Polynomial Complexity Research Articles

Assessing and Restoring “Traffic-State Order” in Open, Irreversible, Dynamically Routed, Zone-Controlled Guidepath-Based Transport Systems

The notion of the ``h-ordered'' traffic state provides an efficient approach for maintaining liveness in an open, irreversible, dynamically routed zone-controlled guidepath-based transport system. The restriction of these transport systems in their class of h-ordered states can be performed with polynomial complexity with respect to the size of these systems, while the resulting supervisory control policy retains high levels of operational latitude. The work presented in this article provides novel efficient algorithms for the following two problems: 1) assessing whether a given traffic state is h-ordered; and 2) bringing the underlying transport system from some general traffic state to the class of its h-ordered states in a way that minimizes a certain measure of ``operational disruption.'' The developed algorithms are motivated by, and find immediate applicability, in the model predictive control (MPC) scheme for the considered transport systems that was developed in

Structural Controllability of Directed Signed Networks

In this article, the structural controllability of multiagent networks defined over directed signed graphs is investigated, where the agents are divided into leaders and followers, and only leaders are manipulated by the external control input directly. The communication digraph contains positive and negative edges, which represent cooperative and antagonistic interactions between agents, respectively. First, a graph-theoretic necessary and sufficient condition for structural controllability of multiagent networks is derived, that is, a multiagent network is structurally controllable if and only if the communication digraph is leader–follower connected. Next, the minimal controllability problem is studied. It is shown that the minimal controllability problem is solvable, and a method of polynomial complexity to accomplish minimal structural controllability is developed. In addition, the structural controllability of specified Cartesian product networks is discussed and a necessary and sufficient condition concerning the factor networks for the structural controllability is established. The application of the theoretical results is demonstrated by several practical examples.

Initial-state detectability and initial-state opacity of unambiguous weighted automata

In this paper, we investigate the verification problem of initial-state detectability (I-detectability) and initial-state opacity (I-opacity) in discrete event systems modeled by unambiguous weighted automata. An I-observer is constructed so as to derive necessary and sufficient conditions for checking strong I-detectability, weak I-detectability, and I-opacity, with exponential complexity. In addition, an approach based on diagnosability analysis is proposed for verifying strong I-detectability. Compared with an I-observer-based approach, the diagnosability-based approach has a lower complexity, and in the case where all the unobservable events in an unambiguous weighted automaton are represented by a unique symbol, the diagnosability-based approach has polynomial complexity.

Fixation probabilities in evolutionary dynamics under weak selection.

In evolutionary dynamics, a key measure of a mutant trait's success is the probability that it takes over the population given some initial mutant-appearance distribution. This "fixation probability" is difficult to compute in general, as it depends on the mutation's effect on the organism as well as the population's spatial structure, mating patterns, and other factors. In this study, we consider weak selection, which means that the mutation's effect on the organism is small. We obtain a weak-selection perturbation expansion of a mutant's fixation probability, from an arbitrary initial configuration of mutant and resident types. Our results apply to a broad class of stochastic evolutionary models, in which the size and spatial structure are arbitrary (but fixed). The problem of whether selection favors a given trait is thereby reduced from exponential to polynomial complexity in the population size, when selection is weak. We conclude by applying these methods to obtain new results for evolutionary dynamics on graphs.

Learning the boundary of inductive invariants

We study the complexity of invariant inference and its connections to exact concept learning. We define a condition on invariants and their geometry, called the fence condition, which permits applying theoretical results from exact concept learning to answer open problems in invariant inference theory. The condition requires the invariant's boundary---the states whose Hamming distance from the invariant is one---to be backwards reachable from the bad states in a small number of steps. Using this condition, we obtain the first polynomial complexity result for an interpolation-based invariant inference algorithm, efficiently inferring monotone DNF invariants with access to a SAT solver as an oracle. We further harness Bshouty's seminal result in concept learning to efficiently infer invariants of a larger syntactic class of invariants beyond monotone DNF. Lastly, we consider the robustness of inference under program transformations. We show that some simple transformations preserve the fence condition, and that it is sensitive to more complex transformations.

