Graph coloring using the reduced quantum genetic algorithm

Genetic algorithms (GA) are computational methods for solving optimization problems inspired by natural selection. Because we can simulate the quantum circuits that implement GA in different highly configurable noise models and even run GA on actual quantum computers, we can analyze this class of heuristic methods in the quantum context for NP-hard problems. This paper proposes an instantiation of the Reduced Quantum Genetic Algorithm (RQGA) that solves the NP-hard graph coloring problem in O(N1/2). The proposed implementation solves both vertex and edge coloring and can also determine the chromatic number (i.e., the minimum number of colors required to color the graph). We examine the results, analyze the algorithm convergence, and measure the algorithm's performance using the Qiskit simulation environment. Our Reduced Quantum Genetic Algorithm (RQGA) circuit implementation and the graph coloring results show that quantum heuristics can tackle complex computational problems more efficiently than their conventional counterparts.

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness for integer due dates to the scheduling problem, where along with precedence constraints given on the set of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set . We prove that the extended scheduling problem with integer release times of the jobs to minimize schedule length may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) to the vertices of the mixed graph such that, if two vertices and are joined by the edge , their colors have to be different. Further, if two vertices and are joined by the arc , the color of vertex has to be no greater than the color of vertex . We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs , have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.

Maintenance of Water ATMs and Selection of locations for the Water ATMs on basis of Internal factors of the Location using Graph Theory

Under the Smart City Project, Guwahatians got facility of Water ATMs. In this research main aim is to find out the optimal locations in the city on the basis of internal factors of the location (like quantity and quality of water etc.) so that maximum number of citizens avail the benefits of water ATM. There are sets of Water ATMs that cannot be taken down together, because they have certain critical functions like -installation of new software, update of existing software, installation of required equipment, raw water supply problem, maintenance of pipelines, change the capacity of water tanks, electricity problem, plumbing or all routine maintenance etc. This is a typical scheduling application of graph coloring problem. It turned out that 3 colors were good enough to color the graph of 12 nodes. So they could install updates in 3 passes.

A NOVEL GREEDY GENETIC ALGORITHM TO SOLVE COMBINATORIAL OPTIMIZATION PROBLEM

In this paper, a modified genetic algorithm based on greedy sequential algorithm is presented to solve combinatorial optimization problem. The algorithm proposed here is a hybrid of heuristic and computational intelligence algorithm where greedy sequential algorithm is used as operator inside genetic algorithm like crossover and mutation. The greedy sequential function is used to correct non realizable solution after crossover and mutation which contribute to increase the rate of convergence and upgrade the population by improving the quality of chromosomes toward the chromatic number. Experiments on a set of 6 well-known DIMACS benchmark instances of graph coloring problem to test this approach show that the proposed algorithm achieves competitive results in comparison with three states of art algorithms in terms of either success rate and solution quality.