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Coherent Ising Machine (CIM) is a network of optical parametric oscillators that solve combinatorial optimization problems by finding the ground state of an Ising Hamiltonian. In CIMs, a problem arises when attempting to realize the Zeeman term because of the mismatch in size between interaction and Zeeman terms due to the variable amplitude of the optical parametric oscillator pulses corresponding to spins. There have been three approaches proposed so far to address this problem for CIM, including the absolute mean amplitude method, the auxiliary spin method, and the chaotic amplitude control (CAC) method. This paper focuses on the efficient implementation of Zeeman terms within the mean-field CIM model, which is a physics-inspired heuristic solver without quantum noise. With the mean-field model, computation is easier than with more physically accurate models, which makes it suitable for implementation in field programmable gate arrays and large-scale simulations. First, we examined the performance of the mean-field CIM model for realizing the Zeeman term with the CAC method, as well as their performance when compared to a more physically accurate model. Next, we compared the CAC method to other Zeeman term realization techniques on the mean-field model and a more physically accurate model. In both models, the CAC method outperformed the other methods while retaining similar performance. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical systems such as coupled oscillators are being actively investigated as Ising machines for solving computationally hard problems in combinatorial optimization. Prior works have established the equivalence between the global minima of the cost function describing the coupled oscillator system and the ground state of the Ising Hamiltonian. However, the properties of the oscillator Ising machine (OIM) from a nonlinear control viewpoint, such as the stability of the OIM solutions, remain unexplored. Therefore, in this work, using nonlinear control-theoretic analysis, we (i) identify the conditions required to ensure the functionality of the coupled oscillators as an Ising machine, (ii) show that all globally optimal phase configurations may not always be stable, resulting in some configurations being more favored over others and, thus, creating a biased OIM, and (iii) elucidate the impact of the stability of locally optimal phase configurations on the quality of the solution computed by the system. Our work, fostered through the unique convergence between nonlinear control theory and analog systems for computing, provides a new toolbox for the design and implementation of dynamical system-based computing platforms. [ABSTRACT FROM AUTHOR]

This paper addresses the triangle width problem, which generalizes the classic two-machine flexible job-shop problem (FJSP) with tooling constraints. This new problem can be studied from three different angles: scheduling, matrix visualization, and vertex ordering in hypergraphs. We prove the equivalence of the different formulations of the problem and use them to establish the N P -Hardness and polynomiality of several of its subcases. This problem allows us to find more elegant (and probably shorter) proofs for several combinatorial problems in our analysis setting. Our study provides an elegant generalization of Johnson's argument for the two-machine flow shop. It also shows the relation between the question: "Is a matrix triangular?" and the "k-visit of a graph". [ABSTRACT FROM AUTHOR]

The traveling salesman problem (TSP) aims to find the shortest tour that visits each node of a given graph exactly once. TSPs have significant importance as numerous practical problems can be naturally formulated as TSPs. Various algorithms have been developed for solving TSPs, including combinatorial optimization algorithms and deep learning-based approaches. However, these algorithms often face a trade-off between providing exact solutions with long running times and delivering fast but approximate solutions. Therefore, achieving both efficiency and solution quality simultaneously remains a major challenge. In this study, we propose a data-driven algorithm called NeuralGLS to address this challenge. NeuralGLS is a hybrid algorithm that combines deep learning techniques with guided local search (GLS). It incorporates a self-adaptive graph convolutional network (GCN) that takes into account neighborhoods of varying sizes, accommodating TSP instances with different graph sizes. This GCN calculates a regret value for each edge in a given TSP instance. Subsequently, the algorithm utilizes a mixed strategy to construct an initial tour and then employs a GLS module to iteratively improve the tour guided by the acquired regret values until a high-quality tour is obtained. Experimental results on diverse benchmark datasets and real-world TSP instances demonstrate the effectiveness of NeuralGLS in generating high-quality solutions within reasonable computation time. Furthermore, when compared to several state-of-the-art algorithms, our NeuralGLS algorithm exhibits superior generalization performance on both real-world and larger-scale TSP instances. Notably, NeuralGLS also outperforms another hybrid algorithm that also incorporates GLS by reducing the mean optimality gap for real-world TSP instances from 1.318% to 0.958%, with both methods achieving results within the same computation time. This remarkable improvement in solution quality amounts to an impressive relative enhancement of 27.31%. [ABSTRACT FROM AUTHOR]

The traveling salesman problem (TSP) involves determining the shortest length (optimal tour) for a complete circuit through an arbitrary number of points. In the special case that the points to be visited are randomly distributed in the plane within a unit square, and travel costs correspond to the Euclidean distance between points, it is known that the length of the optimal tour divided by the square root of the number of points (n) asymptotically approaches a constant (βTSP) as n becomes large. If individual points are randomly drawn from a uniform distribution, the optimal lengths for a sufficiently large number of TSP instances of the same size (n) comprise a normal distribution whose mean approaches βTSP as n increases. Moreover, as n approaches infinity, this distribution gradually narrows such that its standard deviation asymptotically approaches zero. Although the precise value of βTSP is unknown, published studies indicate its magnitude is approximately 0.712. In this paper, possibly for the first time, precise formulae are proposed for the parameters of the underlying normal distribution and βTSP, based on intuition, mathematical reasoning, and empirical fits to both published and experimentally derived data. [ABSTRACT FROM AUTHOR]

We propose a heuristic method of using network centralities for constructing small-weight Steiner trees in this paper. The Steiner tree problem in graphs is one of the practical NP-hard combinatorial optimization problems. Given a graph and a set of vertices called terminals in the graph, the objective of the Steiner tree problem in graphs is to find a minimum weight Steiner tree that is a tree containing all the terminals. Conventional construction methods make a Steiner tree based on the shortest paths between terminals. If these shortest paths are overlapped as much as possible, we can obtain a small-weight Steiner tree. Therefore, we proposed to use network centralities to distinguish which edges should be included to make a small-weight Steiner tree. Experimental results revealed that using the vertex or the edge betweenness centralities contributes to making small-weight Steiner trees. [ABSTRACT FROM AUTHOR]

