Analysis of Iterated Greedy Heuristic for Vertex Clique Covering

This paper presents a new probabilistie primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.

This paper proposes a tabu mechanism improved iterated greedy (TMIIG) algorithm to solve the no-wait flow-shop scheduling problem with makespan criterion. The motivation of seeking for further improvement in the iterated greedy (IG) algorithm framework is based on the observation that the construction phase of the original IG algorithm may lead to repeated search when applying the insertion neighborhood search. To overcome the drawback, we modified the IG algorithm by a tabu-based reconstruction strategy to enhance its exploitation ability. A powerful neighborhood search method which involves insert, swap, and double-insert moves is then applied to obtain better soluions from the reconstructed solution in the previous step. Numerical computations verified the advantages of utilizing the new reconstruction scheme. In addition, comparisons with other high-performing algorithms demonstrated the effectiveness and robustness of the proposed algorithm.

• Examines scheduling on m identical parallel machines with a common server. • Considers sequence-dependent setup times in parallel machine scheduling. • Introduces an arc-based MILP formulation for makespan minimization. • Presents a GRASP-enhanced Iterated Greedy algorithm with strong performance. • Reports 21 new best-known solutions across benchmark test instances. The identical parallel machine scheduling problem with a single server and sequence-dependent setup times is a challenging optimization problem with important applications in manufacturing and service industries. In such environments, several machines depend on a common server to perform setup operations before production can begin, which creates strong interdependencies and demands more effective scheduling strategies. This characteristic highlights the practical relevance of the problem. The interaction between machine availability and server operations often becomes a critical bottleneck. This study introduces two complementary approaches. The first is an exact method based on a novel arc-based mixed-integer linear programming (ABF) model, which extends the modeling capability of existing formulations by capturing server-related constraints more effectively. The second is an approximation method built on an Iterated Greedy (IG) algorithm. The IG procedure is improved by two evaluation mechanisms: one model-based evaluation derived from the proposed ABF model, and another employing a greedy randomized adaptive search procedure (GRASP)-based strategy that integrates greedy selection, randomization, and reconstruction to enhance solution quality. Computational experiments are conducted on existing benchmark instances. The results show that the proposed ABF model performs well on small and medium-sized instances compared to existing exact methods, while the IG variants, particularly the proposed GRASP-based version, deliver strong performance against state-of-the-art metaheuristics developed for this problem. In addition, 21 new best-known solutions are reported, further demonstrating the effectiveness of the proposed approaches.

In recent years, the distributed permutation flowshop scheduling problem (DPFSP) has been widely studied. In this paper, we extended the DPFSP by considering preventive maintenance (PM) operation to prevent machines from breaking down after the long process. An iterated greedy (IG) algorithm is developed to minimize total flowtime. A heuristic with swapping operator is proposed to initialize the IG. After that, the destruction phase and construction phase are modified to fit our problem. A local search is then applied to further improve the solution generated in the construction stage. At last, a simple simulated annealing-like acceptance criterion is used to prevent local optimal situations. Comparison with three state-of-the art algorithms in the recent literature based on 225 instances shows the high performance of our IG algorithm for solving the DPFSP with PM operation.

We present an algorithm for solving 3SAT instances. Several algorithms have been proved to work whp (with high probability) for various SAT distributions. However, an algorithm that works whp has a drawback. Indeed for typical instances it works well, however for some rare inputs it does not provide a solution at all. Alternatively, one could require that the algorithm always produce a correct answer but perform well on average. Expected polynomial time formalizes this notion. We prove that for some natural distribution on 3CNF formulas, called planted 3SAT, our algorithm has expected polynomial (in fact, almost linear) running time. The planted 3SAT distribution is the set of satisfiable 3CNF formulas generated in the following manner. First, a truth assignment is picked uniformly at random. Then, each clause satisfied by it is included in the formula with probability p. Extending previous work for the planted 3SAT distribution, we present, for the first time for a satisfiable SAT distribution, an expected polynomial time algorithm. Namely, it solves all 3SAT instances, and over the planted distribution (with p = d/n2, d > 0 a sufficiently large constant) it runs in expected polynomial time. Our results extend to k-SAT for any constant k.

The antiferromagnetic Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a connectivity (or degree) distribution P(k) ~ k^(- gamma). The disorder present in these complex networks frustrates the antiferromagnetic spin ordering, giving rise to a spin-glass (SG) phase at low temperature. The paramagnetic-SG transition temperature T_c has been studied as a function of the parameter gamma and the minimum degree present in the networks. T_c is found to increase when the exponent gamma is reduced, in line with a larger frustration caused by the presence of nodes with higher degree.

Iterated reference greedy algorithm for solving distributed no-idle permutation flowshop scheduling problems

Although widely applied in optimisation, relatively little has been proven rigorously about the role and behaviour of populations in randomised search processes. This paper presents a new method to prove upper bounds on the expected optimisation time of population-based randomised search heuristics that use non-elitist selection mechanisms and unary variation operators. Our results follow from a detailed drift analysis of the population dynamics in these heuristics. This analysis shows that the optimisation time depends on the relationship between the strength of the selective pressure and the degree of variation introduced by the variation operator. Given limited variation, a surprisingly weak selective pressure suffices to optimise many functions in expected polynomial time. We derive upper bounds on the expected optimisation time of non-elitist evolutionary algorithms (EA) using various selection mechanisms, including fitness proportionate selection. We show that EAs using fitness proportionate selection can optimise standard benchmark functions in expected polynomial time given a sufficiently low mutation rate. As a second contribution, we consider an optimisation scenario with partial information, where fitness values of solutions are only partially available. We prove that non-elitist EAs under a set of specific conditions can optimise benchmark functions in expected polynomial time, even when vanishingly little information about the fitness values of individual solutions or populations is available. To our knowledge, this is the first runtime analysis of randomised search heuristics under partial information.

