Hard Instances Research Articles

New hard benchmark for the 2-stage multi-machine assembly scheduling problem: Design and computational evaluation

The assembly scheduling problem is a common layout with many applications in real manufacturing scenarios. Despite the high number of studies dealing with this problem, no benchmark has been proposed up-to-now in the literature generating neither hard nor balanced instances. In this paper we present two extensive sets of instances for two variants of the 2-stage assembly scheduling problem. The first set is composed of 240 instances for the variant with one assembly machine in the second stage, while in the second set 960 instances are proposed for the variant with several assembly machines. An exhaustive experimental procedure, generating several preliminary testbeds with different processing times and number of jobs and machines, is carried out in order to identify the most representative instances of the problem under study. A total of 120,000 instances are generated and, among them, 1,200 are selected ensuring that the new benchmarks satisfy the desired characteristics of any benchmark: adequacy, empirical hardness, exhaustiveness, and amenity for statistical analysis. Finally, two computational evaluations are performed comparing and evaluating the existing heuristics in the literature, thus establishing the set of efficient heuristics for this assembly problem.

Fast classification rates without standard margin assumptions

Abstract We consider the classical problem of learning rates for classes with finite Vapnik–Chervonenkis (VC) dimension. It is well known that fast learning rates up to $O\left (\frac{d}{n}\right )$ are achievable by the empirical risk minimization (ERM) algorithm if low noise or margin assumptions are satisfied. These usually require the optimal Bayes classifier to be in the class, and it has been shown that when this is not the case, the fast rates cannot be achieved even in the noise free case. In this paper, we further investigate the question of the fast rates in the agnostic setting, when the Bayes classifier is not in the class and the noise in the labeling is allowed. First, we consider classification with a reject option, namely Chow’s reject option model, and show that by slightly lowering the impact of hard instances, a learning rate of order $O\left (\frac{d}{n}\log \frac{n}{d}\right )$ is always achievable in the agnostic setting by a specific learning algorithm. Similar results were only known under special versions of margin assumptions. As an auxiliary result, we show that under the Bernstein assumption the performance of the proposed algorithm is never worse than the performance of ERM even if some of the labels are predicted at random. Based on those results, we derive the necessary and sufficient conditions for classification (without a reject option) with fast rates in the agnostic setting achievable by improper learners. This simultaneously extends the work by Massart & Nédélec (2006, Ann. Stat., 34, 2326–2366), which studied this question in the case where the Bayesian optimal rule belongs to the class, and the work by Ben-David and Urner (2014, COLT, pp. 527–542 ), which allows the misspecification but is limited to the no noise setting. Our result also provides the first general setup in statistical learning theory in which an improper learning algorithm may significantly improve the learning rate for non-convex losses.

Solving Bin Packing Problems Using VRPSolver Models

We propose branch-cut-and-price algorithms for the classic bin packing problem and also for the following related problems: vector packing, variable sized bin packing and variable sized bin packing with optional items. The algorithms are defined as models for VRPSolver, a generic solver for vehicle routing problems. In that way, a simple parameterization enables the use of several branch-cut-and-price advanced elements: automatic stabilization by smoothing, limited-memory rank-1 cuts, enumeration, hierarchical strong branching and limited discrepancy search diving heuristics. As an original theoretical contribution, we prove that the branching over accumulated resource consumption (Gelinas et al. 1995), that does not increase the difficulty of the pricing subproblem, is sufficient for those bin packing models. Extensive computational results on instances from the literature show that the VRPSolver models have a performance that is very robust over all those problems, being often superior to the existing exact algorithms on the hardest instances. Several instances could be solved to optimality for the first time.

Algorithms for gerrymandering over graphs

We initiate the systematic algorithmic study for gerrymandering over graphs that was recently introduced by Cohen-Zemach, Lewenberg and Rosenschein. Namely, we study a strategic procedure for a political districting designer to draw electoral district boundaries so that a particular target candidate can win in an election. We focus on the existence of such a strategy under the plurality voting rule, and give interesting contrasts which classify easy and hard instances with respect to polynomial-time solvability. For example, we prove that the problem for trees is strongly NP-complete (thus unlikely to have a pseudo-polynomial-time algorithm), but has a pseudo-polynomial-time algorithm when the number of candidates is constant. Another example is to prove that the problem for complete graphs is NP-complete when the number of electoral districts is two, while is solvable in polynomial time when it is more than two.