The Quantum Approximate Optimization Algorithm (QAOA) is a highly promising variational quantum algorithm that aims to solve combinatorial optimization problems that are classically intractable. This comprehensive review offers an overview of the current state of QAOA, encompassing its performance analysis in diverse scenarios, its applicability across various problem instances, and considerations of hardware-specific challenges such as error susceptibility and noise resilience. Additionally, we conduct a comparative study of selected QAOA extensions and variants, while exploring future prospects and directions for the algorithm. We aim to provide insights into key questions about the algorithm, such as whether it can outperform classical algorithms and under what circumstances it should be used. Towards this goal, we offer specific practical points in a form of a short guide. [ABSTRACT FROM AUTHOR]

The team orienteering problem (TOP) is a well-studied optimization challenge in the field of Operations Research, where multiple vehicles aim to maximize the total collected rewards within a given time limit by visiting a subset of nodes in a network. With the goal of including dynamic and uncertain conditions inherent in real-world transportation scenarios, we introduce a novel dynamic variant of the TOP that considers real-time changes in environmental conditions affecting reward acquisition at each node. Specifically, we model the dynamic nature of environmental factors—such as traffic congestion, weather conditions, and battery level of each vehicle—to reflect their impact on the probability of obtaining the reward when visiting each type of node in a heterogeneous network. To address this problem, a learnheuristic optimization framework is proposed. It combines a metaheuristic algorithm with Thompson sampling to make informed decisions in dynamic environments. Furthermore, we conduct empirical experiments to assess the impact of varying reward probabilities on resource allocation and route planning within the context of this dynamic TOP, where nodes might offer a different reward behavior depending upon the environmental conditions. Our numerical results indicate that the proposed learnheuristic algorithm outperforms static approaches, achieving up to 25 % better performance in highly dynamic scenarios. Our findings highlight the effectiveness of our approach in adapting to dynamic conditions and optimizing decision-making processes in transportation systems. [ABSTRACT FROM AUTHOR]

We contribute to the efficient approximation of the Pareto-set for the classical NP -hard multiobjective minimum spanning tree problem (moMST) adopting evolutionary computation. More precisely, by building upon preliminary work, we analyze the neighborhood structure of Pareto-optimal spanning trees and design several highly biased sub-graph-based mutation operators founded on the gained insights. In a nutshell, these operators replace (un)connected sub-trees of candidate solutions with locally optimal sub-trees. The latter (biased) step is realized by applying Kruskal's single-objective MST algorithm to a weighted sum scalarization of a sub-graph. We prove runtime complexity results for the introduced operators and investigate the desirable Pareto-beneficial property. This property states that mutants cannot be dominated by their parent. Moreover, we perform an extensive experimental benchmark study to showcase the operator's practical suitability. Our results confirm that the sub-graph-based operators beat baseline algorithms from the literature even with severely restricted computational budget in terms of function evaluations on four different classes of complete graphs with different shapes of the Pareto-front. [ABSTRACT FROM AUTHOR]

The compositional data are interrelated, and analyzing the evolution of each component is crucial for understanding population dynamics. However, the complex structure and tedious process of modeling pose challenges to the reasonable construction of grey compositional models for analyzing multi-trend mixed data. To address this, a novel enriched nonlinear grey compositional model with global multi-parameter combinatorial optimization is firstly proposed. Secondly, two types of Monte Carlo simulations are designed to validate the performances, modeling characteristics and noise levels of our model. Finally, using the bioenergy power generation structure of China as a case study, the practicability of our approach is verified. The results demonstrate that our model significantly outperforms traditional mainstream models in multi-trend mixed sequences, and the interrelationships among components are effectively verified. Our model not only enriches the methodological base but also broadens the application scope of grey compositional model. • A novel non-linear dynamic GM-Markov compositional model is constructed. • Our model is capable of achieving global multi-parameter combinatorial optimization. • Our model can accurately fit compositional data exhibiting fluctuations and multi-trend. • Our model might be useful in forecasting bioenergy power generation structures. [ABSTRACT FROM AUTHOR]

Transportation networks frequently employ hub-and-spoke network architectures to route flows between many origin and destination pairs. Hub facilities work as switching points for flows in large networks. In this study, we focus on a problem, called the single allocation hub-and-spoke network design problem. In the problem, the goal is to allocate each non-hub node to exactly one of the given hub nodes so as to minimize the total transportation cost. The problem is essentially equivalent to another combinatorial optimization problem, called the metric labeling problem. The metric labeling problem was first introduced by Kleinberg and Tardos in 2002, motivated by application to segmentation problems in computer vision and related areas. In this study, we deal with a case where the set of hubs forms a star, which is encountered especially in telecommunication networks. We propose a polynomial-time randomized approximation algorithm for the problem, whose approximation ratio is less than 5.281. Our algorithms solve a linear relaxation problem and apply dependent rounding procedures. [ABSTRACT FROM AUTHOR]

The dominating set problem (DSP) is one of the most famous problems in combinatorial optimization. It is defined as follows. For a given graph G = (V , E) , a dominating set of G is a subset S ⊆ V such that every vertex in V ∖ S is adjacent to at least one vertex in S. Furthermore, the DSP is the problem of finding a minimum-size dominating set and the corresponding minimum size, the domination number of G. In this, work we investigate a variant of the DSP, the super dominating set problem (SDSP), which has attracted much attention during the last years. A dominating set S is called a super dominating set of G , if for every vertex u ∈ S ¯ = V ∖ S , there exists a v ∈ S such that N (v) ∩ S ¯ = N (v) ∖ S = { u }. Analogously, the SDSP is to find a minimum-size super dominating set, and the corresponding minimum size, the super domination number of G. The decision variants of both the DSP and the SDSP have been shown to be NP -hard. In this paper, we present tight bounds for the super domination number of the neighbourhood corona product, r -clique sum, and the Hajós sum of two graphs. Additionally, we present infinite families of graphs attaining our bounds. Finally, we give the exact number of minimum size super dominating sets for some graph classes. In particular, the number of super dominating sets for cycles has quite surprising properties as it varies between values of the set { 4 , n , 2 n , 5 n 2 − 10 n 8 } based on n mod 4. [ABSTRACT FROM AUTHOR]

