Computational Complexity Perspective Research Articles

On probabilistic and causal reasoning with summation operators

Abstract Ibeling et al. (2023, Annals of Pure and Applied Logic, 103339) axiomatize increasingly expressive languages of causation and probability, and Mossé et al. (2024, The Review of Symbolic Logic, 17, 106–131) show that reasoning (specifically the satisfiability problem) in each causal language is as difficult, from a computational complexity perspective, as reasoning in its merely probabilistic or ‘correlational’ counterpart. Introducing a summation operator to capture common devices that appear in applications—such as the $do$-calculus of Pearl (Causality. Cambridge University Press, 2009) for causal inference, which makes ample use of marginalization—van der Zander et al. (The hardness of reasoning about probabilities and causality. In Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI-23, E. Elkind, ed., pp. 5730–5738. International Joint Conferences on Artificial Intelligence Organization. Main Track. 2023) extend these earlier complexity results to causal and probabilistic languages with marginalization. We complete this extension, fully characterizing the complexity of probabilistic and causal reasoning with summation, demonstrating that these again remain equally difficult. Surprisingly, allowing free variables for random variable values results in a system that is undecidable, so long as the ranges of these random variables are unrestricted. We finally axiomatize these languages featuring marginalization (or more generally summation), resolving open questions posed by Ibeling et al. (2023, Annals of Pure and Applied Logic, 103339).

Placing Green Bridges Optimally, with a Multivariate Analysis

We study the problem of placing wildlife crossings, such as green bridges, over human-made obstacles to challenge habitat fragmentation. The main task herein is, given a graph describing habitats or routes of wildlife animals and possibilities of building green bridges, to find a low-cost placement of green bridges that connects the habitats. We develop three problem models for this task and study them from a computational complexity and parameterized algorithmics perspective.

Priority-based bin packing with subset constraints

This paper is concerned with a new variant of bin packing called priority-based bin packing with subset constraints (PBBP-SC). PBBP-SC is a variant of traditional bin packing (TBP). In a TBP instance, we are given a collection of items {t1,t2,…tn}, where each item ti has size si∈(0,1). The goal of TBP is to pack the items in as few unit-sized bins as possible. In a PBBP-SC instance, we are given a collection of n unit-size items and a collection of m bins of varying capacities. Associated with each item is a positive integer which is called its priority. The priority of an item indicates its importance in being included in a (possibly infeasible) packing. As with the traditional case, these items need to be packed in the fewest number of bins. What complicates the problem is the fact that each item can be assigned to only one of a select set of bins, i.e., the bins are not interchangeable. We investigate several problems associated with PBBP-SC. The first problem we study is the feasibility problem. This is the problem of checking if there is a feasible assignment to a given PBBP-SC instance. The second problem we study is the priority maximization problem. This is the problem of finding a maximum priority assignment of an infeasible PBBP-SC instance. The third problem we study is the bin minimization problem. This is the problem of finding an assignment with the fewest number of bins to pack all of the items in a feasible PBBP-SC instance. We derive a number of results from both the algorithmic and computational complexity perspectives for these problems. In particular, we show that both the feasibility and priority maximization problems are solvable in polynomial time. We also show that the bin minimization problem is log-APX-complete and cannot be solved in time o(1.41m) unless the Strong Exponential Time Hypothesis fails.

Swarm Control for Distributed Construction: A Computational Complexity Perspective

Over the last 20 years, human interaction with robot swarms has been investigated as a means to mitigate problems associated with the control and coordination of such swarms by either human teleoperation or completely autonomous swarms. Ongoing research seeks to characterize those situations in which such interaction is both viable and preferable. In this article, we contribute to this effort by giving the first computational complexity analyses of problems associated with algorithm, environmental influence, and leader selection methods for the control of swarms performing distributed construction tasks. These analyses are done relative to a simple model in which swarms of deterministic finite-state robots operate in a synchronous error-free manner in 2D grid-based environments. We show that all three of our problems are polynomial-time intractable in general and remain intractable under a number of plausible restrictions (both individually and in many combinations) on robot controllers, environments, target structures, and sequences of swarm control commands. We also give the first restrictions relative to which these problems are tractable, as well as discussions of the implications of our results for both the design and deployment of swarm control assistance software tools and the human control of swarms.

A Computational Complexity Perspective on Segmentation as a Cognitive Subcomputation.

