k-Efficient domination: Algorithmic perspective

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

Trapping the Ultimate Success

We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, or the event that exactly one success is yet to occur, or may choose any proper range of future times (a trap). When a trap is chosen, the gambler wins if the last success epoch is the only one that falls in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success. We use this connection to analyse the best-choice problem with random arrivals generated by a Pólya-Lundberg process.

Multicriteria Decision Support Model for Selection of Fiberglass Suppliers

This chapter presents a real case of a decision problem in supplier selection of one of the main raw materials of a wind blades industry. The study considered all currently qualified suppliers according to considerably rigorous standards and specifications and one in qualification process. It is a complex choice, given the strategic importance of the product and the multiplicity of criteria to be considered, both quantitative and qualitative. The strong competitiveness requires a special attention which concerns the supplier selection; not only the price matters; in fact, a day of stoppage due to failure in a delivery, for example, corresponds to high losses that would have justified the purchase from a supplier with a higher price but with no delivery failures. In order to contribute to the problem resolution, the methodologies PROMETHEE and AHP were applied, whose results allow the authors to stablish a ranking of the considered suppliers. The results will support the company on the selection of fiberglass suppliers and in some cases clarify where they can find the main trade-offs.

Combined optimisation and MCDA based solution of the tram depot location problem

This paper deals with an issue of technical facilities location in a public transport system. The decision problem is formulated as a selection of the most advantageous alternative, i.e. the location of a new tram depot among the already existing facilities of this type. The selection is preceded by the evaluation of the alternatives. The assessment is not a trivial task, because there are many groups of interest with usually contradictory points of view. Therefore, the evaluation of the new tram depot locations should represent different aspects, e.g., economical, technical, environmental, and organizational. To handle such a complex decision problem the authors propose a methodology, which is a composition of the optimisation and multiple criteria evaluation techniques. The developed methodology is experimentally applied to the selection of one out of five tram depot locations in the public transport system of the city of Poznan, Poland. All the computational experiments are performed by means of optimization and multiple criteria decision aiding (MCDA) methods and tools, i.e. a linear optimization engine Solver Premium Platform and AHP method with its application AHORNsimple. The calculations are the basis for recommending the location of a new depot in the central part of the transport system network, which is a reasonable solution taking into account, e.g. the proximity of the main railway line, the possibility of triple distribution of the transport means from depot. The proposed methodology of the decision problem solution gives also an opportunity to create the hierarchy of considered tram depot locations as well as to compare the position in the ranking of the best solution with the existing one. Since the proposed methodology assumes the selection of the most suitable MCDA method to the problem under consideration and the decision maker’s preferen¬ces, it guarantees that the result of analysis becomes reliable and the decision aiding process is credible.

Quantum absentminded driver problem revisited

The aim of the paper is to study the problem of absentminded driver in the quantum domain. In the classical case, it is a well-known example of a decision problem with imperfect recall that exhibits lack of equivalence between mixed and behavioral strategies. The optimal payoff outcome is significantly lower than the maximum payoff appearing in the game. This raises the question whether a quantum approach to the problem can increase the strategic position of the decision maker. The results that we present in the paper clearly reveal the benefits from playing the absentminded problem with the aid of quantum objects. Through appropriately chosen initial quantum state, the unitary strategies enable the decision maker to obtain the maximum possible payoff. At the same time, our scheme comes down to the classical problem with a suitable restriction of unitary strategies.

Computing some role assignments of Cartesian Product of Graphs

In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. As a simple graph models a social network role assignment rises to the decision problem called r -Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r -Role Assignment restricted to Cartesian product of graphs, for r = 2,3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.

Is your back-up IT infrastructure in a safe location? A multi-criteria approach to location analysis for business continuity facilities

Building redundant capacity into an organization’s information technology (IT) infrastructure is a standard part of business continuity planning (BCP). Traditionally, cost concerns have dominated the decision of where to locate the redundant facilities. However; recently managers are becoming more aware of the fact that the very issues that make the main IT facilities vulnerable to disruption (i.e. man-made or natural disasters) are likely to impact the redundant (back-up facilities as well. This complicates the process of selecting redundant facility location(s). The problem is essentially a multi-criteria decision problem, and can be addressed using the location analysis techniques that have been used in other domains in the past. Meanwhile, what make this context somewhat unique are the decision criteria and the rather subjective nature of the decision process. This paper provides a simple decision model for the problem, and illustrates the model with a case where relevant decision criteria are identified and the solution is obtained using a mix of objective and subjective decision techniques. We believe the paper is valuable because it presents an actionable methodology for practitioners involved in BCP. <br>

Is your back-up IT infrastructure in a safe location? A multi-criteria approach to location analysis for business continuity facilities

Building redundant capacity into an organization’s information technology (IT) infrastructure is a standard part of business continuity planning (BCP). Traditionally, cost concerns have dominated the decision of where to locate the redundant facilities. However; recently managers are becoming more aware of the fact that the very issues that make the main IT facilities vulnerable to disruption (i.e. man-made or natural disasters) are likely to impact the redundant (back-up facilities as well. This complicates the process of selecting redundant facility location(s). The problem is essentially a multi-criteria decision problem, and can be addressed using the location analysis techniques that have been used in other domains in the past. Meanwhile, what make this context somewhat unique are the decision criteria and the rather subjective nature of the decision process. This paper provides a simple decision model for the problem, and illustrates the model with a case where relevant decision criteria are identified and the solution is obtained using a mix of objective and subjective decision techniques. We believe the paper is valuable because it presents an actionable methodology for practitioners involved in BCP. <br>

Sampling from Complex Probability Distributions: A Monte Carlo Primer for Engineers

Models which are constructed to represent the uncertainty arising in engineered systems can often be quite complex to ensure they provide a reasonably faithful reflection of the real-world system. As a result, even computation of simple expectations, event probabilities, variances, or integration over utilities for a decision problem can be analytically intractable. Indeed, such models are often sufficiently high dimensional that even traditional numerical methods perform poorly. However, access to random samples drawn from the probability model under study typically simplifies such problems substantially. The methodologies to generate and use such samples fall under the stable of techniques usually referred to as ‘Monte Carlo methods’. This chapter provides a motivation, simple primer introduction to the basics, and sign-posts to further reading and literature on Monte Carlo methods, in a manner that should be accessible to those with an engineering mathematics background. There is deliberately informal mathematical presentation which avoids measure-theoretic formalism. The accompanying lecture can be viewed at https://www.louisaslett.com/Courses/UTOPIAE/.

Cooperative Attack-Defense Decision-Making of Multi-UAV Using Satisficing Decision-Enhanced Wolf Pack Search Algorithm

Unmanned Aerial Vehicles (UAVs) have shown their superiority for applications in complicated military missions. A cooperative attack-defense decision-making method based on satisficing decision-enhanced wolf pack search (SDEWPS) algorithm is developed for multi-UAV air combat in this paper. Firstly, the multi-UAV air combat mathematical model is provided and the attack-defense decision-making constraints are defined. Besides the traditional air combat situation, the capability of UAVs and target information including target type and target intention are all considered in this paper to establish the air combat superiority function. Then, the wolf pack search (WPS) algorithm is used to solve the attack decision problem. In order to improve efficiency, the satisficing decision theory is employed to enhance the WPS to obtain the satisficing solution rather than optimal solution. The simulation results show that the developed method can realize the cooperative attack decision-making.