New Results on the Storage-Retrieval Tradeoff in Private Information Retrieval Systems

In a private information retrieval (PIR) system, the user needs to retrieve one of the possible messages from a set of storage servers, but wishes to keep the identity of the requested message private from any given server. Existing efforts in this area have made it clear that the efficiency of the retrieval will be impacted significantly by the amount of the storage space allowed at the servers. In this work, we consider the tradeoff between the storage cost and the retrieval cost. We first present three fundamental results: 1) a regime-wise approximate characterization of the optimal tradeoff with a factor of two, 2) a cyclic permutation lemma that can produce more sophisticated codes from simpler base codes, and 3) a relaxed entropic linear program (LP) lower bound that has a polynomial complexity. Equipped with the cyclic permutation lemma, we then propose two novel code constructions, and by applying the lemma, obtain new storage-retrieval points. Furthermore, we derive more explicit lower bounds by utilizing only a subset of the constraints in the relaxed entropic LP in a systematic manner. Though the new upper bound and lower bound do not lead to a better approximation factor uniformly, they are significantly tighter than the existing art in some regimes.

A Phase Estimation Algorithm for Quantum Speed-Up Multi-Party Computing

Security and privacy issues have attracted the attention of researchers in the field of IoT as the information processing scale grows in sensor networks. Quantum computing, theoretically known as an absolutely secure way to store and transmit information as well as a speed-up way to accelerate local or distributed classical algorithms that are hard to solve with polynomial complexity in computation or communication. In this paper, we focus on the phase estimation method that is crucial to the realization of a general multi-party computing model, which is able to be accelerated by quantum algorithms. A novel multi-party phase estimation algorithm and the related quantum circuit are proposed by using a distributed Oracle operator with iterations. The proved theoretical communication complexity of this algorithm shows it can give the phase estimation before applying multi-party computing efficiently without increasing any additional complexity. Moreover, a practical problem of multi-party dating investigated shows it can make a successful estimation of the number of solution in advance with zero communication complexity by utilizing its special statistic feature. Sufficient simulations present the correctness, validity and efficiency of the proposed estimation method.

Joint Radar-Communication Transmission: A Generalized Pareto Optimization Framework

Dual-functional radar-communication (DFRC) has been envisioned as a key enabler for a number of emerging applications. Nevertheless, the intrinsic difference between communication and sensing functionalities has imposed challenges in fully realizing the promise of the technology. In this paper, we propose a Pareto optimization framework for a multi-antenna DFRC system, where a single transmitter with multiple antennas communicates with multiple users and detects radar targets simultaneously. To begin with, we define an achievable performance region that is built upon the difference between peak and sidelobe (DPSL) of the radar spatial beampattern and the signal-to-interference-plus-noise ratio (SINR) of each communication user, which is proved to be compact and normal. Then, we formulate a fairness-profile optimization problem with a lowest acceptable performance level of dual functionalities and a pre-defined direction to attain the Pareto boundary of the achievable performance region. Also, we provide a bisection search algorithm to solve the Pareto optimization problem with polynomial complexity, and prove its optimality can be achieved by single-stream transmission. As a further step, we extend the discussion to the DFRC beamforming design given hardware impairments. Finally, we evaluate the effectiveness of the proposed design and the impact brought by the impairments by numerical simulations.

An empirical study on decoupling PNLSS models illustrated on an airplane

Abstract This paper illustrates a combined nonparametric and parametric system identification framework for modeling nonlinear vibrating structures. First step is the analysis: multiple-input multiple-output measurements are (semi-automatically) preprocessed, and a nonparametric Best Linear Approximation (BLA) method is performed. The outcome of the BLA analysis results in nonparametric frequency response function, noise and nonlinear distortion estimates. Based on this information, a linear parametric (state-space) model is built. This model is used to initialize a high complexity Polynomial Nonlinear State-Space PNLSS model. The nonlinear part of a PNLSS model is manifested as a combination of high-dimensional multivariate polynomials. The last step in the proposed approach is the decoupling: transforming multivariate polynomials into a simplified, alternative basis, thereby dramatically reducing the number of parameters. In this work a novel filtered canonical polyadic decomposition (CPD) is used. The proposed methodology is illustrated on, but of course not limited to, a ground vibration testing measurement of an air fighter.

An Artificial-Noise-Based Approach for the Secrecy Rate Maximization of MISO VLC Wiretap Channel With Multi-Eves

In this paper, we consider improving the secure performance of multiple-input single-output visible light communication channel in the presence of multiple eavesdroppers with multiple photodiodes. Our goal is to design an optimal artificial-noise (AN) aided transmission strategy to maximize the achievable secrecy rate subject to both sum power constraint and peak amplitude constraint. We consider a joint optimization of the transmit covariance and AN covariance for the non-convex secrecy rate maximization (SRM) problem. In order to solve it, the SRM problem is transformed into a series of single-variable semidefinite programming (SDP) problems without losing any optimality, and a one-dimensional search based algorithm is proposed to handle the converted problem, with polynomial complexity. By exploiting Karush-Kuhn-Tucker conditions of the problem, beamforming is found to be optimal for the confidential information transmission. Simulation results show the superior performance of the proposed AN-aided method compared with two other AN-aided methods and no AN-aided method.