Considerable advances have been made recently in applying reinforcement learning (RL) to packing problems. However, most of these methods lack scalability and cannot be applied in dynamic environments. To address this research gap, we propose a hybrid algorithm called GraphPack to solve the strip packing problem. Two graph neural networks are designed to fully incorporate the problem's structure and enhance generalization performance. SkylineNet encodes the geometry of free space as the context feature, while PackNet, supporting the symmetry of rectangles, chooses the next rectangle to pack from the remaining rectangles at each timestep. We conduct fixed-scale, variable rectangle number and variable strip width experiments to test our method. The experimental results show that our method outperforms classical heuristic methods as well as previous RL methods. Notably, our method exhibits strong generalization ability and produces stable results even when the number of rectangles or strip width differs from that during training. [ABSTRACT FROM AUTHOR]

Probabilistic computing can solve complex combinatorial optimization problems more efficiently than conventional deterministic computing. A probabilistic bit (p‐bit) with an n‐p‐n bistable resistor (biristor) is demonstrated for probabilistic computing. It is fabricated on an 8‐inch wafer with complementary metal–oxide–semiconductor (CMOS) compatible technologies. Its stochastic behavior of threshold switching, which is based on the phenomenon of a single transistor latch, provides output with a Boltzmann distribution. The p‐bit is composed of a biristor, a serial resistor, and a comparator. The output probability of the biristor‐based p‐bits shows a sigmoidal relationship with the input voltage, showing typical p‐bit characteristics. Invertible Boolean logic operations with p‐bits are demonstrated, and weighted maximum Boolean satisfiability problems are solved with high energy efficiency and accuracy. The biristor‐based p‐bits with perfect CMOS compatibility show sufficient device stability, demonstrating the possibility of large‐scale integration with a p‐bit array for complex optimization solvers. [ABSTRACT FROM AUTHOR]

The photonic Ising machine is a new paradigm of optical computing that takes advantage of the unique properties of light wave propagation, parallel processing, and low-loss transmission. Thus, the process of solving combinatorial optimization problems can be accelerated through photonic/optoelectronic devices, but implementing photonic Ising machines that can solve arbitrary large-scale Ising problems with fast speed remains challenging. In this work, we have proposed and demonstrated the Phase Encoding and Intensity Detection Ising Annealer (PEIDIA) capable of solving arbitrary Ising problems on demand. The PEIDIA employs the heuristic algorithm and requires only one step of optical linear transformation with simplified Hamiltonian calculation by encoding the Ising spins on the phase term of the optical field and performing intensity detection during the solving process. As a proof of principle, several 20 and 30-spin Ising problems have been solved with high ground state probability (≥0.97/0.85 for the 20/30-spin Ising model). Photonic Ising machines exploit the parallelism and high propagation speed of light to solve combinatorial optimization tasks. The authors propose and demonstrate a photonic Ising machine with a fully reconfigurable optical vector-matrix transformation system and a modified algorithm based on simulated annealing, solving 20 and 30-spin Ising problems with high ground state probability. [ABSTRACT FROM AUTHOR]

Solving combinatorial optimization problems (COPs) poses a significant challenge in various application domains. The NP-hardness of many COPs necessitates the integration of meta-heuristics to effectively tackle these problems by leveraging the strengths of each meta-heuristic. However, hybrid meta-heuristics lack a mechanism to determine when to activate specific techniques during the search process. To address these limitations and enhance the overall generality, this study proposes a novel agent-based hyper-heuristic framework, referred to as IAFCO, for solving COPs. The proposed framework imbues the search strategy with greater intelligence and information, resulting in improved performance. The proposed framework comprises 4 coalitions that leverage different search perspectives to enhance the search process: a global coalition, a local coalition, a learning coalition, and a perturbation coalition. Additionally, a control agent is introduced to coordinate the coalitions. Using a reinforcement learning mechanism, the control agent autonomously selects the most suitable coalition of agents based on the search state. The set of states employed in this study is problem-independent and encompasses all possible states that may arise during the search process. Furthermore, some agents adopt chaotic numbers instead of random numbers to maintain diversity. To showcase the effectiveness of the proposed framework, its performance is evaluated on the software modularization problem (SMP) -also known as software clustering- and community detection problem (CDP) in complex networks as 2 case studies. The experiments on SMP's software systems, including 10 applications and 10 folders of Mozilla Firefox, and comparison with 21 modularization algorithms and experiments on CDP's networks, including 9 real-world networks, and comparison with 22 state-of-the-art community detection algorithms, indicate that the proposed framework is comparable to other existing methods. The source codes and datasets of this paper are available at https://github.com/mahjoubeh-t/IAFCO. [ABSTRACT FROM AUTHOR]

