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In this paper, we consider a testing setting where the set of possible definitions of the Implementation Under Test (IUT), as well as the behavior of each of these definitions in all possible interactions, are extensionally defined, i.e., on an element-by-element and case-by-case basis. Under this setting, the problem of finding the minimum testing strategy such that collected observations will necessarily let us decide whether the IUT is correct or not (i.e., whether it necessarily belongs to the set of possible correct definitions or not) is studied in four possible problem variants: with or without non-determinism; and with or without more than one possible definition in the sets of possible correct and incorrect definitions. The computational complexity of these variants is studied, and properties such as PSPACE-completeness and Log-APX-hardness are identified. [ABSTRACT FROM AUTHOR]

Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other "path-related" temporal notions, only few attempts have been made to investigate "non-path" temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions. [ABSTRACT FROM AUTHOR]

Threats on the stability of a financial system may severely affect the functioning of the entire economy, and thus considerable emphasis is placed on the analyzing the cause and effect of such threats. The financial crisis in the current and past decade has shown that one important cause of instability in global markets is the so-called financial contagion, namely the spreadings of instabilities or failures of individual components of the network to other, perhaps healthier, components. This leads to a natural question of whether the regulatory authorities could have predicted and perhaps mitigated the current economic crisis by effective computations of some stability measure of the banking networks. Motivated by such observations, we consider the problem of defining and evaluating stabilities of both homogeneous and heterogeneous banking networks against propagation of synchronous idiosyncratic shocks given to a subset of banks. We formalize the homogeneous banking network model of Nier et al. (J. Econ. Dyn. Control 31:2033-2060, ) and its corresponding heterogeneous version, formalize the synchronous shock propagation procedures outlined in (Nier et al. J. Econ. Dyn. Control 31:2033-2060, ; M. Eboli Mimeo, ), define two appropriate stability measures and investigate the computational complexities of evaluating these measures for various network topologies and parameters of interest. Our results and proofs also shed some light on the properties of topologies and parameters of the network that may lead to higher or lower stabilities. [ABSTRACT FROM AUTHOR]

Abstract: Given a graph of interactions, a module (also called a community or cluster) is a subset of nodes whose fitness is a function of the statistical significance of the pairwise interactions of nodes in the module. The topic of this paper is a model-based community finding approach, commonly referred to as modularity clustering, that was originally proposed by Newman (Leicht and Newman, 2008 [25]) and has subsequently been extremely popular in practice (e.g., see Agarwal and Kempe, 2008 [1], Guimer‘a et al., 2007 [20], Newman, 2006 [28], Newman and Girvan, 2004 [30], Ravasz et al., 2002 [32]). Various heuristic methods are currently employed for finding the optimal solution. However, as observed in Agarwal and Kempe (2008) [1], the exact computational complexity of this approach is still largely unknown. To this end, we initiate a systematic study of the computational complexity of modularity clustering. Due to the specific quadratic nature of the modularity function, it is necessary to study its value on sparse graphs and dense graphs separately. Our main results include a -inapproximability for dense graphs and a logarithmic approximation for sparse graphs. We make use of several combinatorial properties of modularity to get these results. These are the first non-trivial approximability results beyond the NP-hardness results in Brandes et al. (2007) [10]. [Copyright &y& Elsevier]

Abstract: We study low degree graph problems such as Maximum Independent Set and Minimum Vertex Cover. The goal is to improve approximation lower bounds for them and for a number of related problems like Max-B-Set Packing, Min-B-Set Cover, and Max-B-Dimensional Matching, . We prove, for example, that it is NP-hard to achieve an approximation factor of for Max-3-DM, and a factor of for Max-4-DM. In both cases the hardness result applies even to instances with exactly two occurrences of each element. [Copyright &y& Elsevier]

The inapproximability of non-NP-hard optimization problems is investigated. Techniques are given to show that the problems LOG DOMINATING SET and LOG HYPERGRAPH VERTEX COVER cannot be approximated to a constant ratio in polynomial time unless the corresponding NP-hard versions are also approximable in deterministic subexponential time. A direct connection is established between non-NP-hard problems and a PCP characterization of NP. Reductions from the PCP characterization show that LOG CLIQUE is not approximable in polynomial time and MAX SPARSE SAT does not have a FPTAS under the assumption that NP⊈DTIME(2O(√ of n log n)). A number of nontrivial approximation-preserving reductions are also presented, making it possible to extend inapproximability results to more natural non-NP-hard problems such as TOURNAMENT DOMINATING SET and RICH HYPERGRAPH VERTEX COVER. [Copyright &y& Elsevier]