Computational feasibility is a widespread concern that guides the framing and modeling of natural and artificial intelligence. The specification of cognitive system capacities is often shaped by unexamined intuitive assumptions about the search space and complexity of a subcomputation. However, a mistaken intuition might make such initial conceptualizations misleading for what empirical questions appear relevant later on. We undertake here computational-level modeling and complexity analyses of segmentation - a widely hypothesized subcomputation that plays a requisite role in explanations of capacities across domains, such as speech recognition, music cognition, active sensing, event memory, action parsing, and statistical learning - as a case study to show how crucial it is to formally assess these assumptions. We mathematically prove two sets of results regarding computational hardness and search space size that may run counter to intuition, and position their implications with respect to existing views on thesubcapacity.

A Hybrid Stochastic/Robust Model for Transmission Expansion Planning under an Ellipsoidal Uncertainty Set

Over the last few years, the concept of robust optimization (RO) and its application to power system problems have been at the center of much recent research due to its successful implementation in handling uncertainties. In this regard, this paper presents a new approach for hybrid stochastic/robust transmission expansion planning (TEP), considering three sources of uncertainties, including demand growth, generation capacity, and thermal capacity of transmission lines. These uncertain parameters are modeled using an ellipsoidal uncertainty set, which can capture the correlation efficiently. The TEP problem is modeled as a bilevel mathematical optimization where the lower level is replaced by its Lagrange dual optimization problem using the duality theorem (DT). One main benefit of using the dual problem is that all uncertainties will appear in the objective function and thus can be converted to convex second-order cone constraints. The entire bilevel problem is then solved using the standard column-and-constraint generation (C&CG) technique by decomposing the mathematical problem into a master and a binary variable-free subproblem. Numerical studies on two test power systems validate the effectiveness of the proposed model. The results indicate that the presented methodology is more efficient than the existing models from the computational complexity perspective.

Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data.

Time-domain backprojection algorithms are widely used in state-of-the-art synthetic aperture radar (SAR) imaging systems that are designed for applications where motion error compensation is required. These algorithms include an interpolation procedure, under which an unknown SAR range-compressed data parameter is estimated based on complex-valued SAR data samples and backprojected into a defined image plane. However, the phase of complex-valued SAR parameters estimated based on existing interpolators does not contain correct information about the range distance between the SAR imaging system and the given point of space in a defined image plane, which affects the quality of reconstructed SAR scenes. Thus, a phase-control procedure is required. This paper introduces extensions of existing linear, cubic, and sinc interpolation algorithms to interpolate complex-valued SAR data, where the phase of the interpolated SAR data value is controlled through the assigned a priori known range time that is needed for a signal to reach the given point of the defined image plane and return back. The efficiency of the extended algorithms is tested at the Nyquist rate on simulated and real data at THz frequencies and compared with existing algorithms. In comparison to the widely used nearest-neighbor interpolation algorithm, the proposed extended algorithms are beneficial from the lower computational complexity perspective, which is directly related to the offering of smaller memory requirements for SAR image reconstruction at THz frequencies.

Rule-based Bayesian regression

We introduce a novel rule-based approach for handling regression problems. The new methodology carries elements from two frameworks: (i) it provides information about the uncertainty of the parameters of interest using Bayesian inference, and (ii) it allows the incorporation of expert knowledge through rule-based systems. The blending of those two different frameworks can be particularly beneficial for various domains (e.g., engineering), where even though the significance of uncertainty quantification motivates a Bayesian approach, there is no simple way to incorporate researcher intuition into the model. We validate our models by applying them to synthetic applications: a simple linear regression problem and two more complex structures based on partial differential equations, and we illustrate their use through two cases derived from real data. Finally, we review the advantages of our methodology, which include the simplicity of the implementation, the uncertainty reduction due to the added information and, in some occasions, the derivation of better point predictions, and we outline limitations, mainly from the computational complexity perspective, such as the difficulty in choosing an appropriate algorithm and the added computational burden.

Computational Complexity of Biased Diffusion-Limited Aggregation

Diffusion-Limited Aggregation (DLA) is a cluster growth model that consists of a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote $k$-DLA the model where the particles move only in $k$ possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid $c$ and a sequence $S$ of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site $c$ when sequence $S$ is realized. The second problem is Realization, where the input is a set of positions of the grid, $P$. The question is whether there exists a sequence $S$ that realizes $P$, i.e. all particles of $S$ exactly occupy the positions in $P$. Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is P-Complete for 2-DLA (thus for 3-DLA). Later, we show that Prediction can be solved much more efficiently for 1DLA. In fact, we show that in that case, the problem is NL-Complete. With respect to Realization, we show that restricted to 2DLA the problem is in P, while for 1DLA is in L.

