Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

The Induced Graph Matching problem asks to find \(k\) disjoint induced subgraphs isomorphic to a given graph  \(H\) in a given graph \(G\) such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in \(k\) on claw-free graphs when \(H\) is a fixed connected graph, and even admits a polynomial kernel when  \(H\) is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove \(\mathsf {W}[1]\) -hardness of Induced Graph Matching on graphs excluding \(K_{1,4}\) as an induced subgraph, for any fixed complete graph \(H\) . In particular, we show that Independent Set is \(\mathsf {W}[1]\) -hard on \(K_{1,4}\) -free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or \(\mathsf {NP}\) -complete, depending on the connectivity of \(H\) and the structure of \(G\) .     

Design patterns percolating to parallel programming framework implementation

Structured parallel programming is recognised as a viable and effective means of tackling parallel programming problems. Recently, a set of simple and powerful parallel building blocks ( \(\mathsf{RISC\text{- }pb^2l}\) ) has been proposed to support modelling and implementation of parallel frameworks. In this work we demonstrate how that same parallel building block set may be used to model both general purpose parallel programming abstractions, not usually listed in classical skeleton sets, and more specialized domain specific parallel patterns. We show how an implementation of \(\mathsf{RISC\text{- }pb^2l}\) can be realised via the FastFlow framework and present experimental evidence of the feasibility and efficiency of the approach.     

The categories L-\mathbf{Top}_{0} and L-\mathbf{Sob} as epireflective hulls

We show that the category \(L\) - \(\mathbf{Top}_{0}\) of \(T_{0}\) - \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}\) of \(L\) -topological spaces and the category \(L\) - \(\mathbf{Sob}\) of sober \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}_{0}\) .     

Fault tolerance of hypercubes and folded hypercubes

Let \(G = (V,E)\) be a connected graph. The conditional edge connectivity \(\lambda _\delta ^k(G)\) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects \(G\) and each component of \(G - F\) has \(\delta \ge k\) . We assume that \(F \subseteq E\) is an edge set, \(F\) is called edge extra-cut, if \(G - F\) is not connected and each component of \(G - F\) has more than \(k\) vertices. The edge extraconnectivity \(\lambda _\mathrm{e}^k(G)\) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.     

Accelerating 2D orthogonal matching pursuit algorithm on GPU

Two-dimensional orthogonal matching pursuit (2D-OMP) algorithm is an extension of the one-dimensional OMP (1D-OMP), whose complexity and memory usage are lower than the 1D-OMP when they are applied to 2D sparse signal recovery. However, the major shortcoming of the 2D-OMP still resides in long computing time. To overcome this disadvantage, we develop a novel parallel design strategy of the 2D-OMP algorithm on a graphics processing unit (GPU) in this paper. We first analyze the complexity of the 2D-OMP and point out that the bottlenecks lie in matrix inverse and projection. After adopting the strategy of matrix inverse update whose performance is superior to traditional methods to reduce the complexity of original matrix inverse, projection becomes the most time-consuming module. Hence, a parallel matrix–matrix multiplication leveraging tiling algorithm strategy is launched to accelerate projection computation on GPU. Moreover, a fast matrix–vector multiplication, a parallel reduction algorithm, and some other parallel skills are also exploited to boost the performance of the 2D-OMP further on GPU. In the case of the sensing matrix of size 128 \(\times \) 256 (176 \(\times \) 256, resp.) for a 256 \(\times \) 256 scale image, experimental results show that the parallel 2D-OMP achieves 17 \(\times \) to 41 \(\times \) (24 \(\times \) to 62 \(\times \) , resp.) speedup over the original C code compiled with the O \(_2\) optimization option. Higher speedup would be further obtained with larger-size image recovery.     

