Combinatorial Network Optimization Research Articles

The Neutrosophic Traveling Salesman problem with Neutrosophic EdgeWeight: Formulation and A Genetic Algorithm

The traveling salesman problem (TSP) is an important and well known classical combinatorial network optimization problem in operation research, where the TSP finds a shortest possible route through a set of n nodes such that each and every node are visited exactly one time except for the starting node. In this problem, the arc lengths are generally considered to represent the traveling time or travelling cost rather than geographical distance. It is not possible to predict the exact arc length because the traveling time or traveling cost fluctuated with payload, weather, traffic conditions and so on. neutrosophic set theory provides a new tool to handle the uncertainties in TSP. In this paper, we concentrate on TSP on a network in which neutrosophic set, Instead of real number is assigned to edge as edge weight. We propose a mathematical model for a TSP with neutrosophic arc lengths. We present the utility of neutrosophic sets as arc length for TSP. An algorithmic method based on Genetic Algorithm (GA) is proposed for solving this problem. We have designed a new heuristic crossover and heuristic mutation our proposed GA. We have used a numerical example to illustrate the effectiveness of our proposed algorithm.

Design of Trustworthy Cyber–Physical–Social Systems With Discrete Bayesian Optimization

Abstract Cyber–physical–social systems (CPSS) with highly integrated functions of sensing, actuation, computation, and communication are becoming the mainstream consumer and commercial products. The performance of CPSS heavily relies on the information sharing between devices. Given the extensive data collection and sharing, security and privacy are of major concerns. Thus, one major challenge of designing those CPSS is how to incorporate the perception of trust in product and systems design. Recently, a trust quantification method was proposed to measure the trustworthiness of CPSS by quantitative metrics of ability, benevolence, and integrity. The CPSS network architecture can be optimized by choosing a subnet such that the trust metrics are maximized. The combinatorial network optimization problem, however, is computationally challenging. Most of the available global optimization algorithms for solving such problems are heuristic methods. In this paper, a surrogate-based discrete Bayesian optimization method is developed to perform network design, where the most trustworthy CPSS network with respect to a reference node is formed to collaborate and share information with. The applications of ability and benevolence metrics in design optimization of CPSS architecture are demonstrated.

Thompson Sampling for Combinatorial Network Optimization in Unknown Environments

Influence maximization, adaptive routing, and dynamic spectrum allocation all require choosing the right action from a large set of alternatives. Thanks to the advances in combinatorial optimization, these and many similar problems can be efficiently solved given an environment with known stochasticity. In this paper, we take this one step further and focus on combinatorial optimization in unknown environments. We consider a very general learning framework called combinatorial multi-armed bandit with probabilistically triggered arms and a very powerful Bayesian algorithm called Combinatorial Thompson Sampling (CTS). Under the semi-bandit feedback model and assuming access to an oracle without knowing the expected base arm outcomes beforehand, we show that when the expected reward is Lipschitz continuous in the expected base arm outcomes CTS achieves O(Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> logT /(p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> )) regret and a O(max{E[m√TlogT/p*], E[m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /p*]}) Bayesian regret, where m denotes the number of base arms, pi and Δi denote the minimum non-zero triggering probability and the minimum suboptimality gap of base arm i respectively, T denotes the time horizon, and p* denotes the overall minimum non-zero triggering probability. We also show that when the expected continuity, CTS achieves O(max{m√TlogT , m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> }) Bayesian reward satisfies the triggering probability modulated Lipschitz regret, and when triggering probabilities are non-zero for all base arms, CTS achieves O(1/p* log(1/p*)) regret independent of the time horizon. Finally, we numerically compare CTS with algorithms based on upper confidence bounds in several networking problems and show that CTS outperforms these algorithms by at least an order of magnitude in majority of the cases.