Quantum algorithmic measurement

There has been recent promising experimental and theoretical evidence that quantum computational tools might enhance the precision and efficiency of physical experiments. However, a systematic treatment and comprehensive framework are missing. Here we initiate the systematic study of experimental quantum physics from the perspective of computational complexity. To this end, we define the framework of quantum algorithmic measurements (QUALMs), a hybrid of black box quantum algorithms and interactive protocols. We use the QUALM framework to study two important experimental problems in quantum many-body physics: determining whether a system’s Hamiltonian is time-independent or time-dependent, and determining the symmetry class of the dynamics of the system. We study abstractions of these problems and show for both cases that if the experimentalist can use her experimental samples coherently (in both space and time), a provable exponential speedup is achieved compared to the standard situation in which each experimental sample is accessed separately. Our work suggests that quantum computers can provide a new type of exponential advantage: exponential savings in resources in quantum experiments.

Complexity of Scorpion Solitaire and applications to Klondike

Scorpion is a puzzle game that falls under the category of Solitaires. The problem of determining whether a starting configuration of a Solitaire game (generalized to contain an arbitrary number of cards) can lead to a winning configuration has been studied from the perspective of Computational Complexity for some of the most popular Solitaires, namely Free Cell [1], Klondike [2] and Spider [3], all resulting in hardness proofs. Scorpion, though, has a unique twist compared to the aforementioned ones: each card can be moved to cover only one other card. This property narrows the set of possible moves and makes a worthwhile issue investigating whether the same problem proved NP-complete for Free Cell, Klondike and Spider falls in P as far as Scorpion is considered. In this paper we prove that the problem of deciding whether an n-card s-suits initial configuration of Scorpion allows for a winning sequence of moves is NP-complete for any s≥1, and it remains so also when the initial configuration contains no face down (locked) cards and under a constant ratio constraint between deck size and tableau size. As a side effect of our results, we have also succeeded in negatively answering a question posed in [2], in that we prove that it is NP-complete to decide whether an n-card one red suit-one black suit configuration of the Klondike Solitaire allows for a winning sequence of moves.

Recognizing Lexicographically Smallest Words and Computing Successors in Regular Languages

For a formal language [Formula: see text], the problem of language enumeration asks to compute the length-lexicographically smallest word in [Formula: see text] larger than a given input [Formula: see text] (henceforth called the [Formula: see text]-successor of [Formula: see text]). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are (in general) necessary and sufficient for an unambiguous finite-state transducer to compute [Formula: see text]-successors. As a byproduct, we obtain that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are sufficient for a DFA to recognize the subset [Formula: see text] of [Formula: see text] composed of its lexicographically smallest words. We give a matching lower bound that holds even if [Formula: see text] is represented as an NFA. It has been known that [Formula: see text]-successors can be computed in polynomial time, even if the regular language is given as part of the input (assuming a suitable representation of the language, such as a DFA). In this paper, we refine this result in multiple directions. We show that if the regular language is given as part of the input and encoded as a DFA, the problem is in [Formula: see text]. If the regular language [Formula: see text] is fixed, we prove that the enumeration problem of the language is reducible to deciding membership to the Myhill-Nerode equivalence classes of [Formula: see text] under [Formula: see text]-uniform [Formula: see text] reductions. In particular, this implies that fixed star-free languages can be enumerated in [Formula: see text], arbitrary fixed regular languages can be enumerated in [Formula: see text] and that there exist regular languages for which the problem is [Formula: see text]-complete.