Parallelizing Complex Streaming Applications on Distributed Scratchpad Memory Multicore Architecture

Multicore processors can provide sufficient computing power and flexibility for complex streaming applications, such as high-definition video processing. For less hardware complexity and power consumption, the distributed scratchpad memory architecture is considered, instead of the cache memory architecture. However, the distributed design poses new challenges to programming. It is difficult to exploit all available capabilities and achieve maximal throughput, due to the combined complexity of inter-processor communication, synchronization, and workload balancing. In this study, we developed an efficient design flow for parallelizing multimedia applications on a distributed scratchpad memory multicore architecture. An application is first partitioned into streaming components and then mapped onto multicore processors. Various hardware-dependent factors and application-specific characteristics are involved in generating efficient task partitions and allocating resources appropriately. To test and verify the proposed design flow, three popular multimedia applications were implemented: a full-HD motion JPEG decoder, an object detector, and a full-HD H.264/AVC decoder. For demonstration purposes, SONY PlayStation \(^{\circledR }\) 3 was selected as the target platform. Simulation results show that, on PS3, the full-HD motion JPEG decoder with the proposed design flow can decode about 108.9 frames per second (fps) in the 1080p format. The object detection application can perform real-time object detection at 2.84 fps at \(1280 \times 960\) resolution, 11.75 fps at \(640 \times 480\) resolution, and 62.52 fps at \(320 \times 240\) resolution. The full-HD H.264/AVC decoder applications can achieve nearly 50 fps.     

A note on the growth factor in Gaussian elimination for accretive-dissipative matrices

This short note proves that if \(A\) is accretive-dissipative, then the growth factor for such \(A\) in Gaussian elimination is less than \(4\) . If \(A\) is a Higham matrix, i.e., the accretive-dissipative matrix \(A\) is complex symmetric, then the growth factor is less than \(2\sqrt{2}\) . The result obtained improves those of George et al. in [Numer. Linear Algebra Appl. 9, 107–114 (2002)] and is one step closer to the final solution of Higham’s conjecture.     

Solving time-invariant differential matrix Riccati equations using GPGPU computing

Differential matrix Riccati equations (DMREs) enable to model many physical systems appearing in different branches of science, in some cases, involving very large problem sizes. In this paper, we propose an adaptive algorithm for time-invariant DMREs that uses a piecewise-linearized approach based on the Padé approximation of the matrix exponential. The algorithm designed is based upon intensive use of matrix products and linear system solutions so we can seize the large computational capability that modern graphics processing units (GPUs) have on these types of operations using CUBLAS and CULATOOLS libraries (general purpose GPU), which are efficient implementations of BLAS and LAPACK libraries, respectively, for NVIDIA \(\copyright \) GPUs. A thorough analysis showed that some parts of the algorithm proposed can be carried out in parallel, thus allowing to leverage the two GPUs available in many current compute nodes. Besides, our algorithm can be used by any interested researcher through a friendly MATLAB \(\copyright \) interface.     

pROST: a smoothed \ell _p-norm robust online subspace tracking method for background subtraction in video

An increasing number of methods for background subtraction use Robust PCA to identify sparse foreground objects. While many algorithms use the \(\ell _1\) -norm as a convex relaxation of the ideal sparsifying function, we approach the problem with a smoothed \(\ell _p\) -quasi-norm and present pROST, a method for robust online subspace tracking. The algorithm is based on alternating minimization on manifolds. Implemented on a graphics processing unit, it achieves realtime performance at a resolution of \(160 \times 120\) . Experimental results on a state-of-the-art benchmark for background subtraction on real-world video data indicate that the method succeeds at a broad variety of background subtraction scenarios, and it outperforms competing approaches when video quality is deteriorated by camera jitter.     