Due to its high spectral efficiency and various other advantages, filter bank multicarrier/offset quadrate amplitude modulation (FBMC/OQAM) has long been considered as a candidate waveform for the fifth generation (5G) and beyond telecommunication technologies. On the other hand, it is possible to both increase the data rate and alleviate the channel fading effects by using the multiple-input multiple-output (MIMO) antenna structure in the FBMC/OQAM transceiver. However, since the symbol detection is an indispensable task to be fulfilled in wireless communication, it is crucial to employ an efficient symbol detector at the MIMO-FBMC/OQAM receiver. Maximum likelihood (ML) detector, which always finds the optimal symbols by trying all of the possible symbol combinations likely to be transmitted, is known for its extremely high computational complexity making it impractical to be used in any system. On the other hand, it is possible to both considerably reduce the ML complexity and achieve the near-ML performance by optimizing the symbol vectors instead of implementing an exhaustive search. Since searching for the optimal symbol combination in discrete space is a combinatorial optimization problem, we developed a novel discrete harmony search (disHS) algorithm to perform this operation. According to the simulation results, the newly developed disHS algorithm not only achieves near-ML performance with lower computational complexity, but also clearly leaves behind the other symbol detectors considered in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the knapsack problem with conflict pair constraints. After a thorough literature survey on the topic, our study focuses on the special case of bipartite conflict graphs. For complete bipartite (multipartite) conflict graphs, the problem is shown to be NP-hard but solvable in pseudo-polynomial time, and it admits an FPTAS. Extensions of these results to more general classes of graphs are also presented. Further, a class of integer programming models for the general knapsack problem with conflict pair constraints is presented, which generalizes and unifies the existing formulations. The strength of the LP relaxations of these formulations is analyzed, and we discuss different ways to tighten them. Experimental comparisons of these models are also presented to assess their relative strengths. This analysis disclosed various strong and weak points of different formulations of the problem and their relationships to different types of problem data. This information can be used in designing special purpose algorithms for KPCC involving a learning component. [ABSTRACT FROM AUTHOR]

Linear assignment problems hold a pivotal role in combinatorial optimization, offering a broad spectrum of applications within the field of data sciences. They consist of assigning "agents" to "tasks" in a way that leads to a minimum total cost associated with the assignment. The assignment is balanced when the number of agents equals the number of tasks, with a one-to-one correspondence between agents and tasks, and it is and unbalanced otherwise. Additional options and constraints may be imposed, such as allowing agents to perform multiple tasks or allowing tasks to be performed by multiple agents. In this paper, we propose a novel framework that can solve all these assignment problems employing methodologies derived from the field of statistical physics. We describe this formalism in detail and validate all its assertions. A major part of this framework is the definition of a concave effective free energy function that encapsulates the constraints of the assignment problem within a finite temperature context. We demonstrate that this free energy monotonically decreases as a function of a parameter β representing the inverse of temperature. As β increases, the free energy converges to the optimal assignment cost. Furthermore, we demonstrate that when β values are sufficiently large, the exact solution to the assignment problem can be derived by rounding off the elements of the computed assignment matrix to the nearest integer. We describe a computer implementation of our framework and illustrate its application to multi-task assignment problems for which the Hungarian algorithm is not applicable. [ABSTRACT FROM AUTHOR]

In image processing, image segmentation is the process of partitioning a digital image into multiple image segments. Among state‐of‐the‐art methods, Markov random fields can be used to model dependencies between pixels and achieve a segmentation by minimising an associated cost function. Currently, finding the optimal set of segments for a given image modelled as a Markov random fields appears to be NP‐hard. The authors aim to take advantage of the exponential scalability of quantum computing to speed up the segmentation of synthetic aperture radar images. For that purpose, the authors propose a hybrid quantum annealing classical optimisation expectation maximisation algorithm to obtain optimal sets of segments. After proposing suitable formulations, the authors discuss the performances and the scalability of authors' approach on the D‐Wave quantum computer. The authors also propose a short study of optimal computation parameters to enlighten the limits and potential of the adiabatic quantum computation to solve large instances of combinatorial optimisation problems. [ABSTRACT FROM AUTHOR]

In recent years, the development of graph neural network (GNN) and deep reinforcement learning provides new methods to solve combinatorial optimization problems. Currently, most of these methods do not consider the problem of algorithm parameter learning. This paper developed an intelligent optimization model based on graph attention networks to solve this problem. The model automatically learned neighborhood search operators and destructive sequence terminators according to a significant amount of problem data, and trained model parameters based on reinforcement learning. This article used standard examples to test the model and conducted three different groups of experiments. The experimental results show that the learned neighborhood search operator has a remarkable ability to optimize and converge, while significantly reducing the training memory occupation. It can solve CVRP problems containing hundreds of nodes in a short time and has the potential for expansion. [ABSTRACT FROM AUTHOR]

We consider the problem of designing lightweight, load-bearing planar frame structures for additive manufacturing, which can be formulated as a nonlinear, non-convex mathematical programming problems. Even using state-of-the-art commercial solvers, exact methods are only capable of solving small unrealistic instances (with very few variables). In this paper, we develop a heuristic method which is fast and capable of solving the design problem for larger-scale, weight-optimized, planar frame structures for additive manufacturing. The approach explicitly considers manufacturability constraints stemming from the use of additive manufacturing technology and leverages the problem structure imposed by these constraints. The proposed heuristic method is based on iteratively resolving a relaxed master problem on a reduced ground structure. The approach differs from the existing methods in two important aspects. First, we consider planar frame optimization master problem directly (instead of simpler but less relevant truss optimization). Secondly, we employ both element and node addition, which allows us to enforce additive manufacturability constraints without using binary variables (and hence, avoiding the need for computationally expensive integer programming). [ABSTRACT FROM AUTHOR]

Protein–ligand docking plays a significant role in structure-based drug discovery. This methodology aims to estimate the binding mode and binding free energy between the drug-targeted protein and candidate chemical compounds, utilizing protein tertiary structure information. Reformulation of this docking as a quadratic unconstrained binary optimization (QUBO) problem to obtain solutions via quantum annealing has been attempted. However, previous studies did not consider the internal degrees of freedom of the compound that is mandatory and essential. In this study, we formulated fragment-based protein–ligand flexible docking, considering the internal degrees of freedom of the compound by focusing on fragments (rigid chemical substructures of compounds) as a QUBO problem. We introduced four factors essential for fragment–based docking in the Hamiltonian: (1) interaction energy between the target protein and each fragment, (2) clashes between fragments, (3) covalent bonds between fragments, and (4) the constraint that each fragment of the compound is selected for a single placement. We also implemented a proof-of-concept system and conducted redocking for the protein–compound complex structure of Aldose reductase (a drug target protein) using SQBM+, which is a simulated quantum annealer. The predicted binding pose reconstructed from the best solution was near-native (RMSD = 1.26 Å), which can be further improved (RMSD = 0.27 Å) using conventional energy minimization. The results indicate the validity of our QUBO problem formulation. [ABSTRACT FROM AUTHOR]

