Minimum constraint removal problem for line segments is NP-hard

In the minimum constraint removal ([Formula: see text]), there is no feasible path to move from a starting point towards the goal, and the minimum constraints should be removed in order to find a collision-free path. It has been proved that [Formula: see text] problem is NP-hard when constraints have arbitrary shapes or even they are in shape of convex polygons. However, it has a simple linear solution when constraints are lines and the problem is open for other cases yet. In this paper, using a reduction from Subset Sum problem, in three steps, we show that the problem is NP-hard for both weighted and unweighted line segments.

Universal Nonlinear Spiking Neural P Systems with Delays and Weights on Synapses

The nonlinear spiking neural P systems (NSNP systems) are new types of computation models, in which the state of neurons is represented by real numbers, and nonlinear spiking rules handle the neuron’s firing. In this work, in order to improve computing performance, the weights and delays are introduced to the NSNP system, and universal nonlinear spiking neural P systems with delays and weights on synapses (NSNP-DW) are proposed. Weights are treated as multiplicative constants by which the number of spikes is increased when transiting across synapses, and delays take into account the speed at which the synapses between neurons transmit information. As a distributed parallel computing model, the Turing universality of the NSNP-DW system as number generating and accepting devices is proven. 47 and 43 neurons are sufficient for constructing two small universal NSNP-DW systems. The NSNP-DW system solving the Subset Sum problem is also presented in this work.

Optimality and Complexity Analysis of a Branch-and-Bound Method in Solving Some Instances of the Subset Sum Problem

In this paper we study the question of parallelization of a variant of Branch-and-Bound method for solving of the subset sum problem which is a special case of the Boolean knapsack problem. The following natural approach to the solution of this question is considered. At the first stage one of the processors (control processor) performs some number of algorithm steps of solving a given problem with generating some number of subproblems of the problem. In the second stage the generated subproblems are sent to other processors for solving (one subproblem per processor). Processors solve completely the received subproblems and return their solutions to the control processor which chooses the optimal solution of the initial problem from these solutions. For this approach we define formally a model of parallel computing (frontal parallelization scheme) and the notion of complexity of the frontal scheme. We study the asymptotic behavior of the complexity of the frontal scheme for two special cases of the subset sum problem.