On the characterization of the domination of a diameter-constrained network reliability model

AbstractLet G=(V,E) be a digraph with a distinguished set of terminal vertices K⊆V and a vertex s∈K. We define the s,K-diameter of G as the maximum distance between s and any of the vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the diameter-constrained s,K-terminal reliability of G, Rs,K(G,D), is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D.The diameter-constrained network reliability is a special case of coherent system models, where the domination invariant has played an important role, both theoretically and for developing algorithms for reliability computation. In this work, we completely characterize the domination of diameter-constrained network models, giving a simple rule for computing its value: if the digraph either has an irrelevant arc, includes a directed cycle or includes a dipath from s to a node in K longer than D, its domination is 0; otherwise, its domination is -1 to the power |E|-|V|+1. In particular this characterization yields the classical source-to-K-terminal reliability domination obtained by Satyanarayana.Based on these theoretical results, we present an algorithm for computing the reliability

On the characterization of the source-to-all-terminal diameter-constrained reliability domination

Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K . We de ne the s;K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s;K-terminal reliability of G, Rs;K(G;D), is de ned as the probability that surviving arcs span a subgraph whose s;K- diameter does not exceed D [5, 11]. A graph invariant called the domination of a graph G was introduced by Satyanarayana and Prabhakar [13] to generate the non-canceling terms of the classical reliability expres- sion, Rs;K(G), based on the same reliability model (i.e. arcs fail randomly and indepen- dently and where nodes are perfect), and de ned as the probability that the surviving arcs span a subgraph of G with unconstrained nite s;K-diameter. This result allowed the generation of rapid algorithms for the computation of Rs;K(G). In this paper we present a characterization of the diameter-constrained s;K-terminal reliability domination of a digraph G = (V;E) with terminal set K = V , and for any diameter bound D, and, as a result, we solve the classical reliability domination, as a speci c case. Moreover we also present a rapid algorithm for the evaluation of Rs;V (G;D).Eje: Teoría (TEOR)Red de Universidades con Carreras en Informática (RedUNCI

Applying ant systems to two real-life assignment problems

Ant Systems (AS) is a recently proposed meta-heuristic inspired on biological behaviors, which has been applied to a variety of combinatorial optimization problems, including the QAP (Quadratic Assignment Problems). In this work, we have studied the adaptation of the Ant Systems meta-heuristic to two different real-life assignment problems, which appear in educational institutions: the timetabling problem (assigning courses to classrooms and times), and the assignment of final proyects to students. There is no standard definition for these problems, as in each institution the rules and objectives are different. We have been successful in adapting. AS to tackle these problems as defined by our institution rules, showing the adaptability of this meta-heuristic to complex, real-life problems. In both cases the AS meta-heuristic obtained good quality solutions (in the timetabling case, at the cost of longer running times)

On the characterization of the source-to-all-terminal diameter-constrained reliability domination

Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K . We de ne the s;K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s;K-terminal reliability of G, Rs;K(G;D), is de ned as the probability that surviving arcs span a subgraph whose s;K- diameter does not exceed D [5, 11]. A graph invariant called the domination of a graph G was introduced by Satyanarayana and Prabhakar [13] to generate the non-canceling terms of the classical reliability expres- sion, Rs;K(G), based on the same reliability model (i.e. arcs fail randomly and indepen- dently and where nodes are perfect), and de ned as the probability that the surviving arcs span a subgraph of G with unconstrained nite s;K-diameter. This result allowed the generation of rapid algorithms for the computation of Rs;K(G). In this paper we present a characterization of the diameter-constrained s;K-terminal reliability domination of a digraph G = (V;E) with terminal set K = V , and for any diameter bound D, and, as a result, we solve the classical reliability domination, as a speci c case. Moreover we also present a rapid algorithm for the evaluation of Rs;V (G;D).Eje: Teoría (TEOR)Red de Universidades con Carreras en Informática (RedUNCI

On the robustness of fishman's bound-based method for the network reliability problem

International audienceStatic network unreliability computation is an NP-hard problem, leading to the use of Monte Carlo techniques to estimate it. The latter, in turn, suffer from the rare event problem, in the frequent situation where the system's unreliability is a very small value. As a consequence, specific rare event event simulation techniques are relevant tools to provide this estimation. We focus here on a method proposed by Fishman making use of bounds on the structure function of the model. The bounds are based on the computation of (disjoint) mincuts disconnecting the set of nodes and (disjoint) minpaths ensuring that they are connected. We analyze the robustness of the method when the unreliability of links goes to zero. We show that the conditions provided by Fishman, based on a bound, are only sufficient, and we provide more insight and examples on the behavior of the method

Diameter constrained network reliability :exact evaluation by factorization and bounds

Consider a network where the links are subject to random, independent failures. The diameter constrained network reliability parameter R(G,K,D) measures the probability that the set K of terminals of the network are linked by operational paths of length less or equal to D. This parameter generalizes the classical network reliability, allowing to reflect performance objectives that restrict the maximum length of a path in the network. This is the case, for example, when the transmissions between every two terminal nodes in the subset K are required to experience a maximum delay D.T (where T is the delay experienced at a single node or link); then the probability that after random failures of the communication links, the surviving network meets the maximum delay requirement is the diameter constrained reliability R(G,K,D). This paper defines the diameter constrained network reliability, and gives a formulation in terms of events corresponding to the operation of the (length constrained) paths of the network. Based on this formulation, the exact value of the diameter constrained reliability is derived, for the special case where K=\{s,t\} and the upper bound D of the path length is 2. For other values of K and D an exact evaluation algorithm based on a factorization approach is proposed. As this algorithm has exponential worst case complexity, upper and lower bounds for K=\{s,t\} are developed, which in some cases may be used instead of the exact valu

Domination Invariant of a Diameter Constrained Network Reliability Model

Let G=(V,E) be a digraph with a distinguished set of terminal vertices K in V and a vertex s in K. We define the s,K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s,K-terminal reliability of G, R_\{s,K\}(G,D) is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D. The Diameter-constrained network reliability is a special case of coherent system models, where the domination invariant has played an important role, both theoretically and for developing algorithms for reliability computation. In this work, we completely characterize the domination of diameter-constrained network models, giving a simple rule for computing its value: if the digraph either has an irrelevant edge, includes a dicycle or includes a dipath from $s$ to a node in K longer than D, its domination is 0; otherwise, its domination is -1 to the power |E|-|V|+1