Network disintegration is a fundamental issue in the field of complex networks, with its core in identifying critical nodes or sets and removing them to weaken network functionality. The research on this problem has significant strategic value and has increasingly attracted attention, including in controlling the spread of diseases and dismantling terrorist organizations. In this paper, we focus on the problem of network disintegration with discrete entity resources from the attack view, that is, optimizing resource allocation to maximize the effect of network disintegration. Specifically, we model the network disintegration problem with limited entity resources as a nonlinear optimization problem and prove its NP-hardness. Then, we design a method based on deep reinforcement learning (DRL), Net-Cracker, which transforms the two-stage entity resource and network node selection task into a single-stage object selection problem. Extensive experiments demonstrate that compared with the benchmark algorithm, Net-Cracker can improve the solution quality by about 8∼62%, while enabling a 30-to-160-fold speed up. Net-Cracker also exhibits strong generalization ability and can find better results in a near real-time manner even when the network scale is much larger than that in training data. [ABSTRACT FROM AUTHOR]

In this paper, we study a robust two-stage concept for combinatorial optimization problems under discrete demand uncertainty. Combinatorial optimization problems are based on a finite set of elements for which we decide whether they are part of a solution. We divide the elements into two types, the so-called fixed and free elements. In a first stage, we irrecoverably decide whether some fixed elements are part of a solution despite uncertain demand. In a second stage, we decide whether some free elements complete a solution after a demand scenario is realized. The objective is to find a robust solution that minimizes the worst-case cost over a finite scenario set. We show the NP -hardness of this robust two-stage version of several specific combinatorial optimization problems under discrete demand uncertainty but provide a polynomial-time solvable special case. In particular, we apply this concept to three combinatorial optimization problems: the representative multi-selection, the shortest path, and the minimum weight perfect b -matching problem. We prove the NP -hardness and present special cases solvable in pseudo-polynomial and polynomial time. [ABSTRACT FROM AUTHOR]

Assigning patients to rooms is a fundamental task in hospitals and, especially, within wards. For this so-called patient-to-room assignment problem (PRA) many heuristics have been proposed with a large variety of different practical constraints. However, a thorough investigation of the problem's structure itself has been neglected so far. In this paper, we present insights about the basic, underlying combinatorial problem of PRA with a focus on minimizing the number of patient transfers which occur if patients have to change rooms during their stay. Particularly, we prove that in the case of double bedrooms, although minimizing the total number of transfers is NP -complete, each patient has to be transferred at most once, and provide upper bounds on the minimum total number of transfers. [ABSTRACT FROM AUTHOR]

Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that, given any initial configuration of pebbles, at least one pebble can be moved to a specified root vertex. Recent lines of inquiry apply computational techniques to pebbling bound generation and improvement. Along these lines, we present a computational framework that produces a set of tree strategy weight functions that are capable of proving pebbling number upper bounds on a connected graph. Our mixed-integer linear programming approach automates the generation of large sets of such functions and provides verifiable certificates of pebbling number upper bounds. The framework is capable of producing verifiable pebbling bounds on any connected graph, regardless of its structure or pebbling properties. We apply the model to the 4th weak Bruhat to prove π (B 4) ≤ 66 and to the Lemke square graph to produce a set of certificates that verify π (L □ L) ≤ 96. [ABSTRACT FROM AUTHOR]

The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1 / 2 -approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continues this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work Konrad and Naidu (Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX/RANDOM 2021, August 16–18, 2021. LIPIcs, vol 207, pp 19:1–19:18, 2021. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.19) shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1 / 2 -approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms were previously known, and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs. The main significance of our results is not in the numerical improvements, but in demonstrating the potential of non-maximal-matching-first algorithms. [ABSTRACT FROM AUTHOR]

We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in R d where the edge cost between two points is given by a pth power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and d ≥ 3 , 1 ≤ p < d . Previous results were limited to the range p < d / 2 . Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques. [ABSTRACT FROM AUTHOR]

We address an inventory-routing problem that arises in a liquid oxygen-producing company. Decisions must be made for the efficient transport of the product from sources to destinations by means of a heterogeneous fleet of trucks. This combinatorial problem has been stated as a constrained minimization one, whose objective function is the quotient of the operating cost divided by the total amount of delivered product. The operating cost comes from the distances traveled, the drivers' salary, and the drivers' overnight accommodation. The constraints include time windows for drivers and destinations, inventory safety levels, lower bounds for the quantity of product delivered to destinations, and maximum driving times. To approximate the optimal solution of this challenging problem, we developed a heuristic algorithm that first finds a feasible solution, and then iteratively improves it by combining the Metropolis criterion with local search. Our results are competitive with the best proposals in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a novel hybrid approach, fusing genetic algorithms (GA) and penguin search optimization (PSeOA), to address the flow shop scheduling problem (FSSP). GA utilizes selection, crossover, and mutation inspired by natural selection, while PSeOA emulates penguin foraging behavior for efficient exploration. The approach integrates GA's genetic diversity and solution space exploration with PSeOA's rapid convergence, further improved with FSSP-specific modifications. Extensive experiments validate its efficacy, outperforming pure GA, PSeOA, and other metaheuristics. [ABSTRACT FROM AUTHOR]