Unified IRS-Aided MIMO Transceiver Designs via Majorization Theory

In this paper, we develop a unified framework for IRS-aided transceiver designs under general power constraints in multiple-input multiple-output (MIMO) systems which implement interference (pre-)subtraction via Tomlinson-Harashima precoding (THP) or Decision Feedback Equalization (DFE) technologies. Armed with majorization theory, two fundamental classes of performance criteria, namely $K$ -increasing Schur-concave and Schur-convex functions of the logarithm of Mean Square Error (MSE) of the data stream, are investigated in depth. Firstly, we propose a simplified counterpart of the optimal transceiver design under general power constraints, with equivalence guaranteed by Pareto optimization theory and Lagrange duality. Moreover, the optimal semi-closed form solution to this simplified transceiver design can be attained using the modified subgradient method. Next, we prove that for any Schur-concave objective, the optimal nonlinear THP (DFE) design is in essence the linear precoding (equalization). For any Schur-convex objective, the optimal transceiver design results in individual data streams with equal MSEs, and thereby reduces to the Gaussian mutual information maximization based design. Based on the above conclusions, we further propose an efficient alternating optimization algorithm to decouple the optimization of the transmit precoder and the IRS reflection coefficients, where the classical successive convex approximation (SCA) technique is applied to fight against non-convex subproblems. From the low computational complexity perspective, a two-stage scheme is also developed inspired by the capability of the IRS in constructing favorable wireless links. Finally, numerical results show the global optimality of the modified subgradient method and the excellent performance of the proposed alternating optimization algorithm and two-stage scheme.

Remarks on black hole complexity puzzle

Recently a certain conceptual puzzle in the AdS/CFT correspondence, concerning the growth of quantum circuit complexity and the wormhole volume, has been identified by Bouland-Fefferman-Vazirani and Susskind. In this note, we propose a resolution of the puzzle and save the quantum Extended Church-Turing thesis by arguing that there is no computational shortcut in measuring the volume due to gravitational backreaction from bulk observers. A certain strengthening of the firewall puzzle from the computational complexity perspective, as well as its potential resolution, is also presented.

Auxiliary Vehicle Positioning Based on Robust DOA Estimation With Unknown Mutual Coupling

As an important branch of the Internet of Vehicles (IoV), vehicle positioning has drawn extensive attention. Traditional positioning systems based on a global positioning system incur long delays, and may fail due to obstructions. In this article, we propose an auxiliary positioning architecture, whose core is to estimate the direction of arrival (DOA) of signals from landmarks, such as wireless access points, utilizing a sensor array in the vehicle. Due to space limitations, the array may be placed in an arbitrary geometry and may suffer from unknown mutual coupling. Most algorithms are only effective for sensor arrays with special geometries, e.g., a uniform linear array or rectangular array. To tackle this problem, an improved multiple signal classification algorithm is derived, which is superior to the state-of-the-art iterative method from the perspective of computational complexity. Detailed analysis concerning identifiability, computational complexity, and Cramér-Rao bounds are given. The simulation results verify the improvement of the proposed DOA estimation algorithm. The proposed architecture can obtain robust self-localization with existing vehicular ad hoc networks, and it can collaborate with other positioning systems to provide a safe driving environment.

An FPGA-Based Simple RGB-HSI Space Conversion Algorithm for Hardware Image Processing

In this paper, a new, low-complexity, easy-to-implement hardware method for color space conversion between the red-green-blue (RGB) and the hue-saturation-intensity (HSI) color spaces called the simple RGB-HSI space conversion (S-SC) algorithm is proposed, which aims to provide more rapid computing due to the need for fewer operations. In the S-SC algorithm, we reconstruct the model of space conversion between the RGB color space and the HSI color space (RGB-HSI) by inverting the conversion from the HSI color space to the RGB color space of the traditional geometric derivation algorithm. As a result, the nonlinear model-realized RGB-HSI color space conversion by the geometric derivation algorithm is transformed into a linear conversion model, which can avoid complicated calculations such as trigonometric and inverse trigonometric functions in the color space conversion process. The model can effectively reduce the computational complexity of the algorithm and facilitate hardware implementation at the same time. To evaluate the performance of the S-SC algorithm, we first compare the S-SC algorithm with the geometric derivation algorithm from the computational complexity perspective. On this basis, we compare the S-SC algorithm with five other RGB-HSI color space conversion algorithms from the perspectives of error and conversion effect. Finally, we use the field programmable gate array (FPGA) hardware platform to analyze and verify the timing sequence and logical resource consumption and verify the effectiveness of the proposed algorithm with experimental results. We show that the S-SC algorithm achieves good performance in terms of conversion accuracy, logical unit resource occupancy, and output timing.

