Tractable Criterion Research Articles

Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-Complexity

We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class C of UCQs, the problem #UCQ (C) provides as input a UCQ Ψ ∈ C and a database D and the problem is to compute the number of answers of Ψ in D. Chen and Mengel [PODS'16] have shown that for any recursively enumerable class C, the problem #UCQ (C) is either fixed-parameter tractable or hard for one of the parameterised complexity classes W[1] or #W[1]. However, their tractability criterion is unwieldy in the sense that, given any concrete class C of UCQs, it is not easy to determine how hard it is to count answers to queries in C. Moreover, given a single specific UCQ Ψ, it is not easy to determine how hard it is to count answers to Ψ. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ Ψ=φ 1 ∨ ... ∨ φ l is the conjunctive query ^ Ψ = φ_1 ∧ ... ∧ φ l . We show that under natural closure properties of C, the problem #UCQ (C) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in C, and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables --- if all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of #UCQ (C) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on C is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ Ψ, the meta problem of deciding whether #UCQ (Ψ) can be solved in time O(|D| d ) is NP-hard for any fixed d ≥ 1. Moreover, we prove that a known exponential-time algorithm for solving the meta problem is optimal under assumptions from fine-grained complexity theory. As a corollary of our reduction, we also establish that approximating the Weisfeiler-Leman-Dimension of a UCQ is NP-hard.

New Results on the Unimodular Equivalence of Multivariate Polynomial Matrices

The equivalence of systems is a crucial concept in multidimensional systems. The Smith normal forms of multivariate polynomial matrices play important roles in the theory of polynomial matrices. In this paper, we mainly study the unimodular equivalence of some special kinds of multivariate polynomial matrices and obtain some tractable criteria under which such matrices are unimodular equivalent to their Smith normal forms. We propose an algorithm for reducing such nD polynomial matrices to their Smith normal forms and present an example to illustrate the availability of the algorithm. Furthermore, we extend the results to the non-square case.

Exponential stability of singularly perturbed systems with mixed impulses

This paper investigates the exponential stability problem for a class of singularly perturbed impulsive systems in which the flow dynamics is unstable and is affected at discrete time instants by impulses that have both destabilizing and stabilizing effects. More precisely the impulses have stabilizing effects on the slow variables but destabilizing effects on the fast ones. Thus, a first contribution of our work is related to stability analysis of singularly perturbed impulsive systems in the case when neither the flow dynamics nor the impulsive one is stable. In order to take full advantage of the jump matrix structure and its stabilizing effects on the slow dynamics, we introduce a new impulse-dependent vector Lyapunov function. This function allows us to better describe the behavior between two consecutive impulses as well as the jumps at impulse instants. Several numerically tractable criteria for stability of singularly perturbed impulsive systems are established based on vector comparison principle. Additionally, upper bounds on the singular perturbation parameter are derived. Finally, the validity of our results is verified by two numerical examples.

Dynamic event-triggered scheduling and control for vehicle active suspension over controller area network

This paper addresses the problem of event-triggered scheduling and control for active suspension of uncertain quarter vehicle suspension systems over resource-constrained controller area network (CAN). The central aim is to develop an efficient dynamic event-triggered scheduling and control co-design approach for improving suspension performance, while significantly alleviating the resource occupancy over CAN in the simultaneous presence of transmission delays and data losses. To this end, two new dynamic event-triggered schedulers (DETSs) are first developed to orchestrate the transmissions of sensor data packets at each sampling time. One salient feature of the DETSs is that the dynamic variables employed in the relevant triggering laws can unremittingly adjust their values over time, which greatly favors the dynamic scheduling of data transmissions over CAN. Then, tractable criteria for stability and suspension performance analysis as well as criteria for co-designing the desired controllers and DETSs are derived. It is shown that the co-design criteria enable comprehensive trade-off analysis between network quality of service and control quality of performance and communication efficiency. Finally, simulation results for a two-degree-of-freedom quarter vehicle model demonstrate possible improvements on ride comfort, good road holding, limited suspension stroke and actuator force, meanwhile saving a large portion of communication resource expenditure.

Characterizing Tractability of Simple Well-Designed Pattern Trees with Projection

We study the complexity of evaluating well-designed pattern trees, a query language extending conjunctive queries with the possibility to define parts of the query to be optional. This possibility of optional parts is important for obtaining meaningful results over incomplete data sources as it is common in semantic web settings. Recently, a structural characterization of the classes of well-designed pattern trees that can be evaluated in polynomial time was shown. However, projection—a central feature of many query languages—was not considered in this study. We work towards closing this gap by giving a characterization of all tractable classes of simple well-designed pattern trees with projection (under some common complexity theoretic assumptions). Since well-designed pattern trees correspond to the fragment of well-designed {AND, OPTIONAL}-SPARQL queries this gives a complete description of the tractable classes of queries with projections in this fragment that can be characterized by the underlying graph structures of the queries. For non-simple pattern trees the tractability criteria for simple pattern trees do not capture all tractable classes. We thus extend the characterization for the non-simple case in order to capture some additional tractable cases.