In this paper, a problem arising in the planning of specialized diagnostic services in a segmented public healthcare system is addressed. The problem consists of deciding which hospitals will provide the service and their capacity levels, the allocation of demand in each institution, and the reallocation of uncovered demand to other institutions or private providers, while minimizing the total equivalent annual cost of investment and operating cost required to satisfy all the demand. An associated mixed-integer linear programming model can be solved by conventional branch and bound for relatively small instances; however, for larger instances the problem becomes intractable. To effectively address larger instances, a hybrid metaheuristic framework combining iterated greedy (IGA) and variable neighborhood descent (VND) components for this problem is proposed. Two greedy construction heuristics are developed, one starting with an infeasible solution and iteratively adding capacity and the other starting with a feasible, but expensive, solution and iteratively decrease capacity. The iterated greedy algorithm includes destruction and reconstruction procedures. Four different neighborhood structures are designed and tested within a VND procedure. In addition, the computation of local search components benefit from an intelligent exploitation of problem structure since, when the first-level location variables (hospital location and capacity) are fixed, the remaining subproblem can be solved efficiently as it is very close to a transshipment problem. All components and different strategies were empirically assessed both individually and within the IGA-VND framework. The resulting metaheuristic is able to obtain near optimal solutions, within 3% of optimality, when tested over a data base of 60- to 300-hospital instances.

Metaheuristics for the permutation flowshop problem with a weighted quadratic tardiness objective

A distributed heterogeneous permutation flowshop scheduling problem with sequence‐dependent setup times (DHPFSP‐SDST) is addressed, which well reflects real‐world scenarios in heterogeneous factories. The objective is to minimise the maximum completion time (makespan) by assigning jobs to factories, and sequencing them within each factory. First, a mathematical model to describe the DHPFSP‐SDST is established. Second, four meta‐heuristics, including genetic algorithms, differential evolution, artificial bee colony, and iterated greedy (IG) algorithms are improved to optimally solve the concerned problem compared with the other existing optimisers in the literature. The Nawaz‐Enscore‐Ham (NEH) heuristic is employed for generating an initial solution. Then, five local search operators are designed based on the problem characteristics to enhance algorithms' performance. To choose the local search operators appropriately during iterations, Q‐learning‐based strategy is adopted. Finally, extensive numerical experiments are conducted on 72 instances using 5 optimisers. The obtained optimisation results and comparisons prove that the improved IG algorithm along with Q‐learning based local search selection strategy shows better performance with respect to its peers. The proposed algorithm exhibits higher efficiency for scheduling the concerned problems.

In this paper we develop new algorithms for factoring polynomials over finite fields by exploring an interesting connection between the algebraic factoring problem and the combinatorial problem of stable coloring of tournaments. We present an algorithm which can be viewed as a recursive refinement scheme through which most cases of polynomials are completely factored in deterministic polynomial time within the first level of refinement, most of the remaining cases are factored completely before the end of the second level refinement, and so on. The algorithm has average polynomial time complexity and (n \(^{\rm log{\it n}}\) logp)O(1) worst case complexity. Under a purely combinatorial conjecture concerning tournaments, the algorithm has worst case complexity (n \(^{\rm loglog{\it n}}\) logp)O(1). Our approach is also useful in reducing the amount of randomness needed to factor a polynomial completely in expected polynomial time. We present a random polynomial time algorithm for factoring polynomials over finite fields which requires only log p random bits. All these results assume the Extended Riemann Hypothesis.KeywordsPolynomial TimeZero DivisorStable ColoringPrimitive IdempotentFactoring PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

During the last decade, many problems in social, biological, and financial networks require finding cliques, or quasi-cliques. Cliques or clique partitions have also been used as clustering or classification tools in data sets represented by networks. These networks can be very large and often massive and therefore external (or semi-external) memory algorithms are needed. We discuss four applications where we identify computational challenges which are both of practical and theoretical interest.

A referenced iterated greedy algorithm for the distributed assembly mixed no-idle permutation flowshop scheduling problem with the total tardiness criterion

The growth in e-commerce sales has been constant and, driven by the Covid-19 pandemic, has exceeded previous expectations. Owing to social isolation, parcel lockers - systems for goods collection - have become widely used to reduce physical contact between people involved in the delivery process. This resource has also become an important strategy for reducing costs, as it enables deliveries from multiple customers to be consolidated in the same location. In the context of smart cities, in which the optimization of urban transport plays a key role in building more sustainable and efficient environments, the inclusion of parcel lockers is essential. Thus, research in this area that considers their use aligns with the concept of smart cities, promoting the use of innovative technologies to improve mobility and urban logistics. Specifically in this context, the Vehicle Routing Problem With Parcel Lockers (VRPPL) is an extension of the Vehicle Routing Problem with TimeWindows (VRPTW), introducing the use of lockers, with the objective to reduce the total distance traveled. In this study, an algorithm based on Iterated Greedy (IG) and Random Variable Neighborhood Descent (RVND) was developed to reduce the costs and computational times presented in the literature. We selected a set of 168 well-known instances for the problem. Computational experiments and comparative analyses were performed. The developed algorithm achieved better results in 167 instances, with an average gap of −1.315%. The best execution times were obtained for all instances, thereby reducing the mean execution time from 240.293s to 0.104s.