Variable Neighborhood Search for Integrated Timetable Based Design of Railway Infrastructure

In this paper we deal with the problem of building new or extending an existing railway infrastructure. The goal is to determine a minimum cost infrastructure fulfilling the requirements defined by an integrated timetable and the operation of the railway system. We first model this planning task as a combinatorial network optimization problem, capturing the essential aspects. We then present a metaheuristic solution method based on general variable neighborhood search that makes use of a dynamic programming procedure for realizing individual connections. Computational experiments indicate that the suggested approach is promising and the analysis of obtained results gives useful hints for future work in this area.

Markov Approximation for Combinatorial Network Optimization

Many important network design problems are fundamentally combinatorial optimization problems. A large number of such problems, however, cannot readily be tackled by distributed algorithms. The Markov approximation framework studied in this paper is a general technique for synthesizing distributed algorithms. We show that when using the log-sum-exp function to approximate the optimal value of any combinatorial problem, we end up with a solution that can be interpreted as the stationary probability distribution of a class of time-reversible Markov chains. Selected Markov chains among this class yield distributed algorithms that solve the log-sum-exp approximated combinatorial network optimization problem. By examining three applications, we illustrate that the Markov approximation technique not only provides fresh perspectives to existing distributed solutions, but also provides clues leading to the construction of new distributed algorithms in various domains with provable performance. We believe the Markov approximation techniques will find applications in many other network optimization problems.

Combinatorial Network Optimization With Unknown Variables: Multi-Armed Bandits With Linear Rewards and Individual Observations

We formulate the following combinatorial multi-armed bandit (MAB) problem: There are <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> random variables with unknown mean that are each instantiated in an i.i.d. fashion over time. At each time multiple random variables can be selected, subject to an arbitrary constraint on weights associated with the selected variables. All of the selected individual random variables are observed at that time, and a linearly weighted combination of these selected variables is yielded as the reward. The goal is to find a policy that minimizes regret, defined as the difference between the reward obtained by a genie that knows the mean of each random variable, and that obtained by the given policy. This formulation is broadly applicable and useful for stochastic online versions of many interesting tasks in networks that can be formulated as tractable combinatorial optimization problems with linear objective functions, such as maximum weighted matching, shortest path, and minimum spanning tree computations. Prior work on multi-armed bandits with multiple plays cannot be applied to this formulation because of the general nature of the constraint. On the other hand, the mapping of all feasible combinations to arms allows for the use of prior work on MAB with single-play, but results in regret, storage, and computation growing exponentially in the number of unknown variables. We present new efficient policies for this problem that are shown to achieve regret that grows logarithmically with time, and polynomially in the number of unknown variables. Furthermore, these policies only require storage that grows linearly in the number of unknown parameters. For problems where the underlying deterministic problem is tractable, these policies further require only polynomial computation. For computationally intractable problems, we also present results on a different notion of regret that is suitable when a polynomial-time approximation algorithm is used.

Component placement optimization in the brain

This computational neuroanatomy study evaluates how well some formalisms derived from combinatorial network optimization theory fit as models for brain structure. At multiple hierarchical levels--brain, ganglion, individual cell--physical placement of neural components appears consistent with a single, simple goal: minimize cost of connections among the components. The most dramatic instance of this "save wire" organizing principle is reported for adjacencies among ganglia in the nematode nervous system; among about 40,000,000 alternative layout orderings, the actual ganglion placement in fact requires the least total connection length. In addition, evidence supports a component placement optimization hypothesis for positioning of individual neurons in the nematode, and also for positioning of mammalian cortical areas.

Local optimization of neuron arbors.

How parsimoniously is brain wiring laid out, that is, how well does a neuron minimize costs of connections among its synapses? Neural optimization of dendritic and axonic arbors can be evaluated using a generalization of the Steiner tree concept from combinatorial network optimization theory. Local branch-junction geometry of neuronal connecting structures fits a volume minimization model well. In addition, volume of the arborizations at this neighborhood level is significantly more strongly minimized than their length, signal propagation speed, or surface area. The mechanism of this local volume optimization resembles those involved in formation of nonliving tree structures such as river junctions and electric-discharge patterns, and appears to govern initial nerve growth-cone behavior through vector-mechanical energy minimization.