Exact Algorithm Research Articles

Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines

Abstract We consider the problem of scheduling jobs with equal lengths and arbitrary sizes on uniform parallel batch machines with different capacities. Each machine can only process the jobs whose sizes are not larger than its capacity. Several jobs can be processed as a batch simultaneously on a machine, as long as their total size does not exceed the machine’s capacity. The objective is to minimize makespan. Under a divisibility constraint, we obtain two efficient exact algorithms. For the general problem, we obtain an efficient 2-approximation algorithm. Previous work has shown that the problem cannot be approximated to within an approximation ratio better than 2, unless P = NP, even when all machines have identical speeds and capacities.

Exact and Heuristic Multi-Robot Dubins Coverage Path Planning for Known Environments.

Coverage path planning (CPP) of multiple Dubins robots has been extensively applied in aerial monitoring, marine exploration, and search and rescue. Existing multi-robot coverage path planning (MCPP) research use exact or heuristic algorithms to address coverage applications. However, several exact algorithms always provide precise area division rather than coverage paths, and heuristic methods face the challenge of balancing accuracy and complexity. This paper focuses on the Dubins MCPP problem of known environments. Firstly, we present an exact Dubins multi-robot coverage path planning (EDM) algorithm based on mixed linear integer programming (MILP). The EDM algorithm searches the entire solution space to obtain the shortest Dubins coverage path. Secondly, a heuristic approximate credit-based Dubins multi-robot coverage path planning (CDM) algorithm is presented, which utilizes the credit model to balance tasks among robots and a tree partition strategy to reduce complexity. Comparison experiments with other exact and approximate algorithms demonstrate that EDM provides the least coverage time in small scenes, and CDM produces a shorter coverage time and less computation time in large scenes. Feasibility experiments demonstrate the applicability of EDM and CDM to a high-fidelity fixed-wing unmanned aerial vehicle (UAV) model.

Finding Experts in Community Question Answering System Using Trie String Matching Algorithm with Domain Knowledge

Different community question-answering systems have been evolving as online fora for knowledge sharing among users across the world. However, finding experts on the questions is a key issue in the community question-answering system. To address this issue, a novel approach is proposed in this paper for finding experts from the community question-answering website using an exact string-matching algorithm along with domain knowledge. The proposed system is designed with three phases: i) User Profile Modeling, ii) Question Preprocessing, and iii) Expert Recommendation. Initially, the matrix factorization model is adopted for user profile modeling where the tags and the answerer’s information are represented in the form of a matrix. Then, the domain-wise grouping of tags is done to minimize the search time of the tags. The question preprocessing is done using a keyword extraction algorithm to extract the keywords. Finally, the expert recommendation is accomplished using the trie string matching algorithm and key-value mapping process. For doing experiments, a stack overflow community question-answering website is utilized in this work. The performance of the system is measured and the results section proved that the system achieved 91.2% accuracy.

Problem characterization of unique shortest path routing

To demonstrate the necessity and importance of solving the unique shortest path routing problem based on and tailored to a complete formulation, the problem characteristics with respect to objective functions, constraints, and solution qualities are studied in this paper, including the impacts of objective functions and lower & upper bounds of the problem. To illustrate the advantage of developing an exact solution approach to the problem, the resulting routing performance from the unique shortest path routing problem is compared with those from using two default methods as well as those from two relaxed problems. It is shown that by solving the problem based on a complete formulation and an exact algorithm, the resulting routing performance is notably improved, relative to that from using the two default methods, and is competitively close to the lower bounds.