Routing Regardless of Network Stability

How effective are interdomain routing protocols, such as the border gateway protocol, at routing packets? Theoretical analyses have attempted to answer this question by ignoring the packets and instead focusing upon protocol stability. To study stability, it suffices to model only the control plane (which determines the routing graph)—an approach taken in the stable paths problem. To analyse packet routing requires modelling the interactions between the control plane and the forwarding plane (which determines where packets are forwarded), and our first contribution is to introduce such a model. We then examine the effectiveness of packet routing in this model for the broad class next-hop preferences with filtering. Here each node \(v\) has a filtering list \(\mathcal {D}(v)\) consisting of nodes it does not want its packets to route through. Acceptable paths (those that avoid nodes in the filtering list) are ranked according to the next-hop, that is, the neighbour of \(v\) that the path begins with. On the negative side, we present a strong inapproximability result. For filtering lists of cardinality at most one, given a network in which an equilibrium is guaranteed to exist, it is NP-hard to approximate the maximum number of packets that can be routed to within a factor of \(n^{1-\epsilon }\) , for any constant \(\epsilon >0\) . On the positive side, we give algorithms to show that, in two fundamental cases, there exist activation sequences under which every packet will route. The first case is when each node’s filtering list contains only itself, that is, \(\mathcal {D}(v)=\{v\}\) ; this is the fundamental case in which a node does not want its packets to cycle. Moreover, every packet will be routed before the control plane reaches an equilibrium. The second case is when all the filtering lists are empty, that is, \(\mathcal {D}(v)=\emptyset \) . Thus, every packet will route even when the nodes do not care if their packets cycle! Furthermore, under these activation sequences, every packet will route even when the control plane has no equilibrium at all. Our positive results require the periodic application of route verification. To our knowledge, these are the first results to guarantee the possibility that all packets get routed without stability. These positive results are tight—for the general case of filtering lists of cardinality one, it is not possible to ensure that every packet will eventually route.     

Multidimensional gravity,flux and black brane solutions governed by polynomials

Two families of composite black brane solutions are overviewed, fluxbrane and black brane ones, in a model with scalar fields and fields of forms. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat “internal” spaces. The solutions are governed by moduli functions \(\mathcal{H}_s \) (for fluxbranes) and H _s (for black branes), obeying nonlinear differential equations with certain boundary conditions. Themaster equations for \(\mathcal{H}_s \) and H _s are equivalent to Toda-like equations and depend on a nondegenerate matrix A related to brane intersection rules. The functions H _s and \(\mathcal{H}_s \) , as was conjectured and confirmed (at least partly) earlier, should be polynomials in proper variables if A is a Cartan matrix of some semisimple finite-dimensional Lie algebra. The fluxbrane polynomials \(\mathcal{H}_s \) were shown to be used for the construction of black brane polynomials H _s . This approach is illustrated by examples of nonextremal electric black p-brane solutions related to Lie algebras A ₂, C ₂, and G ₂.     

Nonlinear features of heart rate variability in paranoid schizophrenic

Cardiovascular mortality is significantly increased in patients suffering from schizophrenia. However, psychotic symptoms are quantified by means of the scale for the assessment of positive and negative symptoms, but many investigations try to introduce new etiology for psychiatric disorders based on combination of biological, psychological and social causes. Classification between healthy and paranoid cases has been achieved by time, frequency, Hilbert–Huang (HH) and a combination between those features as a hybrid features. Those features extracted from the Hilbert–Huang transform for each intrinsic mode function (IMF) of the detrended time series for each healthy case and paranoid case. Short-term ECG recordings of 20 unmedicated patients suffering from acute paranoid schizophrenia and those obtained from healthy matched peers have been utilized in this investigation. Frequency features: very low frequency (VLF), low frequency (LF), high frequency (HF) and HF/LF (ratio) produced promising success rate equal to 97.82 % in training and 97.77 % success rate in validation by means of IMF1 and ninefolds. Time–frequency features [LF, HF and ratio, mean, maximum (max), minimum (min) and standard deviation (SD)] provided 100 % success in both training and validation trials by means of ninefolds for IMF1 and IMF2. By utilizing IMF1 and ninefolds, frequency and Hilbert–Hang features [LF, HF, ratio, mean value of envelope ( \(\bar{a}\) )] produced 96.87 and 95.5 % for training and validation, respectively. By analyzing the first IMF and using ninefolds, time and Hilbert–Hang features [mean, max, min, SD, median, first quartile (Q1), third quartile (Q3), kurtosis, skewness, Shannon entropy, approximate entropy and energy, ( \(\bar{a}\) ), level of envelope variation ([ \(\dot{a}\) (t)]^2), central frequency \((\bar{W})\) and number of zero signal crossing \((\left| {\bar{W}} \right|)\) ] produced a 100 % success rate in training and 90 % success rate in validation. Time, frequency and HH features [energy, VLF, LF, HF, ratio and ( \(\bar{a}\) )] provided 97.5 % success rate in training and 95.24 % success rate in validation using IMF1 and sixfolds. However, frequency features have produced promising classification success rate, but hybrid features emerged the highest classification success rate than using features in each domain separately.     