Embedded Insertion Functions for Opacity Enforcement

We investigate the enforcement of opacity, an information-flow privacy property, using insertion sequences that modify the output of the system by event insertions. Previous work considered the problem of enforcing the opacity under the assumption that the insertion functions were based on the observed system strings. Now, we investigate the more powerful method of insertion sequences based on the exact system states and events. In this case, the insertion function would be embedded into the system itself, rather than being an output interface. In this article, we develop methods that verify if a valid insertion function exists in this setting; if one exists, synthesize one using a computationally effective algorithm; and investigate a special case where it is possible to verify and synthesize a valid embedded insertion function with polynomial complexity in the size of the system.

An interior point-proximal method of multipliers for convex quadratic programming

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.

EdgeDASH: Exploiting Network-Assisted Adaptive Video Streaming for Edge Caching

With increasing demand for video streaming applications, exploiting cached content at the network edge becomes paramount to prevent congestion in the link between the wireless access network and the content providers. However, it is often challenging to exploit the caches in current client-driven video streaming solutions due to two key reasons. First, even those clients interested in the same content might request different quality levels as a video content is encoded into multiple qualities to match a wide range of network conditions and device capabilities. Second, the clients, who select the quality of the next chunk to request, are unaware of the cached content at the network edge. Hence, it becomes imperative to develop network-side solutions to exploit caching, in particular for the scenarios in which multiple video clients compete for some bottleneck capacity. In this article, we propose EdgeDASH which is a network-side control logic running at a WiFi AP to facilitate the use of cached video content. In particular, an AP can assign a client station to a video quality different than its request, in case the alternative quality provides a better utility. This includes, for example, a function of bits delivered from the cache, video bit rate, and the buffer stalls. We formulate the quality assignment problem as an optimization problem and develop several heuristics with polynomial complexity. Our simulations show that EdgeDASH facilitates significant cache hits and decreases the buffer stalls only by changing the client’s request by one quality level, however with some increase in session instability. From our analysis, we conclude that EdgeDASH benefits are more visible especially when the clients with identical interests compete for a bottleneck link’s capacity, over the baseline where the clients determine the quality adaptation.

Recovering Data Permutations From Noisy Observations: The Linear Regime

This article considers a noisy data structure recovery problem. The goal is to investigate the following question: given a noisy observation of a permuted data set, according to which permutation was the original data sorted? The focus is on scenarios where data is generated according to an isotropic Gaussian distribution, and the noise is additive Gaussian with an arbitrary covariance matrix. This problem is posed within a hypothesis testing framework. The objective is to study the linear regime in which the optimal decoder has a polynomial complexity in the data size, and it declares the permutation by simply computing a permutation-independent linear function of the noisy observations. The main result of this article is a complete characterization of the linear regime in terms of the noise covariance matrix. Specifically, it is shown that this matrix must have a very flat spectrum with at most three distinct eigenvalues to induce the linear regime. Several practically relevant implications of this result are discussed, and the error probability incurred by the decision criterion in the linear regime is also characterized. A core technical component consists of using linear algebraic and geometric tools, such as Steiner symmetrization.

Critical pipeline of the acyclic wave processor

This paper is devoted to studying the performance of an acyclic wave processor consisting of heterogeneous processor elements having various delays. Data channels have sufficiently large buffers. The main result is the proof of the conjecture that for an arbitrary amount of input data, an acyclic wave processor contains a pipeline whose operating time is equal to the data processing time using the entire wave processor. This pipeline is called critical. For partially testing the conjecture, a multi-threaded application simulating the work of a two-dimensional pipeline has been developed. Earlier, the author proposed an algorithm for calculating the input processing time, the complexity of which depends not only on the number of stages but also on the amount of data. The proven conjecture eliminates this drawback, it allows building algorithms for calculating data processing time, the complexity of which does not depend on the amount of data. In particular, for any class of acyclic wave processors that have some algorithm of polynomial complexity for listing maximum pipelines, it is easy to construct a polynomial algorithm for calculating the data processing time for an arbitrary volume.