Rule Extraction Model Based on Decision Dependency Degree

Rule extraction is the core in rough set. Two procedures are contained in rule extraction: one is attribute reduction and another is attribute value reduction. It was proved through computational complexity perspective that obtaining all the reduction, minimum attribute reduction, and minimum attribute value reduction is an NP problem. So, generally, a heuristic reduction method is used to solve attribute reduction and attribute value reduction. However, for most heuristic methods, it is hard to put into practice and has high cost on computational complexity. Moreover, part of the methods extracted redundant rules. To approach a quick and effective model for rule extraction in decision systems, against the concept of distinguishable relation, relevant concepts and basic theorems of rule extraction are proposed. In order to get concise and accurate rules quickly, algorithms for finding conflict object set, finding duplicate object set, and finding redundant rules are given. After that, using decision dependency degree as attribute importance to determine the importance of each attribute in rule object, a new rule extraction model based on decision dependency degree is proposed in this paper. Compared with the previous models, this model does not generate matrix; instead, it finds conflict object set and duplicate object set by equivalence class, and consequently, improves the time performance tomaxOCU,OC2U/C, andOREDU/C/RED2. The theoretical analysis and experimental research show that the new model more accurately and effectively reduces the redundant data and extracts more concise decision rules from dataset.

Joint Radio Resource Allocation and SIC Ordering in NOMA-Based Networks Using Submodularity and Matching Theory

In this paper, we propose a novel joint radio resource allocation and successive interference cancellation (SIC) ordering algorithm for the downlink scenario of a power-domain sparse code multiple access (PSMA) based wireless network. To this end, we formulate a novel joint radio and SIC ordering optimization problem. Our aim is to maximize sum rate over codebook assignment, transmit power, and SIC ordering variables. Since the proposed optimization problem is NP-hard and mathematically intractable, we exploit alternate search method (ASM). Based on this algorithm, the proposed optimization problem is decomposed into three sub-problems, namely, codebook assignment, power allocation, and SIC ordering. To solve the codebook assignment sub-problem, we exploit matching theory and submodularity and compared them from the performance and complexity perspectives. Due to the fact that PD-NOMA-based networks adopt SIC at the receiver side, the proposed SIC ordering algorithm is also studied for this type of networks. Moreover, the proposed algorithms are investigated from the performance, convergence, and computational complexity perspectives. Numerical results show that the proposed SIC ordering method outperforms the conventional one by approximately 48% in the studied NOMA-based networks.

Direction finding in MIMO radar with large antenna arrays and nonorthogonal waveforms

As an important research branch in multiple-input multiple-output (MIMO) radar, orthogonal waveforms are always desirable. In practice, however, the orthogonality may not be always guaranteed. In this paper, a computationally efficient parallel factor (PARAFAC) estimator is proposed, which is suitable for direction-of-arrival (DOA) estimation in colocated MIMO radar with large antenna arrays as well as nonorthogonal waveforms. Firstly, the spatially colored noise caused by the nonorthogonal waveforms is eliminated via temporal cross-correlation of the measurement. Thereafter, a third-order PARAFAC (or called trilinear decomposition) model is established by exploiting the multidimensional structure as well as the low rank property of the cross covariance matrix. The DOA estimation problem is then linked to PARAFAC decomposition and finally obtained via least squares technique. Compared with the existing matrix completion-based algorithm, the proposed estimator is attractive from the perspective of computational complexity and estimation accuracy, especially with large antenna arrays scenario. In addition, the stochastic Cramér-Rao bound on DOA estimation with waveform imperfectly is derived. Numerical simulations are provided to verify the effectiveness of the proposed estimator.

Enhanced DFT-Based Channel Estimator for Leakage Effect Mitigation in OFDM Systems

The leakage effect occurs in a DFT-based channel estimator (CE) due to the presence of guard bands, where the energy of the channel impulse response (CIR) is dispersed to other estimated taps. Furthermore, some of the CIR energy leaks outside of the estimation range and fails to be captured in the channel estimation. This letter presents an enhanced DFT-based CE for orthogonal frequency-division multiplexing (OFDM) systems. The proposed method is composed of two techniques ( spectral extrapolation and circular shift ) to mitigate the leakage effect, followed by a denoising post-processing to enhance the estimation accuracy. Through performance analysis, it is demonstrated that the spectral extrapolation eliminates the leakage effect corresponding to the first CIR tap, and the circular shift improves estimation accuracy by incorporating the leaked CIR energy in the channel estimation. The simulation results show that the proposed method has superior performance compared with conventional CEs. In addition, the proposed method is feasible from a computational complexity perspective.