Polynomial algorithms for p-dispersion problems in a 2d Pareto Front

Having many best compromise solutions for bi-objective optimization problems, this paper studies p-dispersion problems to select $p\geqslant 2$ representative points in the Pareto Front(PF). Four standard variants of p-dispersion are considered. A novel variant, denoted Max-Sum-Neighbor p-dispersion, is introduced for the specific case of a 2d PF. Firstly, it is proven that $2$-dispersion and $3$-dispersion problems are solvable in $O(n)$ time in a 2d PF. Secondly, dynamic programming algorithms are designed for three p-dispersion variants, proving polynomial complexities in a 2d PF. The Max-Min p-dispersion problem is proven solvable in $O(pn\log n)$ time and $O(n)$ memory space. The Max-Sum-Min p-dispersion problem is proven solvable in $O(pn^3)$ time and $O(pn^2)$ space. The Max-Sum-Neighbor p-dispersion problem is proven solvable in $O(pn^2)$ time and $O(pn)$ space. Complexity results and parallelization issues are discussed in regards to practical implementation

Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

In this paper, we present a quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is $O(\sqrt{\hat{n}m}\log \hat{n})$, and the running time of the best known deterministic algorithm is $O(n+m)$, where $n$ is the number of vertices, $\hat{n}$ is the number of vertices with at least one outgoing edge; $m$ is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluating. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.Comment: UCNC2019 Conference pape

Analytical solutions to nonlinear mechanical oscillation problems

In this paper the Max-Min Method is utilized for solving the nonlinear oscillation problems. The proposed approach is applied to three systems with complex nonlinear terms in their motion equations. By means of this method the dynamic behavior of oscillation systems can be easily approximated using He Chengtian’s interpolation. The comparison of the obtained results from Max-Min method with time marching solution and the results achieved from literature verifies its convenience and effectiveness. It is predictable that He's Max-Min Method will find wide application in various engineering problems as indicated in the following cases