Deviation Analysis for the Max-Product Fuzzy Relation Inequalities

Max-product fuzzy relation inequalities have been widely applied in foodstuff management, wireless communication base-station system, and wireless server-to-client network system. Under such application background, any solution of the max-product fuzzy relation system represents a feasible scheme. In order to reflect the stability of a given feasible scheme in this article, the concepts of allowable deviation and maximum deviation are defined for a given solution. Moreover, the centralized solution is further defined as the solution, which owns the biggest maximum deviation in a given max-product system. The centralized solution could be viewed as the most stable solution in the system. Two algorithms with polynomial complexities are proposed for obtaining the maximum deviation of a given solution and the centralized solution of a given max-product system, respectively. Numerical examples illustrate the feasibility and efficiency of our proposed algorithms.

Efficient Feasibility Analysis for Graph-Based Real-Time Task Systems

The demand bound function (DBF) is a powerful abstraction to analyze the feasibility/schedulability of real-time tasks. Computing the DBF for expressive system models, such as graph-based tasks, is typically very expensive. In this article, we develop new techniques to drastically improve the DBF computation efficiency for a representative graph-based task model, digraph real-time tasks (DRT). First, we apply the well-known quick processor-demand analysis (QPA) technique, which was originally designed for simple sporadic tasks, to the analysis of DRT. The challenge is that existing analysis techniques of DRT have to compute the demand for each possible interval size, which is contradictory to the idea of QPA that aims to aggressively skip the computation for most interval sizes. To solve this problem, we develop a novel integer linear programming (ILP)-based analysis technique for DRT, to which we can apply QPA to significantly improve the analysis efficiency. Second, we improve the task utilization computation (a major step in DBF computation for DRT) efficiency from pseudo-polynomial complexity to polynomial complexity. Experiments show that our approach can improve the analysis efficiency by dozens of times.

Deep Rényi entropy graph kernel

Graph kernels are applied heavily for the classification of structured data. In this paper, we propose a deep Rényi entropy graph kernel for this purpose. We gauge the deep information through a family of h-layer expansion subgraphs rooted at a vertex, and define a h-layer depth-based second-order Rényi entropy representation for each vertex. The second-order Rényi entropy representation is used together with Euclidean distance to build a deep second-order Rényi entropy graph kernel (SREGK). For graphs with n vertices, the time complexity for our kernel is O(n3). This low-order polynomial complexity enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state-of-the-art graph kernels. Experimental results on fourteen real world graph datasets are shown to demonstrate the overall superior performance of our approach over a number of state-of-the-art methods.

DRAG: Deep Reinforcement Learning Based Base Station Activation in Heterogeneous Networks

Heterogeneous Network (HetNet), where Small cell Base Stations (SBSs) are densely deployed to offload traffic from macro Base Stations (BSs), is identified as a key solution to meet the unprecedented mobile traffic demand. The high density of SBSs are designed for peak traffic hours and consume an unnecessarily large amount of energy during off-peak time. In this paper, we propose a deep reinforcement-learning based SBS activation strategy that activates the optimal subset of SBSs to significantly lower the energy consumption without compromising the quality of service. In particular, we formulate the SBS on/off switching problem into a Markov Decision Process that can be solved by Actor Critic (AC) reinforcement learning methods. To avoid prohibitively high computational and storage costs of conventional tabular-based approaches, we propose to use deep neural networks to approximate the policy and value functions in the AC approach. Moreover, to expedite the training process, we adopt a Deep Deterministic Policy Gradient (DDPG) approach together with a novel action refinement scheme. Through extensive numerical simulations, we show that the proposed scheme greatly outperforms the existing methods in terms of both energy efficiency and computational efficiency. We also show that the proposed scheme can scale to large system with polynomial complexities in both storage and computation.

Digital quantum simulation of hadronization in Yang–Mills theory

A quantum algorithm of SU([Formula: see text]) Yang–Mills theory is formulated in terms of quantum circuits. It can nonperturbatively calculate the Dyson series and scattering amplitudes with polynomial complexity. The gauge fields in the interaction picture are discretized on the same footing with the lattice fermions in momentum space to avoid the fermion doubling and the gauge symmetry breaking problems. Applying the algorithm to the quantum simulation of quantum chromodynamics, the quark and gluon’s wave functions evolved from the initial states by the interactions can be observed and the information from wave functions can be extracted at any discrete time. This may help us understand the natures of the hadronization which has been an outstanding question of significant implication on high energy phenomenological studies.