Łukasiewicz logic and Riesz spaces

We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from \([0,1]\) . Extending Mundici’s equivalence between MV-algebras and \(\ell \) -groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C \(^*\) -algebras. The propositional calculus \({\mathbb R}{\mathcal L}\) that has Riesz MV-algebras as models is a conservative extension of ?ukasiewicz \(\infty \) -valued propositional calculus and is complete with respect to evaluations in the standard model \([0,1]\) . We prove a normal form theorem for this logic, extending McNaughton theorem for ? ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in \({\mathbb R}{\mathcal L},\) and relate them with the analogue of de Finetti’s coherence criterion for \({\mathbb R}{\mathcal L}\) .     

On requirement verification for evolving Statecharts specifications

Software development processes have been evolving from rigid, pre-specified, and sequential to incremental, and iterative. This evolution has been dictated by the need to accommodate evolving user requirements and reduce the delay between design decision and feedback from users. Formal verification techniques, however, have largely ignored this evolution and even when they made enormous improvements and found significant uses in practice, like in the case of model checking, they remained confined into the niches of safety-critical systems. Model checking verifies if a system’s model \(\mathcal{M}\) satisfies a set of requirements, formalized as a set of logic properties \(\Phi\) . Current model-checking approaches, however, implicitly rely on the assumption that both the complete model \(\mathcal{M}\) and the whole set of properties \(\Phi\) are fully specified when verification takes place. Very often, however, \(\mathcal{M}\) is subject to change because its development is iterative and its definition evolves through stages of incompleteness, where alternative design decisions are explored, typically to evaluate some quality trade-offs. Evolving systems specifications of this kind ask for novel verification approaches that tolerate incompleteness and support incremental analysis of alternative designs for certain functionalities. This is exactly the focus of this paper, which develops an incremental model-checking approach for evolving Statecharts. Statecharts have been chosen both because they are increasingly used in practice natively support model refinements.     

Master Lovas–Andai and equivalent formulas verifying the $$\frac{8}{33}$$ two-qubit Hilbert–Schmidt separability probability and companion rational-valued conjectures

We begin by investigating relationships between two forms of Hilbert–Schmidt two-rebit and two-qubit “separability functions”—those recently advanced by Lovas and Andai (J Phys A Math Theor 50(29):295303, 2017), and those earlier presented by Slater (J Phys A 40(47):14279, 2007). In the Lovas–Andai framework, the independent variable \(\varepsilon \in [0,1]\) is the ratio \(\sigma (V)\) of the singular values of the \(2 \times 2\) matrix \(V=D_2^{1/2} D_1^{-1/2}\) formed from the two \(2 \times 2\) diagonal blocks (\(D_1, D_2\)) of a \(4 \times 4\) density matrix \(D= \left||\rho _{ij}\right||\). In the Slater setting, the independent variable \(\mu \) is the diagonal-entry ratio \(\sqrt{\frac{\rho _{11} \rho _ {44}}{\rho _ {22} \rho _ {33}}}\)—with, of central importance, \(\mu =\varepsilon \) or \(\mu =\frac{1}{\varepsilon }\) when both \(D_1\) and \(D_2\) are themselves diagonal. Lovas and Andai established that their two-rebit “separability function” \(\tilde{\chi }_1 (\varepsilon )\) (\(\approx \varepsilon \)) yields the previously conjectured Hilbert–Schmidt separability probability of \(\frac{29}{64}\). We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we newly find its two-qubit, two-quater[nionic]-bit and “two-octo[nionic]-bit” counterparts, \(\tilde{\chi _2}(\varepsilon ) =\frac{1}{3} \varepsilon ^2 \left( 4-\varepsilon ^2\right) \), \(\tilde{\chi _4}(\varepsilon ) =\frac{1}{35} \varepsilon ^4 \left( 15 \varepsilon ^4-64 \varepsilon ^2+84\right) \) and \(\tilde{\chi _8} (\varepsilon )= \frac{1}{1287}\varepsilon ^8 \left( 1155 \varepsilon ^8-7680 \varepsilon ^6+20160 \varepsilon ^4-25088 \varepsilon ^2+12740\right) \). These immediately lead to predictions of Hilbert–Schmidt separability/PPT-probabilities of \(\frac{8}{33}\), \(\frac{26}{323}\) and \(\frac{44482}{4091349}\), in full agreement with those of the “concise formula” (Slater in J Phys A 46:445302, 2013), and, additionally, of a “specialized induced measure” formula. Then, we find a Lovas–Andai “master formula,” \(\tilde{\chi _d}(\varepsilon )= \frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left( -\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right) }{\Gamma \left( \frac{d}{2}+1\right) ^2}\), encompassing both even and odd values of d. Remarkably, we are able to obtain the \(\tilde{\chi _d}(\varepsilon )\) formulas, \(d=1,2,4\), applicable to full (9-, 15-, 27-) dimensional sets of density matrices, by analyzing (6-, 9, 15-) dimensional sets, with not only diagonal \(D_1\) and \(D_2\), but also an additional pair of nullified entries. Nullification of a further pair still leads to X-matrices, for which a distinctly different, simple Dyson-index phenomenon is noted. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence. A two-qubit separability probability of \(1-\frac{256}{27 \pi ^2}\) is obtained based on the operator monotone function \(\sqrt{x}\), with the use of \(\tilde{\chi _2}(\varepsilon )\).     

