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A variety of probabilistic notions of network reliability of graphs and digraphs have been proposed and studied since the early 1950s. Although grounded in the engineering and logistics of network design and analysis, the research also spans pure and applied mathematics, with connections to areas as diverse as combinatorics and graph theory, combinatorial enumeration, optimization, probability theory, real and complex analysis, algebraic topology, commutative algebra, the design and analysis of algorithms, and computational complexity. In this paper we describe the landscape of various notions of network reliability, the roads well traveled, and some that appear likely to lead to meaningful and important journeys. [ABSTRACT FROM AUTHOR]

A new algebraic system, test algebra ($\mathsf{TA}$), is proposed for identifying faults in combinatorial testing for software-as-a-service (SaaS) applications. In the context of cloud computing, SaaS is a new software delivery model, in which mission-critical applications are composed, deployed, and executed on cloud platforms. Testing SaaS applications is challenging because new applications need to be tested once they are composed, and prior to their deployment. A composition of components providing services yields a configuration providing an SaaS application. While individual components in the configuration may have been thoroughly tested, faults still arise due to interactions among the components composed, making the configuration faulty. When there are $k$ components, combinatorial testing algorithms can be used to identify faulty interactions with $t$ or fewer components, for some threshold $2 \leq t \leq k$ on the size of interactions considered. In general, these methods do not identify specific faults, but rather indicate the presence or absence of some faults. To identify specific faults, an adaptive testing regime repeatedly constructs and tests configurations in order to determine, for each interaction of interest, whether it is faulty or not. In order to perform such testing in a loosely coupled distributed environment such as the cloud, it is imperative that testing results can be combined from many different servers. The $\mathsf{TA}$ defines rules to permit results to be combined, and to identify the faulty interactions. Using the $\mathsf{TA}$ , configurations can be tested concurrently on different servers and in any order. The $\mathsf{TA}$ always keeps the high reduction rate of potential faulty configurations in fault location analysis. [ABSTRACT FROM AUTHOR]

Software behavior depends on many factors. Combinatorial testing (CT) aims to generate small sets of test cases to uncover defects caused by those factors and their interactions. Covering array generation, a discrete optimization problem, is the most popular research area in the field of CT. Particle swarm optimization (PSO), an evolutionary search-based heuristic technique, has succeeded in generating covering arrays that are competitive in size. However, current PSO methods for covering array generation simply round the particle’s position to an integer to handle the discrete search space. Moreover, no guidelines are available to effectively set PSOs parameters for this problem. In this paper, we extend the set-based PSO, an existing discrete PSO (DPSO) method, to covering array generation. Two auxiliary strategies (particle reinitialization and additional evaluation of gbest) are proposed to improve performance, and thus a novel DPSO for covering array generation is developed. Guidelines for parameter settings both for conventional PSO (CPSO) and for DPSO are developed systematically here. Discrete extensions of four existing PSO variants are developed, in order to further investigate the effectiveness of DPSO for covering array generation. Experiments show that CPSO can produce better results using the guidelines for parameter settings, and that DPSO can generate smaller covering arrays than CPSO and other existing evolutionary algorithms. DPSO is a promising improvement on PSO for covering array generation. [ABSTRACT FROM PUBLISHER]

Necessary conditions for existence of a resolvable group divisible design (GDD) with block size 3 and type [ABSTRACT FROM AUTHOR]

Abstract: The construction of covering arrays with the fewest rows remains a challenging problem. Most computational and recursive constructions result in extensive repetition of coverage. While some is necessary, some is not. By reducing the repeated coverage, metaheuristic search techniques typically outperform simpler computational methods, but they have been applied in a limited set of cases. Time constraints often prevent them from finding an array of competitive size. We examine a different approach. Having used a simple computation or construction to find a covering array, we employ a post-optimization technique that repeatedly adjusts the array in an attempt to reduce its number of rows. At every stage the array retains full coverage. We demonstrate its value on a collection of previously best known arrays by eliminating, in some cases, 10% of their rows. In the well-studied case of strength two with twenty factors having ten values each, post-optimization produces a covering array with only 162 rows, improving on a wide variety of computational and combinatorial methods. We identify certain important features of covering arrays for which post-optimization is successful. [Copyright &y& Elsevier]

Abstract: The deterministic construction of measurement matrices for compressive sensing is a challenging problem, for which a number of combinatorial techniques have been developed. One of them employs a widely used column replacement technique based on hash families. It is effective at producing larger measurement matrices from smaller ones, but it can only preserve the strength (level of sparsity supported), not increase it. Column replacement is extended here to produce measurement matrices with larger strength from ingredient arrays with smaller strength. To do this, a new type of hash family, called a strengthening hash family, is introduced. Using these hash families, column replacement is shown to increase strength under two standard notions of recoverability. Then techniques to construct strengthening hash families, both probabilistically and deterministically, are developed. Using a variant of the Stein–Lovász–Johnson theorem, a deterministic, polynomial time algorithm for constructing a strengthening hash family of fixed strength is derived. [Copyright &y& Elsevier]

Abstract: A Hanani triple system of order , , is a decomposition of the complete graph into sets of disjoint triangles and one set of disjoint triangles. A nearly Kirkman triple system of order , , is a decomposition of into sets of disjoint triangles; here is a one-factor of . The Hanani triple systems of order and the nearly Kirkman triple systems of order can be classified using the classification of the Steiner triple systems of order . This is carried out here for : There are 3787983639 isomorphism classes of s and 25328 isomorphism classes of s. Several properties of the classified systems are tabulated. In particular, seven of the s have orthogonal resolutions, and five of the s admit a pair of resolutions in which the almost parallel classes are orthogonal. [Copyright &y& Elsevier]

Test response compaction for integrated circuits (ICs) with scan-based design-for-testability (DFT) support in the presence of unknown logic values (Xs) is investigated from a combinatorial viewpoint. The theoretical foundations of X-codes, employed in an X-tolerant compaction technique called X-compact, are examined. Through the formulation of a combinatorial model of X-compact, novel design techniques are developed for X-codes to detect a specified maximum number of errors in the presence of a specified maximum number of unknown logic values, while requiring only small fan-out. The special class of X-codes that results leads to an avoidance problem for configurations in combinatorial designs. General design methods and nonconstructive existence theorems to estimate the compaction ratio of an optimal X-compactor are also derived. [ABSTRACT FROM AUTHOR]

Abstract: A -design on points embeds a Steiner triple system (STS) if there is a subset of points on which the graphs of the design induce the blocks of a STS. It is established that , and that when equality is met that such a minimum embedding of an exists, except when . [Copyright &y& Elsevier]

Deterministic construction of measurement matrices for compressive sensing can be effected by first constructing a relatively small matrix explicitly, and then inflating it using a column replacement technique to form a large measurement matrix that supports at least the same level of sparsity. In particular, using easily developed null space conditions for \ell_0- and \ell_1-recoverability, properties of the pattern matrix used to select columns lead to well-studied matrices, separating and distributing hash families. Two-stage compression and recovery techniques are developed that employ more computationally intensive \ell_0-recoverability for small matrices and simpler \ell_1-recoverability for one larger matrix; this can reduce the number of measurements required. [ABSTRACT FROM PUBLISHER]