The development of information technology, wireless communications have become prevalent in every other field we can imagine. Sensor nodes are the fundamental element of wireless communication networks. When they are deployed, they gather environmental data from the surrounding area and send it to the base station for further analysis. This paper focused on creating a routing technique that improved wireless communication energy conservation through the use of fuzzy logic and ant colony optimisation in order to extend battery life. Ant colony optimisation algorithm (HFACO) and hybrid fuzzy logic were used in the construction of the routing protocol. Numerous derived techniques have been adapted to dynamic problems in real variables, stochastic problems, multi-targets, and parallel implementations. Ant colony optimisation algorithms have been applied to a wide range of combinatorial optimisation problems, from quadratic assignment to protein folding or routing vehicles. Several techniques that are based on how ant colonies forage have been used recently to solve challenging discrete optimisation problems.By taking into account the node's energy and traffic load, fuzzy logic was utilised to determine the total node cost to the gateway. Ant colony optimisation (ACO) was used to find the shortest path between the source and destination sensor nodes, and the path's value was determined by measuring its shortest distance. When compared to the ACO's performance in the same conditions, the Matlab simulation's results showed superior energy conservation performance. This improved routing technique can be applied to Wi-Fi sensor networks in industrial environments. [ABSTRACT FROM AUTHOR]

Symmetry is an important principle and characteristic that is prevalent in nature and artificial environments. In the three-dimensional packing problem, leveraging the inherent symmetry of goods and the symmetry of the packing space can enhance packing efficiency and utilization.The three-dimensional packing problem is an NP-hard combinatorial optimization problem in the field of modern logistics, with high computational complexity. This paper proposes an improved genetic algorithm by incorporating a fusion tabu search strategy to address this problem. The algorithm employs a three-dimensional loading mathematical model and utilizes a wall-building method under residual space constraints for stacking goods. Furthermore, adaptation of fitness variation strategy, chromosome adjustment, and tabu search algorithm are introduced to balance the algorithm's global and local search capabilities, as well as to enhance population diversity and convergence speed. Through testing on benchmark cases such as Bischoff and Ratcliff, the improved algorithm demonstrates an average increase of over 3% in packing space utilization compared to traditional genetic algorithms and other heuristic algorithms, validating its feasibility and effectiveness. The proposed improved genetic algorithm provides new insights for solving three-dimensional packing problems and optimizing logistics loading schedules, offering promising prospects for various applications. [ABSTRACT FROM AUTHOR]

The global market competition is becoming increasingly fierce, and manufacturing enterprises need to invest and expand. However, the traditional financial optimization of manufacturing enterprises has faced problems such as low efficiency and inaccurate search for the optimal solution, which has made manufacturing enterprises likely to face financial risks. The investment portfolio can enable enterprises to obtain the maximum profit on a certain risk level, or reduce their investment risk on a certain return level as far as possible. If the combinatorial optimization is realized, it can be applied to the optimal selection of manufacturing enterprises' financialization. This article analyzed the respective characteristics of Genetic Algorithm (GA) and Simulated Annealing (SA) algorithms, and analyzed the combination of GA and SA algorithms to solve the optimal investment portfolio through GA-SA algorithm, thereby helping manufacturing enterprises to make the optimal choice for financialization. The experimental results of this article indicated that the GA-SA algorithm solved the problem of GA algorithm easily falling into local optima, SA algorithm's initial temperature and generation mechanism, and improved the efficiency of finding the optimal solution. Meanwhile, the experimental results showed that the average optimal solutions of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms for 36 stock portfolios in Enterprise 1 were 69, 69, and 107, respectively. The average optimal solutions of the three algorithms for 36 stock portfolios in Enterprise 2 were 73, 90, and 112, respectively. This proves that the number of optimal solutions searched by GA-SA algorithm is higher than that of GA and SA algorithm, and also proves that it is effective to use GA-SA algorithm to optimize investment portfolio and help manufacturing enterprises to make optimal financial choices. [ABSTRACT FROM AUTHOR]

Optimal sensor placement (OSP) is a challenging combinatorial problem commonly addressed using Genetic algorithms (GAs), which are well suited to discrete problems. However, coding the problem can be difficult and often requires manual modifications during optimization. On the other hand, applying optimization methods designed for continuous problems to OSP is problematic due to its discrete nature. In this study, we propose a novel triple-structure coding approach that transforms OSP into a permutation and then a continuous optimization problem. This solves gene duplication in GAs and enables direct employment of all suitable methods for continuous problems in sensor placement optimization without any manual intervention. We evaluated the proposed method by implementing the encoding scheme with GA and mutated particle swarm optimization (MPSO) algorithms, two of the most renowned evolutionary algorithms. Additionally, we integrate modal identification within the optimization process for addressing the practicality of mode shape identification in a high-rise structure and a steel dome truss. The proposed coding reduces the cost of GA by 7 to 10 percent and MPSO by 25 to 54 percent, showcasing advancements in cost reduction within the context of sensor placement optimization. Moreover, the percentage of shared nodes in placements obtained from analytical and modal identification dropped to 34% in certain scenarios for the high-rise structure and 26% for the steel dome truss. This emphasizes the substantial distinctions in placements resulting from modal identification using structural responses compared to those obtained exclusively from analytical mode shapes. [ABSTRACT FROM AUTHOR]

Ultra-high-pressure (UHP) water jetting is commonly used to remove rust from ship hulls. However, few studies have been undertaken to optimize the topology of reaction rotating sprinklers (RRS) using UHP. This work proposes two contributions aimed at optimizing the RRS topology and maximizing the efficiency of rust removal. The first contribution is a novel accumulative impinging-duration model, which can accurately quantify the spiral trajectory density of multiple water jets and provide a practical means to evaluate the feasibility of RRS topology. The second contribution is the development of a new meta-heuristic algorithm called adaptive genetic algorithm (AGA), which offers high-quality solutions for the combinatorial topology optimization problem associated with RRS. The AGA converges significantly faster than the conventional genetic algorithm (CGA) and avoids being trappped at a local optimum. The effectiveness of the proposed topology optimization method is demonstrated through two case studies involving RRS with different model specifications. The AGA significantly increases the utilization ratio of water-jet energy by more than 50.6 %, outperforming CGA. The rust-removal efficiency of AGA-optimized RRS is theoretically predicted by analyzing the maximal translation speed and further validated through field experiments. Results show that the rust-removal efficiency of AGA-optimized RRS for ship hulls is greatly improved, with improvements of over 50 % and 43 % in theoretical and practical verification, respectively. This research introduces a unique quantization method and an efficient optimization algorithm for practical RRS design. [ABSTRACT FROM AUTHOR]

