Limit of the Maximum Random Permutation Set Entropy

The Random Permutation Set (RPS) is a new type of set proposed recently, which can be regarded as the generalization of evidence theory. To measure the uncertainty of RPS, the entropy of RPS and its corresponding maximum entropy have been proposed. Exploring the maximum entropy provides a possible way of understanding the physical meaning of RPS. In this paper, a new concept, the envelope of entropy function, is defined. In addition, the limit of the envelope of RPS entropy is derived and proved. Compared with the existing method, the computational complexity of the proposed method to calculate the envelope of RPS entropy decreases greatly. The result shows that when $N \to \infty$, the limit form of the envelope of the entropy of RPS converges to $e \times (N!)^2$, which is highly connected to the constant $e$ and factorial. Finally, numerical examples validate the efficiency and conciseness of the proposed envelope, which provides a new insight into the maximum entropy function.

When does the Physarum Solver Distinguish the Shortest Path from other Paths: the Transition Point and its Applications

Physarum solver, also called the physarum polycephalum inspired algorithm (PPA), is a newly developed bio-inspired algorithm that has an inherent ability to find the shortest path in a given graph. Recent research has proposed methods to develop this algorithm further by accelerating the original PPA (OPPA)'s path-finding process. However, when does the PPA ascertain that the shortest path has been found? Is there a point after which the PPA could distinguish the shortest path from other paths? By innovatively proposing the concept of the dominant path (D-Path), the exact moment, named the transition point (T-Point), when the PPA finds the shortest path can be identified. Based on the D-Path and T-Point, a newly accelerated PPA named OPPA-D using the proposed termination criterion is developed which is superior to all other baseline algorithms according to the experiments conducted in this paper. The validity and the superiority of the proposed termination criterion is also demonstrated. Furthermore, an evaluation method is proposed to provide new insights for the comparison of different accelerated OPPAs. The breakthrough of this paper lies in using D-path and T-point to terminate the OPPA. The novel termination criterion reveals the actual performance of this OPPA. This OPPA is the fastest algorithm, outperforming some so-called accelerated OPPAs. Furthermore, we explain why some existing works inappropriately claim to be accelerated algorithms is in fact a product of inappropriate termination criterion, thus giving rise to the illusion that the method is accelerated.

The Capacity Constraint Physarum Solver

Physarum polycephalum inspired algorithm (PPA), also known as the Physarum Solver, has attracted great attention. By modelling real-world problems into a graph with network flow and adopting proper equations to calculate the distance between the nodes in the graph, PPA could be used to solve system optimization problems or user equilibrium problems. However, some problems such as the maximum flow (MF) problem, minimum-cost-maximum-flow (MCMF) problem, and link-capacitated traffic assignment problem (CTAP), require the flow flowing through links to follow capacity constraints. Motivated by the lack of related PPA-based research, a novel framework, the capacitated physarum polycephalum inspired algorithm (CPPA), is proposed to allow capacity constraints toward link flow in the PPA. To prove the validity of the CPPA, we developed three applications of the CPPA, i.e., the CPPA for the MF problem (CPPA-MF), the CPPA for the MCFC problem, and the CPPA for the link-capacitated traffic assignment problem (CPPA-CTAP). In the experiments, all the applications of the CPPA solve the problems successfully. Some of them demonstrate efficiency compared to the baseline algorithms. The experimental results prove the validation of using the CPPA framework to control link flow in the PPA is valid. The CPPA is also very robust and easy to implement since it could be successfully applied in three different scenarios. The proposed method shows that: having the ability to control the maximum among flow flowing through links in the PPA, the CPPA could tackle more complex real-world problems in the future.