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Feder and Vardi have conjectured that all constraint satisfaction problems to a fixed structure (constraint language) are polynomial or NP-complete. This so-called dichotomy conjecture remains open, although it has been proved in a number of special cases. Most recently, Bulatov has verified the conjecture for conservative structures, i.e., structures which contain all possible unary relations. We explore three different implications of Bulatov's result. First, the above dichotomy can be extended to so-called inclusive structures, corresponding to conservative constraint satisfaction problems in which each variable comes with its own domain. (This has also been independently observed by Bulatov.) We prove a more general version, extending the dichotomy to so-called three-inclusive structures, i.e., structures which contain, with any unary relation $R$, all unary relations $R'$ for subsets $R' \\subseteq R$ with at most three elements. For the constraint satisfaction problems in this generalization we must restrict the instances to so-called $1$-full structures, in which each variable is involved in a unary constraint. This leads to our second focus, which is on restrictions to more general kinds of \"full\" input structures. For any set $W$ of positive integers, we consider a restriction to $W$-full input structures, i.e., structures in which, for each $w \\in W$, any $w$ variables are involved in a $w$-ary constraint. We identify a class of structures (the so-called $W$-set-full structures) for which the restriction to $W$-full input structures does not change the complexity of the constraint satisfaction problem, and hence the family of these restricted problems also exhibits dichotomy. The general family of three-inclusive constraint satisfaction problems restricted to $W$-full input structures contains examples which we cannot seem to prove either polynomial or NP-complete. Nevertheless, we are able to use our result on the dichotomy for three-inclusive constraint satisfaction problems, to deduce the fact that all three-inclusive constraint satisfaction problems restricted to $W$-full input structures are NP-complete or \"quasi-polynomial\" (of order $n^{O(\\log n)}$). Our third focus deals with bounding the number of occurrences of a variable, which we call the degree. We conjecture that the complexity classification of three-inclusive constraint satisfaction problems extends to the case where all degrees are bounded by three. Using previous results, we are able to verify this conjecture in a number of special cases. Conservative, inclusive, and three-inclusive constraint satisfaction problems can be viewed as problems in which each variable is restricted to a \"list\" of allowed values. This point of view of lists is frequently encountered in the study of graph colorings, graph homomorphisms, and graph partitions. Our results presented here, in all three areas, were strongly motivated by these results on graphs.

Constraints, although ubiquitous in production and distribution planning, scheduling and control, often lead to inconsistencies in the decision-making process. The constraint-based modeling helps circumvent many organization-impacting issues. To address this, we developed a multi-level approach to the modeling and solving of combinatorial optimization problems. It is versatile and effective owing to the use of multi-level presolving and multiple paradigms, such as constraint programming, logic programming, mathematical programming and fuzzy logic, for their complementary strengths. The capability of this framework and its advantage over mathematical programming alone or over hybrid frameworks is shown in the illustrative example, in which combinatorial optimization is used as a benchmark to prove the effectiveness of the proposed approach. Knowledge of the problem is stored in the form of facts.

The next few years will be exciting as prototype universal quantum processors emerge, enabling the implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation and which have the potential to significantly expand the breadth of applications for which quantum computers have an established advantage. A leading candidate is Farhi et al.’s quantum approximate optimization algorithm, which alternates between applying a cost function based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the quantum alternating operator ansatz, is the consideration of general parameterized families of unitaries rather than only those corresponding to the time evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach, in the spirit of the quantum approximate optimization algorithm, to a wide variety of approximate optimization, exact optimization, and sampling problems. In addition to introducing the quantum alternating operator ansatz, we lay out design criteria for mixing operators, detail mappings for eight problems, and provide a compendium with brief descriptions of mappings for a diverse array of problems.

Nature is rife with networks that are functionally optimized to propagate inputs to perform specific tasks. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning interactions between nodes. Here we explore this behavior in two contexts: strain propagation in mechanical networks and pressure redistribution in flow networks. By adding and removing links, we are able to optimize both types of networks to perform specific functions. We define a single function as a tuned response of a single “target” link when another, predetermined part of the network is activated. Using network structures generated via such optimization, we investigate how many simultaneous functions such networks can be programed to fulfill. We find that both flow and mechanical networks display qualitatively similar phase transitions in the number of targets that can be tuned, along with the same robust finite-size scaling behavior. We discuss how these properties can be understood in the context of constraint–satisfaction problems.

Due to recent booming of unmanned air vehicles (UAVs) technologies, these are being used in many fields involving complex tasks. Some of them involve a high risk to the vehicle driver, such as fire monitoring and rescue tasks, which make UAVs excellent for avoiding human risks. Mission planning for UAVs is the process of planning the locations and actions (loading/dropping a load, taking videos/pictures, acquiring information) for the vehicles, typically over a time period. These vehicles are controlled from ground control stations (GCSs) where human operators use rudimentary systems. This paper presents a new multi-objective genetic algorithm for solving complex mission planning problems involving a team of UAVs and a set of GCSs. A hybrid fitness function has been designed using a constraint satisfaction problem to check whether solutions are valid and Pareto-based measures to look for optimal solutions. The algorithm has been tested on several datasets, optimizing different variables of the mission, such as the makespan, the fuel consumption, and distance. Experimental results show that the new algorithm is able to obtain good solutions; however, as the problem becomes more complex, the optimal solutions also become harder to find.