We determine the sensitivity of a current optimal solution to a combinatorial optimization problem to cost changes in a set of elements. In a recent study, the concept of regular set tolerances has been introduced for a combinatorial optimization problem and for three types of cost functions, namely sum, product, and bottleneck. A regular set tolerance is the supremum amount of cost changes that can be distributed in the most favorable way to multiple elements such as not to change the current optimal solution. In this paper, we introduce an alternative concept, namely the reverse set tolerance , which is a measure of the infimum amount of cost changes to multiple elements such that the current optimal solution becomes non-optimal. We characterize the specific cases in which reverse set upper and lower tolerances have positive values and in which they are infinite. We also show a criterion for the uniqueness of an optimal solution. Furthermore, we present bounds and exact formulas for reverse set upper and lower tolerances using the relation to their corresponding single tolerance counterparts. We discuss the similarities and differences in the results between regular and reverse set tolerances. Finally, we motivate this new concept by analyzing them for special combinatorial optimization problems with important practical applications. [ABSTRACT FROM AUTHOR]

In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s , visit every city exactly once, and finally end his trip at another given city t. In this paper we show that a special case of the Path-TSP with a Demidenko distance matrix is solvable in polynomial time. Demidenko distance matrices fulfill a particular condition abstracted from the convex Euclidian special case by Demidenko (1979) as an extension of an earlier work of Kalmanson (1975). We identify a number of crucial combinatorial properties of the optimal solution and design a dynamic programming approach with time complexity O (n 6). [ABSTRACT FROM AUTHOR]

In classical graph theory, the minimal spanning tree (MST) is a subgraph that lacks cycles and efficiently connects every vertex by utilizing edges with the minimum weights. The computation of a minimum spanning tree for a graph has been a pervasive problem over time. However, in practical scenarios, uncertainty often arises in the form of fuzzy edge weights, leading to the emergence of the Fuzzy Minimum Spanning Tree (FMST). This specialized approach is adept at managing the inherent uncertainty present in edge weights within a fuzzy graph, a situation commonly encountered in real-world applications. This study introduces the initial optimization approach for the Minimum Spanning Tree Problem within the context of interval-valued fermatean neutrosophic domain. The proposed solution involves the adaptation of the Dhouib-Matrix-MSTP (DM-MSTP) method, an innovative technique designed for optimal resolution. The DM-MSTP method operates by employing a column-row navigation strategy through the adjacency matrix. To the best of our knowledge, instances of this specific problem have not been addressed previously. To address this gap, a case study is generated, providing a comprehensive application of the novel DM-MSTP method with detailed insights into its functionality and efficacy. [ABSTRACT FROM AUTHOR]

Steel slabs are key in-process products in the steel production process. The slab allocation problem is to allocate slabs to suitable orders by considering complicated technical restrictions and multiple management requirements. In practice, thousands of slabs in multiple production lines should be allocated to orders. Due to the complexity and large scale of the problem, it takes planners a long time to make decisions, and the obtained allocation schemes are usually ineffective because of the myopic nature of the rule-based methods. In this article, we present a learning-based solution method for solving a practical slab allocation problem in multiple hot rolling lines. We formulate the problem as an integer programming model, then a data-based method is adopted to evaluate the mismatching cost between slabs and orders. To effectively solve the large-sized problem, a learning-based decomposition strategy is proposed to decompose the original problem into several small-sized subproblems. Then, a branch-and-price algorithm is proposed to optimally solve the subproblems. To further speed up the solution process, a primal heuristics based on column generation is designed to solve the large-sized subproblems. The solution methods have been implemented in a steel company and effectively increased slab utilization and reduced production cost. [ABSTRACT FROM AUTHOR]

A 2‐factor in a graph G $G$ is a subset of edges M $M$ such that every node of G $G$ is incident with exactly two edges of M $M$. Many results are known concerning 2‐factors including a polynomial‐time algorithm for finding 2‐factors and a characterization of those graphs that have a 2‐factor. The problem of finding a 2‐factor in a graph is a relaxation of the NP‐hard problem of finding a Hamilton cycle. A stronger relaxation is the problem of finding a triangle‐free 2‐factor, that is, a 2‐factor whose edges induce no cycle of length 3. In this paper, we present a polynomial‐time algorithm for the problem of finding a triangle‐free 2‐factor as well as a characterization of the graphs that have such a 2‐factor and related min–max and augmenting path theorems. [ABSTRACT FROM AUTHOR]

Identifying the rank of species in a complex ecosystem is a difficult task, since the rank of each species invariably depends on the interactions stipulated with other species through the adjacency matrix of the network. A common ranking method in economic and ecological networks is to sort the nodes such that the layout of the reordered adjacency matrix looks maximally nested with all nonzero entries packed in the upper left corner, called Nestedness Maximization Problem (NMP). Here we solve this problem by defining a suitable cost-energy function for the NMP which reveals the equivalence between the NMP and the Quadratic Assignment Problem, one of the most important combinatorial optimization problems, and use statistical physics techniques to derive a set of self-consistent equations whose fixed point represents the optimal nodes' rankings in an arbitrary bipartite mutualistic network. Concurrently, we present an efficient algorithm to solve the NMP that outperforms state-of-the-art network-based metrics and genetic algorithms. Eventually, our theoretical framework may be easily generalized to study the relationship between ranking and network structure beyond pairwise interactions, e.g. in higher-order networks. Species forming complex ecological or economic ecosystems are organized in hierarchies and the ranks of such species are determined by the adjacency matrix of their interaction network. We introduce a framework to calculate the ranks of species by finding the optimal permutation of rows and columns that makes the adjacency matrix maximally nested. [ABSTRACT FROM AUTHOR]

