The sensor coverage problem is a typical combinatorial optimization problem, which is favored by researchers due to its wide practical applications. This article studies the sensor coverage problem, where each sensor has communication, sensing, and computing capabilities to collaboratively cover a certain area. First, we treat each sensor as a player, and establish a potential game for the sensor coverage problem, where the global objective is presented by the potential objective, and individual's utility is designed as a wonderful life utility. Second, we prove that under the established potential game, a Nash equilibrium can guarantee at least 50% optimality. Third, we design a synchronous game learning (SGL) distributed algorithm, where each player has a memory length$m$. Fourth, we prove that our designed SGL algorithm can guarantee that strategies of all sensors converge to a Nash equilibrium, and analyze its complexity. Finally, we demonstrate the effectiveness and superiority of our designed SGL algorithm by comparing with the existing representative optimization algorithms via numerical simulations. In addition, we also find that by employing our designed SGL algorithm, a tradeoff between solution quality and runtime could be achieved via adjusting players' memory length.

As a typical combinatorial optimization problem, the 3-path vertex cover problem has wide applications in practice. To solve the 3-path vertex cover problem from the perspective of distributed optimization, we treat each vertex as an agent (i.e., player) with computation, and decision-making capabilities. First, we establish a 3-player symmetric game model to describe the 3-path vertex cover problem, and design the corresponding cost function for each player. Then, we prove that under the established game model, strict Nash equilibriums (SNEs) act as the basis of the connection between 3-path vertex cover states and minimum 3-path vertex cover states. Next, we propose a novel memory-based synchronous learning (MSL) algorithm, where the initial profile strategy generation of players relies on the designed degree preference rule, and each player has a memory length for recording strategies and independently update their strategies concurrently based on the accessed local information. After that, we prove that our proposed MSL algorithm can guarantee that any strategy profile converges to an SNE, and provide a theoretical analysis of the algorithm’s complexity. Finally, we present numerous numerical simulations to demonstrate the performance of our proposed algorithm on various networks. Moreover, we find that increasing the memory length and adopting the degree preference initialization can yield a better SNE.

A stochastic Boolean network (SBN) emerges as a more realistic model for gene regulatory networks than a deterministic Boolean network (BN). In order to reduce output sampling while ensuring a given estimation accuracy, this article proposes an event-triggered sampling strategy for the state estimation of SBNs. Under this strategy, the output is sampled when the one-step prediction mean error exceeds a prespecified threshold. An iterative algorithm for the state probability distribution is proposed based on the algebraic form of SBNs, which determines the optimal state estimation. A matrix inequality method is proposed to calculate the worst-case mean estimation error based on its monotonicity with time. Then, the range of sampling triggering thresholds that minimize the worst-case mean estimation error is obtained. This article demonstrates that the event-triggered sampling strategy can make a tradeoff between estimation error and sampling rate. It explains that the full sampling estimator is a special event-triggered sampling estimator. Finally, the proposed method is applied to BN models of the lac operon in Escherichia coli to analyze the relationship among the sampling triggering threshold, the sampling rate, and the estimation error.