A two-objective optimization of ship itineraries for a cruise company

This paper deals with the problem of cruise itinerary planning which plays a central role in worldwide cruise ship tourism. In particular, the Day-by-day Cruise Itinerary Optimization (DCIO) problem is considered. Assuming that a cruise has been planned in terms of homeports and journey duration, the DCIO problem consists in determining the daily schedule of each itinerary so that some Key Performance Indicators are optimized. The schedule of an itinerary, i.e. the sequence of visited ports, the arrival and departure time at each port, greatly affect cruise operative costs and attractiveness. We propose a Mixed Integer Linear Programming (MILP) formulation of the problem with the objective of minimizing the itinerary cost due to fuel and port costs, while maximizing an itinerary attractiveness index. This latter is strongly related to the ports visited as well as to the overall schedule of the itinerary. Therefore the problem turns out to be a bi-objective optimization problem. We provide its solution in terms of Pareto optimal solution points. Each single objective MILP problem which arises is solved by using an exact algorithm, implemented in a commercial solver. We consider the day-by-day itineraries of a major luxury cruise company in many geographical areas all over the world. Here we report, as illustrative examples, the results obtained on some of these real instances.

Replica symmetry breaking for Ulam's problem

We study increasing subsequences (IS) for an ensemble of sequences given by permutation of numbers {1,2,...,n}. We consider a Boltzmann ensemble at temperature T. Thus each IS appears with the corresponding Boltzmann probability where the energy is the negative length -l of the IS. For T -> 0, only ground states, i.e. longest IS (LIS) contribute, also called Ulam's problem. We introduce an algorithm which allows us to directly sample IS in perfect equilibrium in polynomial time, for any given sequence and any temperature. Thus, we can study very large sizes. We obtain averages for the first and second moments of number of IS as function of $n$ and confirm analytical predictions. Furthermore, we analyze for low temperature $T$ the sampled ISs by computing the distribution of overlaps and performing hierarchical cluster analyses. In the thermodynamic limit the distribution of overlaps stays broad and the configuration landscape remains complex. Thus, Ulam's problem exhibits replica symmetry breaking. This means it constitutes a model with complex behavior which can be studied numerically exactly in a highly efficient way, in contrast to other RSB-showing models, like spin glasses or NP-hard optimization problems, where no fast exact algorithms are known.

Review of Research on Vehicle Routing Problem and Related Algorithms

This paper briefly introduces the relevant theories and wide applications of vehicle routing problem, and then summarizes some common exact algorithms and heuristic algorithms for solving the vehicle routing problem. Finally, two algorithms are summarized for solving the vehicle routing problem of different scales. For the vehicle routing problem of smaller scale, the exact algorithm can directly solve the exact optimal solution of the problem; However, the vehicle routing problem with large scale is generally solved by heuristic algorithm.

Comparing Direct Measurements and Three-Dimensional (3D) Scans for Evaluating Facial Soft Tissue

The inspection of patients' soft tissues and the effects of various dental procedures on their facial physiognomy are quite challenging. To minimise discomfort and simplify the process of manual measuring, we performed facial scanning and computer measurement of experimentally determined demarcation lines. Images were acquired using a low-cost 3D scanner. Two consecutive scans were obtained from 39 participants, to test the scanner repeatability. An additional ten persons were scanned before and after forward movement of the mandible (predicted treatment outcome). Sensor technology that combines red, green, and blue (RGB) data with depth information (RGBD) integration was used for merging frames into a 3D object. For proper comparison, the resulting images were registered together, which was performed with ICP (Iterative Closest Point)-based techniques. Measurements on 3D images were performed using the exact distance algorithm. One operator measured the same demarcation lines directly on participants; repeatability was tested (intra-class correlations). The results showed that the 3D face scans were reproducible with high accuracy (mean difference between repeated scans <1%); the actual measurements were repeatable to some extent (excellent only for the tragus-pogonion demarcation line); computational measurements were accurate, repeatable, and comparable to the actual measurements. Three dimensional (3D) facial scans can be used as a faster, more comfortable for patients, and more accurate technique to detect and quantify changes in facial soft tissue resulting from various dental procedures.