Prompt optimization is a crucial task for improving the performance of large language models for downstream tasks. In this paper, a prompt is a sequence of n-grams selected from a vocabulary. Consequently, the aim is to select the optimal prompt concerning a certain performance metric. Prompt optimization can be considered as a combinatorial optimization problem, with the number of possible prompts (i.e., the combinatorial search space) given by the size of the vocabulary (i.e., all the possible n-grams) raised to the power of the length of the prompt. Exhaustive search is impractical; thus, an efficient search strategy is needed. We propose a Bayesian Optimization method performed over a continuous relaxation of the combinatorial search space. Bayesian Optimization is the dominant approach in black-box optimization for its sample efficiency, along with its modular structure and versatility. We use BoTorch, a library for Bayesian Optimization research built on top of PyTorch. Specifically, we focus on Hard Prompt Tuning, which directly searches for an optimal prompt to be added to the text input without requiring access to the Large Language Model, using it as a black-box (such as for GPT-4 which is available as a Model as a Service). Albeit preliminary and based on "vanilla" Bayesian Optimization algorithms, our experiments with RoBERTa as a large language model, on six benchmark datasets, show good performances when compared against other state-of-the-art black-box prompt optimization methods and enable an analysis of the trade-off between the size of the search space, accuracy, and wall-clock time. [ABSTRACT FROM AUTHOR]

The mathematical side of applied problems in multiple subject areas (biology, pattern recognition, etc.) is reduced to the problem of discrete optimization in the following mathematical method. We were provided a network and graphs in its leaves, for which we needed to find a rearrangement of graphs by non-leaf nodes, in which the given functional reached its minimum. Such a problem, even in the simplest case, is NP-hard, which means unavoidable restrictions on the network, on graphs, or on the functional. In this publication, this problem is addressed in the case of all graphs being so-called "structures", meaning directed-loaded graphs consisting of paths and cycles, and the functional as the sum (over all edges in the network) of distances between structures at the endpoints of every edge. The distance itself is equal to the minimal length of sequence from the fixed list of operations, the composition of which transforms the structure at one endpoint of the edge into the structure at its other endpoint. The list of operations (and their costs) on such a graph is fixed. Under these conditions, the given discrete optimization problem is called the reconstruction problem. This paper presents novel algorithms for solving the reconstruction problem, along with full proofs of their low error and low polynomial complexity. For example, for the network, the problem is solved with a zero error algorithm that has a linear polynomial computational complexity; and for the tree the problem is solved using an algorithm with a multiplicative error of at most two, which has a second order polynomial computational complexity. [ABSTRACT FROM AUTHOR]

Deep learning has been widely used in various research fields. However, researchers have discovered that deep learning models are vulnerable to adversarial attacks. Existing word-level attacks can be seen as a combinatorial optimization problem to effectively conduct textual adversarial attacks, but inappropriate search spaces and search methods may affect attack effectiveness. Sentence-level attacks are successfully used in the field of reading comprehension, but the generated examples sometimes lead to semantic deviation. To address these issues, we propose a hybrid textual adversarial attack method that effectively enhances the performance of textual adversarial attacks. To the best of our knowledge, we are the first to conduct textual adversarial attacks by hybridizing Whale Optimization Algorithm (WOA) with style transfer from multiple sentence and word levels. The WOA is improved by incorporating data characteristics and the Metropolis criterion to escape from local optima and by leveraging the mutation operator to increase population diversity. The improved WOA and style transfer algorithm are fused in a parallel and vertical way. Style transfer can increase population diversity and expand the search space, usually without destroying the semantics and syntax of sentences. The parallel combination improves attack performance by attacking from both word-level and sentence-level perspectives. As a black-box attack model, our method can attack without knowing the internal structure of the model. Compared with the state-of-the-art method, our framework can improve the attack success rate by 6.8%. Additionally, further experiments on grammatical error increase rates, semantic consistency, and transferability demonstrate that our model has excellent performance in many respects. [ABSTRACT FROM AUTHOR]

• A graph neural network is proposed for learning multi-satellite scheduling policy. • It analyzes the differential impact of various constraints in scheduling. • It models the multi-satellite scheduling problem as a heterogeneous graph. • It designs a variant transformer to learn various relations within graph. • It designs a two-stage model to learn a multi-satellite scheduling policy. In this paper, we investigate the multi-satellite scheduling (MSS) problem, a combinatorial optimization problem (COP) that aims to enhance satellite utilization. Numerous scholars have conducted in-depth studies on this matter. Handcrafted policies are commonly employed in various algorithms. Despite the use of graphical representations, designing policies manually becomes increasingly difficult, particularly as the number of satellites and tasks grows. Consequently, we propose a novel approach for automatically summarizing graph structures and learning policies for MSS problem. The proposed approach is developed based on graph neural network, Transformer, and sequence to sequence framework. First, MSS problem is modeled as a heterogeneous graph following various constraints. Next, the encoder, which is based on GNN and Transformer, is designed to summarize crucial information from the aforementioned graph. Finally, a two-stage decoder is proposed taking into account dynamic information in stepwise solution generation. Since obtaining optimal solutions in practical applications is challenging, we train the model using reinforcement learning. The results of the experiment demonstrate that our proposed method performs effectively in various scenarios. The quality of the solutions generated through this method transcends that of conventional construction methods. Additionally, the method generates solutions quickly, which are comparable in quality to those produced by meta-heuristic algorithms. [ABSTRACT FROM AUTHOR]