrative extensive evaluation are provided. It is noticed and demonstrated that the main advantage of the concept suggested is that arc costs, the origin-destination (s-t) pair setting, and the graph topology are external commands representing the inputs (and also the coefficients/parameters) of the ODE-processor model (RNN). This enables a high flexibility and a full re-configurability of the developed shortest path problem solver concept and thereby without any re-training need. Further, this RNN based SP solver concept can handle even graphs with negative arc weights as well as graphs with nonlinear path cost models. To stress test this new RNN shortest path problem solver concept, a benchmarking is performed, whereby a detailed comparison of its performance with one of the currently best amongst the related traditional neural network based concepts (i.e. the Dynamic Neural Network (DNN) based SP solver) is performed. The superiority of the novel RNN SPP solver is convincingly demonstrated in an extensive benchmark while considering computational efficiency, robustness, convergence, flexibility, and re-configurability as performance metrics. Finally, it is demonstrated that the concept developed is applicable to solving shortest path tree (SPT) problems. A case study (SPT) is considered and various results obtained using the concept developed are compared with the results provided by several algorithms namely the semi-dynamic Dijkstra algorithms, the semi-dynamic MBallString algorithms, and the fully-dynamic MFP algorithm. The efficiency of the concept developed for solving SPT problems in both static and dynamically reconfigurable graphs networks is clearly demonstrated through multiple application examples.
['Jean Chamberlain Chedjou', 'Kyandoghere Kyamakya']
Benchmarking a recurrent neural network based efficient shortest path problem (SPP) solver concept under difficult dynamic parameter settings conditions