Popular Combinatorial Problem Research Articles

Hamiltonian path problem solution using DNA computing

In this article we study DNA computing, a method which is based on working with DNA molecules in a laboratory. That approach is implemented in solving one of the most popular combinatorial problem — the Hamiltonian path problem. Related to recent improvements in the biophysics methods, which are needed for DNA computing, we propose to change some steps in the classical algorithm to increase accuracy of this method. The branch-and-bound method, the most popular method which is realized on a computer, is also shown in this paper to compare its performance with the time consumption of DNA computing. The results of that comparison prove that it becomes inefficient to use the branch-and-bound method from the counted number of vertices because of its exponentially growing complexity, while DNA computing works parallel and has linearly growing time consumption.

Hamiltonian path problem: the performance comparison deoxyribonucleic acid computing and the branch-and-bound method

In this article different approaches to one of the most popular combinatorial problem — the Hamiltonian path problem – are illustrated and compared between each other. It is shown that it becomes inefficient to use branch-and-bound method, the most popular method which is realized on a computer, from the counted number of vertices because of its exponentially growing complexity, one more algorithm which is based on working with deoxyribonucleic acid (DNA) molecules in a laboratory is analysed. That method works parallel and has linearly growing time consumption. Due to the improvements in the biophysics methods, which are needed for DNA computing, that algorithm became much faster than it was several years ago and it is now possible to add some new stages in DNA computing, which are shown in this paper.

On some combinatorial problems in cographs

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for connectivity augmentation problems and its variants in cographs, preserving the cograph property. Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs, our framework yields polynomial-time algorithms for all three problems.

A comparative evaluation of swarm intelligence techniques for solving combinatorial optimization problems

This article presents a critical evaluation of swarm intelligence techniques for solving combinatorial optimization problems. Since, unarguably, the traveling salesman’s problem is the most developed, studied, and popular combinatorial problem, this study uses it as a benchmark. After a number of experimental investigations involving 24 popular but complex benchmark symmetric traveling salesman’s problem instances and 15 asymmetric traveling salesman’s problem of the 19 instances available in TSPLIB95, the African buffalo optimization proved to be the best algorithm in terms of efficiency and effectiveness in solving the problems under investigation.

Integer Programming Model for Two-Centered Double Traveling Salesman Problem

Traveling Salesman Problem (TSP) is among the most popular combinatorial problems and has been widely studied with many extensions in the literature. There have been integer programming formulations and solution approaches for TSP and its variations. One of the most popular topics is the multiple TSP (m-TSP). It has been started to work on the last decades. Generally, m-TSP has a single depot and more than one tour. However, some types have more than one depot. Besides, if seeking, many encounter with double traveling salesman problem (d-TSP). As inferred from the literature, d-TSP is a variation of m-TSP in which two salespersons operate in parallel. They start and end either in a single or two depots. Apart from the literature, a new TSP model has been constructed in this study. The model analyzes the behaviors of traveling salesmen both in the main and secondary tour. Tours have been aggregated via single node existing in the secondary tour. In the application part, Simulated Annealing (SA) was used to optimize the traveling paths of salesmen. The objective parameters of both tours have been explored within a numerical example. According to them, several cost values have been found. The optimum parameters and costs of both tours are to be determined and some practical issues relevant to the behaviors of traveling salesmen have been given. Results suggest that alternative travel plans of traveling salesmen could be possible. Also, findings give hints about both tours and their characteristics.

A Comparative Study of Prominent Particle Swarm Optimization Based Methods to Solve Traveling Salesman Problem

Computational methods inspired by natural phenomenon have gain much interest in the recent years. Among the developed algorithms, particle swarm optimization (PSO), mimicking behavior of bird flocking or fish schooling, seems the most famous method due to its simplicity as well as performance. A variant number of PSO based methods was developed for traveling salesman problem (TSP), the most popular combinatorial problem. The aim of the study is to make a comparative study of several prominent PSO based methods in solving TSP. The study is important because different PSO based methods have been developed by different researchers and tested on different sets of problems. Therefore, the description of the prominent PSO based methods in a similar fashion reveals distinct features of individuals. Moreover, experimental results on a common benchmark TSP data set will reveal performance of each method. In this study, the methods have been tested on a large number of benchmark TSPs and outcomes compared among themselves as well as ant colony optimization (ACO), the prominent method to solve TSP. Experimental results revealed that enhanced self-tentative PSO (ESTPSO) and velocity tentative PSO (VTPSO) outperformed ACO; and self-tentative PSO (STPSO) is competitive to ACO. On the other hand, experimental analysis revealed that ESTPSO is computationally heavier than others and VTPSO took least time to solve a benchmark problem. The reasons behind performance and time requirement of each individual method are explained and VTPSO is found most effective PSO based method to solve TSP.

Logic programming for combinatorial problems

Combinatorial problems appear in many areas in science, engineering, biomedicine, business, and operations research. This article presents a new intelligent computing approach for solving combinatorial problems, involving permutations and combinations, by incorporating logic programming. An overview of applied combinatorial problems in various domains is given. Such computationally hard and popular combinatorial problems as the traveling salesman problem are discussed to illustrate the usefulness of the logic programming approach. Detailed discussions of implementation of combinatorial problems with time complexity analyses are presented in Prolog, the standard language of logic programming. These programs can be easily integrated into other systems to implement logic programming in combinatorics.