Geophysical joint inversion based on mixed structural and rock-physics coupling constraints

Joint inversion algorithms continue to receive a lot of interest in the application of geophysical interpretation due to the reduction of uncertainty of the inverse solutions. Over the past decade, several coupling constraints, structural or rock physics, have been introduced to the joint inversion of multiple geophysical data sets. However, a single coupling constraint may not provide the optimum solution for all geologic scenarios. To improve the application of the multiple geophysical data sets, in which some rock-physics information is available, we have developed a new joint inversion algorithm based on a mixed structural and rock-physics coupling constraints. The presented algorithm incorporates the cosine dot product gradient function, a structural coupling constraint, with a guided fuzzy c-means clustering strategy, which is a rock-physics coupling constraint. The former imposes the structural similarity among different model parameters, whereas the latter improves the linear or nonlinear correlation among model parameters. We use the exact structural resemblance algorithm to numerically solve the mixed coupling constraint joint inversion objective function. The developed algorithm is applied on 2D synthetic and field data examples. Our results find a significant improvement in the resolution of the reconstructed models, in comparison with traditional single coupling algorithms, even when the rock-physics information is not complete. This algorithm is more conducive to obtaining clear geologic unit boundaries and provides a favorable basis for delineating metallogenic target areas.

Butterfly counting and bitruss decomposition on uncertain bipartite graphs

AbstractUncertain butterflies are one of, if not the, most important graphlet structures on uncertain bipartite networks. In this paper, we examine the uncertain butterfly structure (in which the existential probability of the graphlet is greater than or equal to a threshold parameter), as well as the global Uncertain Butterfly Counting Problem (to count the total number of these instances over an entire network). To solve this task, we propose a non-trivial exact baseline (UBFC), as well as an improved algorithm (IUBFC) which we show to be faster both theoretically and practically. We also design two sampling frameworks (UBS and PES) which can sample either a vertex, edge or wedge from the network uniformly and estimate the global count quickly. Furthermore, a notable butterfly-based community structure which has been examined in the past is the k-bitruss. We adapt this community structure onto the uncertain bipartite graph setting and introduce the Uncertain Bitruss Decomposition Problem (which can be used to directly answer any k-bitruss search query for any k). We then propose an exact algorithm (UBitD) to solve our problem with three variations in deriving the initial uncertain support. Using a range of networks with different edge existential probability distributions, we validate the efficiency and effectiveness of our solutions.

An Interactive Application for Learning and Analyzing Different Graph Vertex Cover Algorithms

This paper deals with an analysis of three algorithms for the graph vertex cover problem. Certain methods for solving this problem are analyzed. In addition, different studies on the problem and some approaches to its solution are discussed as well. An exact algorithm (based on the backtracking approach) is presented. Calculating the average time for execution of this algorithm is consistent with the multitasking way of work of the operating system. For this purpose, four different starts of the algorithm are made and then the average time of all of them is calculated. The exact algorithm found the optimal solutions for all analyzed graphs. Besides this algorithm, two other heuristic algorithms for solving the problem are discussed. For this study, an interactive application is developed to visualize the performance of the three algorithms and display the obtained results. The results show that for small graphs with no more than 25 vertices the exact algorithm can be used to solve optimally the graph vertex cover problem. For the largest graphs, none of the two heuristic algorithms found the optimal solutions, but these algorithms generated solutions that are very close to the optimal ones. In summary, when the size of the graph increases linearly, the execution time of the heuristic algorithms increases linearly, while the execution time of the exact algorithm increases exponentially.

A Hybrid Algorithm for the Heterogeneous Fixed Fleet Open Vehicle Routing Problem with Time Windows

Nowadays, using a rental fleet to transport goods for delivering to customers at a particular time frame is very important in the services and industry. That is why, in this study, we consider the heterogeneous fixed fleet open vehicle routing problem with time windows, which is one version of the vehicle routing problem with time windows. The problem has not attracted attention so much in the operational research literature than the usual vehicle routing problem. The problem consists of determining the minimum cost routes for a fleet of a fixed number of vehicles with various capacities in order to fulfil the demands of the customer population. Moreover, the vehicles start at the headquarters and terminate at one of the customers. In this study, we introduce a mixed integer programming model and then integrate an exact algorithm to solve this model. Furthermore, a hybrid algorithm (HA) based on modified rank-based ant system is developed and then its efficiency is compared with the exact method and some metaheuristic methods on some standard instances in literature. The results proved the effectiveness of our proposed HA.