This study proposes an adaptive approximation-based control approach for non-linear pure-feedback systems in the presence of full state constraints. Completely non-affine non-linear functions are considered and assumed to be unknown. The dynamic surface design based on integral barrier Lyapunov functionals is provided to achieve both the desired tracking performance and the constraints satisfaction, in consideration of the full-state-constrained non-affine non-linearities. In this design procedure, simple sufficient conditions for choosing control gains, which can be checked off-line, are established to guarantee the feasibility of the controller. The function approximation technique is employed to estimate unknown non-linearities induced from the controller design procedure where the adaptive laws using the projection operator are designed to ensure the boundedness of the function approximators in the feasibility conditions. It is shown that all the signals in the closed-loop system are uniformly ultimately bounded and the tracking error converges to an adjustable neighbourhood of the origin while all state variables always remain in the constrained state space.

This study presents the development of a robust second‐order sliding mode controller (SOSMC) that incorporates position error constraints into the control design. To achieve this, a barrier Lyapunov function (BLF) is employed, enabling the enforcement of predefined bounds on tracking errors throughout the system's evolution. This approach ensures that errors remain within previously known limits, thereby improving safety and reliability in practical applications, particularly robotic systems. The proposed control scheme guarantees the existence of a sliding mode and achieves exponential convergence of the tracking errors, even in the presence of bounded disturbances and model uncertainties. Hence, the main contribution of this study is the integration of the BLF into the SOSMC framework, which not only maintains robustness but also addresses the critical issue of constraint satisfaction. This is often overlooked in traditional sliding mode designs. The effectiveness and improved performance of the proposed controller are validated through simulation studies conducted on a three‐degree‐of‐freedom robotic manipulator and through experiments on a six‐degree‐of‐freedom robotic manipulator. Comparative results demonstrate that, unlike a conventional SOSMC without error constraints, the proposed controller successfully maintains position errors within the specified limits while preserving fast convergence and robustness. These findings highlight the significant benefits of incorporating barrier Lyapunov functions in sliding mode control strategies for systems with strict performance and safety requirements.

This paper aims to discuss the approach of constrained modified feedback linearization model predictive control for the spacecraft simulator. By utilizing the high accuracy and constrained properties of model predictive control (MPC), an optimum MPC is designed for the spacecraft feedback linearized system. The composite controller has the ability to control both the attitude and angular velocity of the reaction wheels (i.e. steering the angular momentum to zero at the end of the maneuver). The simulation and experimental results demonstrate that the proposed hybrid controller has an insignificant calculative cost and facilitates the spacecraft to perform regulation maneuver with sufficient precision in the presence of external torques and actuator saturations. This paper aims to discuss the approach of constrained modified feedback linearization model predictive control (CMFLMPC) for the spacecraft simulator. The simulation and experimental results demonstrate that the proposed hybrid controller has an insignificant calculative cost and facilitates the spacecraft to perform the regulation maneuver with sufficient precision in the presence of external torques and actuator saturations.

An ordering constraint satisfaction problem (OCSP) is defined by a family F of predicates mapping permutations on { 1 , … , k } to { 0 , 1 } . An instance of Max-OCSP( F ) on n variables consists of a list of constraints, each consisting of a predicate from F applied on k distinct variables. The goal is to find an ordering of the n variables that maximizes the number of constraints for which the induced ordering on the k variables satisfies the predicate. OCSPs capture well-studied problems including ‘maximum acyclic subgraph’ (MAS) and “maximum betweenness”. In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every F , Max-OCSP( F ) is approximation-resistant to o( n )-space streaming algorithms, i.e., algorithms using o( n ) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of MAS, our result shows that for every ϵ > 0 , MAS is not ( 1 / 2 + ϵ ) -approximable in o( n ) space. The previous best inapproximability result, due to Guruswami & Tao (2019), only ruled out 3/4-approximations in o ( n ) space. Our results build on recent works of Chou et al. (2022b, 2024) who provide a tight, linear-space inapproximability theorem for a broad class of “standard” (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate standard CSPs (one for every alphabet size q ) from any given OCSP and applying their theorem to this family of CSPs. To convert the resulting hardness results for standard CSPs back to our OCSP, we show that the hard instances from this earlier theorem have the following “partition expansion” property with high probability: For every partition of the n variables into small blocks, for most of the constraints, all variables are in distinct blocks.

High‐speed trains (HSTs) often face challenges associated with car‐body hunting, resulting from factors such as complex operating environments and wheel‐rail contact degradation. These issues have a significant impact on the ride quality and operational safety. Passive suspension systems are inadequate to provide satisfactory dynamic performance for HSTs in such circumstances. However, active suspension systems can provide an effective solution. To address the car‐body hunting stability of HSTs, this study proposes a novel control approach called displacement inequality constraint following control (ICFC‐BIS) for the active suspension of HSTs. The ICFC‐BIS leverages the beneficial nonlinearity of bio‐inspired structures (BIS) to indirectly achieve excellent dynamic characteristics of the BIS through actuators installed in the suspension of HSTs, eliminating the need for actual BIS installation. Numerical simulation results demonstrate that the proposed ICFC‐BIS can effectively suppress car‐body hunting motion, thereby enhancing the ride comfort of HSTs. Consequently, the operational safety and ride comfort indices of HSTs equipped with active suspension systems are significantly improved. To address the issue of anomalous car‐body hunting motion, this study proposes a constraint‐following control method that leverages actuators installed in the suspension system to achieve the beneficial non‐linearity of the bio‐inspired structure. This method overcomes the limitation of directly installing the bio‐inspired structure in the HST suspension system. Additionally, a displacement inequality constraint following control strategy is introduced to minimize constraint errors and enhance control robustness, thereby improving HST dynamics performance.