Dynamic Output Feedback H∞ Control for Linear Parameter-varying Systems with Time-delay

In this paper, we address a synthesis problem of the parameter-dependent output feedback H∞ control for linear parameter-varying systems with time-delay. The scheme adopts the parameter-dependent past state information to construct the dynamic output feedback controller. In this case, on basis of the quadratic Lyapunov functional with parameter-dependence, we analyze the parameter-dependent H∞ stability conditions for the closed-loop time-delayed linear parameter-varying system in terms of linear matrix inequalities. However, this stability condition is of an infinite-dimension. To derive computationally tractable criteria for the dynamic output feedback controller, several slack variables and a convex relaxation technique are employed to have the infinite-dimensional condition of linear matrix inequalities cast into a finite dimensional convex optimization problem. By solving the convex optimization problem, we harvest the dynamic output feedback controller with memory for the time-delayed linear parameter-varying system. Finally, two examples are included to illustrate the effectiveness of the proposed approach.

Deterministic construction of array QC CS measurement matrices based on Singer perfect difference sets

Low-density parity-check (LDPC) codes and compressed sensing (CS) share many common environments. In this study, a novel approach for constructing a new class of deterministic sparse sensing matrices based on array quasi-cyclic (QC) LDPC codes via Singer perfect difference sets is proposed. In contrast to random and the other deterministic matrices, the proposed framework would be highly desirable as it is generated based on circulant permutation matrices, which requires less memory for storage and lower computational cost for sensing. Since the restricted isometric property is very difficult to verify, then the mutual coherence and the girth are two computationally tractable criteria that the authors used to assess the CS recovery capabilities of sensing matrices. In addition, inspired by LDPC codes, they extract a necessary condition for the proposed measurement matrix to have effective values for girth as large as g ≥ 6 and 8 . Comprehensive one-dimensional (1D) and 2D simulations verify that their proposed sensing matrix has minimum coherence and superior CS recovery abilities in comparison with the corresponding random Gaussian, Bernoulli, and the other deterministically generated matrices. Furthermore, the required physical storage space and the complexity of the hardware implementation are greatly reduced due to being sparse and QC in structure.

Input Design for Kernel-Based System Identification From the Viewpoint of Frequency Response

This paper discusses a method for designing input sequences for kernel-based system identification methods from the frequency perspective. The goal of this paper is to minimize the posterior uncertainty of the spectrum over the frequency band of the interest. A tractable criterion for this purpose is proposed, which is related to the so-called Bayesian A-optimality. An online algorithm that gives a suboptimal input for this criterion is proposed. Moreover, it is shown that the optimal solution can be obtained in an offline manner under a certain condition. The effectiveness of these methods is demonstrated through numerical simulations.

Smelling objects

Objects are central to perception and our interactions with the world. We perceive the world as parsed into discrete entities that instantiate particular properties, and these items capture our attention and shape how we interact with the environment. Recently there has been some debate about whether the sense of smell allows us to perceive odours as discrete objects, with some suggesting that olfaction is aspatial and doesn’t allow for object-individuation. This paper offers two empirically tractable criteria for assessing whether particular objects are exhibited in perceptual experience—(1) susceptibility to figure-ground segregation and (2) perceptual constancies—and argues that these criteria are fulfilled by olfactory perception, and thus there are olfactory objects. I argue that there are, in fact, two different ways that olfaction allows for figure-ground segregation. First, I look at various Gestalt grouping principles, which are thought to govern when features are perceived as grouped into structured wholes, segregated from everything around them. I argue that these principles apply to olfactory experience, providing evidence of non-spatial figure-ground segregation. Second, I defend the contentious idea that a spatial variety of figure-ground segregation can also occur in olfaction. To see this, however, we need to look to empirical evidence showing that tactile stimulation and bodily movements play a crucial role in olfactory phenomenology. Finally, I draw on empirical evidence and olfactory phenomenology to argue that there are perceptual constancies in olfactory experience, allowing us to perceive odours as coherent objects that survive shifts in our perspectives on the world.