A note on exact algorithms and heuristics for generation of V- and [formula omitted]-shaped sequences

We present two groups of algorithms for generation of V- and Λ-shaped sequences, and discuss their applications in scheduling. The first group includes two recursive exact algorithms which generate all V- and Λ-shaped sequences composed of the elements of a given set. The second group includes two heuristics which generate suboptimal V-shaped schedules for a single machine time-dependent scheduling problem with deteriorating jobs. The results of extensive computational experiments with C++, C# and Python implementations of these algorithms show that the exact algorithms can be used to solve single machine scheduling problems with V- or Λ-shaped optimal schedules, while the heuristics generate very good suboptimal schedules for the time-dependent scheduling problem.

A general purpose exact solution method for mixed integer concave minimization problems

In this article, we discuss an exact algorithm for solving mixed integer concave minimization problems. A piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem. The bilevel program is reduced to a single level formulation with the help of Karush–Kuhn–Tucker (KKT) conditions. Incorporating the KKT conditions lead to complementary slackness conditions that are linearized using BigM, for which we identify a tight value for general problems. Multiple bilevel programs, when solved over iterations, guarantee convergence to the exact optimum of the original problem. Though the algorithm is general and can be applied to any optimization problem with concave function(s), in this paper, we solve two common classes of operations and supply chain problems; namely, the concave knapsack problem, and the concave production-transportation problem. The computational experiments indicate that our proposed approach outperforms the customized methods that have been used in the literature to solve the two classes of problems by an order of magnitude in most of the test cases.

Grouped variable selection with discrete optimization: Computational and statistical perspectives

We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. While there exist several appealing approaches based on convex relaxations and nonconvex heuristics, we focus on optimal solutions for the ℓ0-regularized formulation, a problem that is relatively unexplored due to computational challenges. Our methodology covers both high-dimensional linear regression and nonparametric sparse additive modeling with smooth components. Our algorithmic framework consists of approximate and exact algorithms. The approximate algorithms are based on coordinate descent and local search, with runtimes comparable to popular sparse learning algorithms. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer programming (MIP) problem to certified optimality. By exploiting the problem structure, our custom BnB algorithm can solve to optimality problem instances with 5×106 features and 103 observations in minutes to hours—over 1000 times larger than what is currently possible using state-of-the-art commercial MIP solvers. We also explore statistical properties of the ℓ0-based estimators. We demonstrate, theoretically and empirically, that our proposed estimators have an edge over popular group-sparse estimators in terms of statistical performance in various regimes. We provide an open source implementation of our proposed framework.

An Optimized LTE-Based Technique for Drone Base Station Dynamic 3D Placement and Resource Allocation in Delay-Sensitive M2M Networks

Drones are expected to facilitate extending wireless networks’ access for both human users and smart machine-type-communication devices (MTCDs) with strict and diverse quality of service (QoS) requirements. In this paper, we propose an optimal solution for the dynamic placement of an LTE drone-mounted base station to maximize the coverage of the MTCDs deployed over a large geographical area. The resulting technique jointly optimizes the drone's 3D positioning to maximize coverage and allocates the network resources in such a way that gives high priority to the delay-sensitive machine-to-machine (M2M) traffic. This optimization algorithm determines the optimal bound of the solution. Since it cannot be used in real-time operations due to its computational complexity, we also introduce a heuristic technique that offers a near-optimal solution with much reduced complexity. We conduct several simulation evaluations to assess the proposed techniques. These results are compared to those of other drone placement approaches. The comparisons show that the proposed techniques offer significantly better results in the communication coverage while fulfilling the diverse QoS requirements of the deployed M2M network. Moreover, the heuristic-based technique is shown to succeed in finding solutions close to the optimal bound with a considerable reduction of complexity over the exact algorithm.