Hermes: a simple and efficient algorithm for building the AOC-poset of a binary relation

Given a relation ?? ? ?? × ?? on a set ?? of objects and a set ?? of attributes, the AOC-poset (Attribute/Object Concept poset), is the partial order defined on the “introducers” of objects and attributes in the corresponding concept lattice. In this paper, we present Hermes, a simple and efficient algorithm for building an AOC-poset which runs in O(m i n{n m, n ^α }), where n is the number of objects plus the number of attributes, m is the size of the relation, and n ^α is the time required to perform matrix multiplication (currently α = 2.376). Finally, we compare the runtime of Hermes with the runtime of other algorithms computing the AOC-poset: Ares, Ceres and Pluton. We characterize the cases where each algorithm is the more relevant.     

Nonperiodic Trigonometric Polynomial Approximation

The most common approach for approximating non-periodic function defined on a finite interval is based on considering polynomials as basis functions. In this paper we will address the non-optimallity of polynomial approximation and suggest to switch from powers of \(x\) to powers of \(\sin (px)\) where \(p\) is a parameter which depends on the dimension of the approximating subspace. The new set does not suffer from the drawbacks of polynomial approximation and by using them one can approximate analytic functions with spectral accuracy. An important application of the new basis functions is related to numerical integration. A quadrature based on these functions results in higher accuracy compared to Legendre quadrature.     

Parameterized complexity of three edge contraction problems with degree constraints

For any graph class \(\mathcal{H}\) , the \(\mathcal{H}\) -Contraction problem takes as input a graph \(G\) and an integer \(k\) , and asks whether there exists a graph \(H\in \mathcal{H}\) such that \(G\) can be modified into \(H\) using at most \(k\) edge contractions. We study the parameterized complexity of \(\mathcal{H}\) -Contraction for three different classes \(\mathcal{H}\) : the class \(\mathcal{H}_{\le d}\) of graphs with maximum degree at most  \(d\) , the class \(\mathcal{H}_{=d}\) of \(d\) -regular graphs, and the class of \(d\) -degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters \(k\) , \(d\) , and \(d+k\) . Moreover, we show that \(\mathcal{H}\) -Contraction admits an \(O(k)\) vertex kernel on connected graphs when \(\mathcal{H}\in \{\mathcal{H}_{\le 2},\mathcal{H}_{=2}\}\) , while the problem is \(\mathsf{W}[2]\) -hard when \(\mathcal{H}\) is the class of \(2\) -degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that \(\mathcal{H}\) -Contraction admits a linear vertex kernel when \(\mathcal{H}\) is the class of cycles.