On the acylindrical hyperbolicity of the tame automorphism group of SL2(C)

We introduce the notion of \"uber-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of non-simplicity, of groups acting on non locally compact spaces of arbitrary dimension. We also give a simple local criterion to construct \"uber-contracting elements for groups acting on complexes with unbounded links of vertices. As an application, we show the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$, a subgroup of the $3$-dimensional Cremona group, through its action on a CAT(0) square complex recently introduced by Bisi-Furter-Lamy.

An Erd\xf6s\u2013R\xe9v\xe9sz type law of the iterated logarithm for reflected fractional Brownian motion

Let B_{H}={B_{H}(t):tin mathbb R} be a fractional Brownian motion with Hurst parameter H ∈ (0,1). For the stationary storage process Q_{B_{H}}(t)=sup _{-infty <sle t}(B_{H}(t)-B_{H}(s)-(t-s)), t ≥ 0, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function f, {mathbb P(Q_{B_{H}}(t) > f(t), text { i.o.})} equals 0 or 1. Using this criterion we find that, for a family of functions fp(t), such that z_{p}(t)=mathbb P(sup _{sin [0,f_{p}(t)]}Q_{B_{H}}(s)>f_{p}(t))/f_{p}(t)=mathcal C(tlog ^{1-p} t)^{-1}, for some mathcal C>0, {mathbb P(Q_{B_{H}}(t) > f_{p}(t), text { i.o.})= 1_{{pge 0}}}. Consequently, with xi _{p} (t) = sup {s:0le sle t, Q_{B_{H}}(s)ge f_{p}(s)}, for p ≥ 0, lim _{tto infty }xi _{p}(t)=infty and limsup _{tto infty }(xi _{p}(t)-t)=0 a.s. Complementary, we prove an Erdös–Révész type law of the iterated logarithm lower bound on ξp(t), i.e., liminf _{tto infty }(xi _{p}(t)-t)/h_{p}(t) = -1 a.s., p > 1; liminf _{tto infty }log (xi _{p}(t)/t)/(h_{p}(t)/t) = -1 a.s., p ∈ (0,1], where hp(t) = (1/zp(t))p loglog t.

Structure Learning of Contextual Markov Networks using Marginal Pseudo‐likelihood

AbstractMarkov networks are popular models for discrete multivariate systems where the dependence structure of the variables is specified by an undirected graph. To allow for more expressive dependence structures, several generalizations of Markov networks have been proposed. Here, we consider the class of contextual Markov networks which takes into account possible context‐specific independences among pairs of variables. Structure learning of contextual Markov networks is very challenging due to the extremely large number of possible structures. One of the main challenges has been to design a score, by which a structure can be assessed in terms of model fit related to complexity, without assuming chordality. Here, we introduce the marginal pseudo‐likelihood as an analytically tractable criterion for general contextual Markov networks. Our criterion is shown to yield a consistent structure estimator. Experiments demonstrate the favourable properties of our method in terms of predictive accuracy of the inferred models.

An Erd\xf6s\u2013R\xe9v\xe9sz Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let {X(t):tin mathbb R_+} be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, mathbb E X(t) = 0, mathbb E X^2(t) = 1 and correlation function satisfying (i) r(t) = 1 - C|t|^{alpha } + o(|t|^{alpha }) as trightarrow 0 for some 0le alpha le 2 and C>0; (ii) sup _{tge s}|r(t)|<1 for each s>0 and (iii) r(t) = O(t^{-lambda }) as trightarrow infty for some lambda >0. For any nge 1, consider n mutually independent copies of X and denote by {X_{r:n}(t):tge 0} the rth smallest order statistics process, 1le rle n. We provide a tractable criterion for assessing whether, for any positive, non-decreasing function f, mathbb P(mathscr {E}_f)=mathbb P(X_{r:n}(t) > f(t), text { i.o.}) equals 0 or 1. Using this criterion we find, for a family of functions f_p(t) such that z_p(t)=mathbb P(sup _{sin [0,1]}X_{r:n}(s)>f_p(t))=O((tlog ^{1-p} t)^{-1}), that mathbb P(mathscr {E}_{f_p})= 1_{{pge 0}}. Consequently, with xi _p (t) = sup {s:0le sle t, X_{r:n}(s)ge f_p(s)}, for pge 0 we have lim _{trightarrow infty }xi _p(t)=infty and limsup _{trightarrow infty }(xi _p(t)-t)=0 a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on xi _p(t), namely, that liminf _{trightarrow infty }(xi _p(t)-t)/h_p(t) = -1 a.s. for p>1 and liminf _{trightarrow infty }log (xi _p(t)/t)/(h_p(t)/t) = -1 a.s. for pin (0,1], where h_p(t)=(1/z_p(t))plog log t.

Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control

In this paper we investigate the stability analysis of nonlinear sampled-data systems, which are affine in the input. We assume that a stabilizing controller is designed using the emulation technique. We intend to provide sufficient stability conditions for the resulting sampled-data system. This allows to find an estimate of the upper bound on the asynchronous sampling intervals, for which stability is ensured. The main idea of the paper is to address the stability problem in a new framework inspired by the dissipativity theory. Furthermore, the result is shown to be constructive. Numerically tractable criteria are derived using linear matrix inequality for polytopic systems and using sum of squares technique for the class of polynomial systems.

A Complete Dichotomy Rises from the Capture of Vanishing Signatures

We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions $\mathcal{F}$ on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions (taking values without a finite modulus). We define and characterize all symmetric vanishing signatures; they turn out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations. A Holant problem defined by a set of constraint functions $\mathcal{F}$ is solvable in polynomial time if it satisfies this tractability criterion, and is #P-hard otherwise. The tractability criterion can be intuitively stated as follows: A set $\mathcal{F}$ is tractable if (1) every function in $\mathcal{F}$ has arity at most two; or (2) $\mathcal{F}$ is transformable to an affine type; or (3) $\mathcal{F}$ is transformable to a product type; o...

Nonnegative Weighted #CSP: An Effective Complexity Dichotomy

We prove a complexity dichotomy theorem for counting constraint satisfaction problems (#CSPs) with nonnegative and algebraic weights. This caps a long series of important results on counting problems including counting unweighted and weighted graph homomorphisms and the celebrated dichotomy theorem for unweighted #CSPs. Our dichotomy theorem gives a succinct criterion for tractability. If a set $\mathcal{F}$ of constraint functions satisfies this criterion, then the problem #CSP$(\mathcal{F})$ defined by $\mathcal{F}$ is solvable in polynomial time; if $\mathcal{F}$ does not satisfy this criterion, then the problem is #P-hard. Furthermore, we show that the question of whether a given $\mathcal{F}$ satisfies the criterion or not is decidable in NP. Surprisingly, our tractability criterion is simpler than the previous criteria for the more restricted classes of counting problems, although when specialized to those classes, they are logically equivalent. Our proof mainly uses linear algebra and represents a departure from universal algebra, the dominant methodology in recent years for the study of #CSPs on large domains.

Hyponormality of bounded‐type Toeplitz operators

In this paper we deal with the hyponormality of Toeplitz operators with matrix‐valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded‐type Toeplitz operators (i.e., the symbol is a matrix‐valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co‐analytic part of the symbol Φ is a left divisor of the analytic part.

Graph Homomorphisms with Complex Values: A Dichotomy Theorem

Each symmetric matrix $\mathbf{A}$ over $\mathbb{C}$ defines a graph homomorphism function $Z_{\bf A}(\cdot)$ on undirected graphs. The function $Z_{\mathbf{A}} (\cdot)$ is also called the partition function from statistical physics, and can encode many interesting graph properties, including counting vertex covers and $k$-colorings. We study the computational complexity of $Z_{\mathbf{A}} (\cdot)$ for arbitrary symmetric matrices $\mathbf{A}$ with algebraic complex values. Building on work by Dyer and Greenhill [Random Structures and Algorithms, 17 (2000), pp. 260--289], Bulatov and Grohe [Theoretical Computer Science, 348 (2005), pp. 148--186], and especially the recent beautiful work by Goldberg et al. [SIAM J. Comput., 39 (2010), pp. 3336--3402], we prove a complete dichotomy theorem for this problem. We show that $Z_{\mathbf{A}} (\cdot)$ is either computable in polynomial-time or \#P-hard, depending explicitly on the matrix $\mathbf{A}$. We further prove that the tractability criterion on $\mathbf{A}$ is polynomial-time decidable.

A Delay-Dependent Approach to Stability of Uncertain Discrete-Time State-Delayed Systems with Generalized Overflow Nonlinearities

This paper addresses the problem of global asymptotic stability of a class of uncertain discrete-time state-delayed systems employing generalized overflow nonlinearities. The systems under investigation involve parameter uncertainties that are assumed to be deterministic and norm bounded. A new computationally tractable delay-dependent criterion for global asymptotic stability of such systems is presented. A numerical example is given to illustrate the effectiveness of the proposed method.

A Complexity Dichotomy for Partition Functions with Mixed Signs

Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colorings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill [Random Structures Algorithms, 17 (2000), pp. 260–289] and Bulatov and Grohe [Theoret. Comput. Sci., 348 (2005), pp. 148–186], we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by Hadamard matrices (these turn out to be central in our proofs) we obtain a simple algebraic tractability criterion, which says that the tractable cases are those “representable” by a quadratic polynomial over the field $\mathbb{F}_2$.