The Component Diagnosability of General Networks

The processor failures in a multiprocessor system have a negative impact on its distributed computing efficiency. Because of the rapid expansion of multiprocessor systems, the importance of fault diagnosis is becoming increasingly prominent. The [Formula: see text]-component diagnosability of [Formula: see text], denoted by [Formula: see text], is the maximum number of nodes of the faulty set [Formula: see text] that is correctly identified in a system, and the number of components in [Formula: see text] is at least [Formula: see text]. In this paper, we determine the [Formula: see text]-component diagnosability of general networks under the PMC model and MM[Formula: see text] model. As applications, the component diagnosability is explored for some well-known networks, including complete cubic networks, hierarchical cubic networks, generalized exchanged hypercubes, dual-cube-like networks, hierarchical hypercubes, Cayley graphs generated by transposition trees (except star graphs), and DQcube as well. Furthermore, we provide some comparison results between the component diagnosability and other fault diagnosabilities.

The Clustered Selected-Internal Steiner Tree Problem

Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.

The Overlap Gap Between Left-Infinite and Right-Infinite Words

We study ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word [Formula: see text] and the prefixes of a right-infinite word [Formula: see text]. The main theorem states that the set of minimum lengths of words [Formula: see text] and [Formula: see text] such that [Formula: see text] or [Formula: see text] is finite, where [Formula: see text] runs over positive integers and [Formula: see text] and [Formula: see text] are respectively the suffix of [Formula: see text] and the prefix of [Formula: see text] of length [Formula: see text], if and only if [Formula: see text] and [Formula: see text] are ultimately periodic words of the form [Formula: see text] and [Formula: see text] for some finite words [Formula: see text], [Formula: see text] and [Formula: see text].

Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees

The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.

NP-Hardness and Approximation Algorithms for Iterative Pricing on Social Networks with Externalities

We study how a monopolist seller should price an indivisible product iteratively to the consumers who are connected by a known link-weighted directed social network. For two consumers [Formula: see text] and [Formula: see text], there is an arc directed from [Formula: see text] to [Formula: see text] if and only if [Formula: see text] is a fashion leader of [Formula: see text]. Assuming complete information about the network, the seller offers consumers a sequence of prices over time and the goal is to obtain the maximum revenue. We assume that the consumers buy the product as soon as the seller posts a price not greater than their valuations of the product. The product’s value for a consumer is determined by three factors: a fixed consumer specified intrinsic value and a variable positive (resp. negative) externality that is exerted from the consumer’s out(resp. in)-neighbours. The setting of positive externality is that the influence of fashion leaders on a consumer is the total weight of links from herself to her fashion leaders who have owned the product, and more fashion leaders of a consumer owning the product will increase the influence (external value) on the consumer. And the setting of negative externalities is that the product’s value of showing off for a consumer is the total weight of links from her followers who do not own the product to herself, and more followers of a consumer owning the product will decrease this external value for the consumer. We confirm that finding an optimal iterative pricing is NP-hard even for acyclic networks with maximum total degree [Formula: see text] and with all intrinsic values zero. We design a greedy algorithm which achieves [Formula: see text]-approximation for networks with all intrinsic values zero and show that the approximation ratio [Formula: see text] is tight. Complementary to the hardness result, we design a [Formula: see text]-approximation algorithm for Barabási–Albert networks.

Unpaired Many-to-Many Disjoint Path Cover of Balanced Hypercubes

A many-to-many [Formula: see text]-disjoint path cover ([Formula: see text]-DPC) of a graph [Formula: see text] is a set of [Formula: see text] vertex-disjoint paths joining [Formula: see text] distinct pairs of source and sink in which each vertex of [Formula: see text] is contained exactly once in a path. The balanced hypercube [Formula: see text], a variant of the hypercube, was introduced as a desired interconnection network topology. Let [Formula: see text] and [Formula: see text] be any two sets of vertices in different partite sets of [Formula: see text] ([Formula: see text]). Cheng et al. in [Appl. Math. Comput. 242 (2014) 127–142] proved that there exists paired many-to-many 2-disjoint path cover of [Formula: see text] when [Formula: see text]. In this paper, we prove that there exists unpaired many-to-many [Formula: see text]-disjoint path cover of [Formula: see text] ([Formula: see text]) from [Formula: see text] to [Formula: see text], which has improved some known results. The upper bound [Formula: see text] is best possible in terms of the number of disjoint paths in unpaired many-to-many [Formula: see text]-DPC of [Formula: see text].

A Pseudo-Random Generator Whose Output is a Normal Sequence

Pseudo-random number generators (PRNGs) are widely used in computer simulation, cryptography, and many other fields. In this paper, we describe a PRNG class, which, firstly, has been successfully tested using the most powerful modern test batteries, and secondly, is proved to consist of generators that generate normal sequences. The latter property means that, for any generated sequence [Formula: see text] and any binary word [Formula: see text], we have [Formula: see text] where [Formula: see text] is the number of occurrences of [Formula: see text] in the sequence [Formula: see text], [Formula: see text].