Performance Evaluation of Decision-Making Units Through Boosting Methods in the Context of Free Disposal Hull: Some Exact and Heuristic Algorithms

This paper aims to show how to calculate different efficiency measures using a technology estimator defined through the adaptation of the Gradient Tree Boosting algorithm. This adaptation shares some features with the standard nonparametric Free Disposal Hull (FDH) approach, but it overcomes data overfitting problems. Nevertheless, from a computational point of view, the new approach presents thousands of decision variables, making it difficult to solve. To tackle this problem, we also propose and check a heuristic approximation to the exact measures. To demonstrate the applicability of the proposed method, the exact and the heuristic approaches are compared through two empirical applications. The main contributions of this paper are as follows: we build a new bridge between machine learning techniques and technical efficiency measurement. In this framework, we show how to determine the output-oriented and input-oriented radial models, the Russell measure of output efficiency and the Russell measure of input efficiency, as well as the directional distance function and the Enhanced Russell Graph measure. We also prove that the new technique is better, in terms of bias and squared mean error, than the standard FDH technique. Furthermore, we show that the new approach may be seen as a possible remedy for solving the curse of dimensionality problem.

Mechanisms for feasibility and improvement for inventory-routing problems

Inventory-routing problems (IRPs) define a class of combinatorial optimization problems, encompassing inventory management and vehicle routing decisions into the same framework. In this article, we propose a new modular mechanism capable of recovering feasibility and improving even partial solutions by reorganizing delivery routes and optimizing inventory flows. It can be embedded into different optimization algorithms, either heuristic or exact ones. We exploit the use of this mechanism to improve a traditional branch-and-cut scheme and evaluate it by solving the multi-vehicle IRP (MIRP) and the multi-depot IRP (MDIRP). The results show that our method is very effective; outperforming other approaches on well-known benchmark instances from the literature. Regarding the MIRP, our algorithm obtains 417 optimal solutions for 638 small instances, the best result among all exact algorithms, with nine new ones. On a large data set, our method finds all optimal solutions for instances with up to 50 customers for the single-vehicle, besides providing 90% of new best-known solutions (BKS) for 100 customers. On the MDIRP, our approach finds 27 new optimal solutions and 73% of new BKS, improving previous BKS by more than 7% on average.

An efficient exact algorithm for identifying hybrids using population genomic sequences.

The identification of individuals that have a recent hybrid ancestry (between populations or species) has been a goal of naturalists for centuries. Since the 1960s, codominant genetic markers have been used with statistical and computational methods to identify F1 hybrids and backcrosses. Existing hybrid inference methods assume that alleles at different loci undergo independent assortment (are unlinked or in population linkage equilibrium). Genomic datasets include thousands of markers that are located on the same chromosome and are in population linkage disequilibrium which violate this assumption. Existing methods may therefore be viewed as composite likelihoods when applied to genomic datasets and their performance in identifying hybrid ancestry (which is a model-choice problem) is unknown. Here, we develop a new program Mongrail that implements a full-likelihood Bayesian hybrid inference method that explicitly models linkage and recombination, generating the posterior probability of different F1 or F2 hybrid, or backcross, genealogical classes. We use simulations to compare the statistical performance of Mongrail with that of an existing composite likelihood method (NewHybrids) and apply the method to analyze genome sequence data for hybridizing species of barred and spotted owls.

Fast and optimal branch-and-bound planner for the grid-based coverage path planning problem based on an admissible heuristic function.

This paper introduces an optimal algorithm for solving the discrete grid-based coverage path planning (CPP) problem. This problem consists in finding a path that covers a given region completely. First, we propose a CPP-solving baseline algorithm based on the iterative deepening depth-first search (ID-DFS) approach. Then, we introduce two branch-and-bound strategies (Loop detection and an Admissible heuristic function) to improve the results of our baseline algorithm. We evaluate the performance of our planner using six types of benchmark grids considered in this study: Coast-like, Random links, Random walk, Simple-shapes, Labyrinth and Wide-Labyrinth grids. We are first to consider these types of grids in the context of CPP. All of them find their practical applications in real-world CPP problems from a variety of fields. The obtained results suggest that the proposed branch-and-bound algorithm solves the problem optimally (i.e., the exact solution is found in each case) orders of magnitude faster than an exhaustive search CPP planner. To the best of our knowledge, no general CPP-solving exact algorithms, apart from an exhaustive search planner, have been proposed in the literature.