Distributionally robust chance-constrained programs with right-hand side uncertainty under Wasserstein ambiguity

We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with large problem sizes, the branch-and-bound based solution processes suffer from large optimality gaps even after hours of computation time. This significantly hinders the practical application of the DR-CCP paradigm. Motivated by these challenges, we conduct a polyhedral study to strengthen these formulations. We reveal several hidden connections between DR-CCP and its nominal counterpart (the sample average approximation), mixing sets, and robust 0–1 programming. By exploiting these connections in combination, we provide an improved formulation and two classes of valid inequalities for DR-CCP. We test the impact of our results on a stochastic transportation problem numerically. Our experiments demonstrate the effectiveness of our approach; in particular our improved formulation and proposed valid inequalities reduce the overall solution times remarkably. Moreover, this allows us to significantly scale up the problem sizes that can be handled in such DR-CCP formulations by reducing the solution times from hours to seconds.

Minimizing maximum tardiness on a single machine with family setup times and machine disruption

Due to the complexity of manufacturing processes, many industries today try to group their production components into families and use a common setup time to process every family. Also, the failure of machinery has an important impact on the planning of production systems. This paper deals with the problem of single machine scheduling with family setup times and randomly machine breakdown. It is assumed that there is disruption along the planning horizon with uncertain start and duration times based on a specific probability distribution function. The goal is to minimize the maximum expected tardiness. For this problem, a hill climbing based heuristic (HCO) and a novel hybrid variable neighborhood search metaheuristic (HVNS) have been developed. A branch and bound (BB) procedure is also proposed to optimally solve this problem using HCO and HVNS algorithms to derive upper bounds. Comparison of the solutions obtained from the presented algorithms with the mathematical programming model of the problem in small-scale instances shows that the solutions obtained from the proposed algorithms have a good quality in terms of the expected value of the objective function and the solving time. The computational results for executing the proposed algorithms on 36 series of various instances show that the BB procedure is able to solve 97.76% of the instances within the specified time frame and HVNS metaheuristic is capable to solve 41.07% of them, optimally. Also, the maximum mean deviation of HVNS solutions from the optimal solution is reported below 5% for hard instances and below 15% for instances with medium complexity. After performing a non-parametric hypothesis test, it is found that at a confidence level 99.9%, the deviation index for HVNS is less than 1.5% for difficult problems and less than 9% for semi-difficult ones. Finally, the results of solving large scale instances, demonstrate the efficiency of proposed metaheuristic.

Range partitioning within sublinear time: Algorithms and lower bounds

Range partitioning is a typical and mostly used data partitioning method and has became a core operation in most of big data computing platforms. Given an input L of N data items admitting a total order, the goal of range partitioning is to divide the whole input into k ranges containing the same number of data items. There is a trivial lower bound Ω(N) for the exact partitioning algorithms, since they need to at least make a full scan of the whole data. In the context of big data computing, even algorithms with O(N) time are not always thought to be efficient enough, the ultimate goal of designing algorithms on big data is usually to solve problems within sublinear time. Therefore, it is well motivated and important to study sublinear algorithms for the range partitioning problem.The paper aims to answer three questions. For the internal memory (RAM) model, since sophisticated sampling based (ϵ,δ)-approximation partitioning algorithm with O(klog⁡(N/δ)ϵ2) time cost has been proposed, the first question is what a lower bound we can obtain for sublinear partitioning algorithms. For the external memory (I/O) model, as far as we know, no previous works give external partitioning algorithms with performance guarantee within sublinear time, therefore the two questions are what the upper bound and the lower bound we can achieve for sublinear external partitioning algorithms. To answer the above questions, based on the RAM and I/O model, the paper studies the lower and upper bounds for the range partitioning problem. For the RAM model, a lower bound Ω(k(1−δ)ϵ2) for the cost of sampling based partitioning algorithms is proved. For the I/O model, two lower bounds of the sampling cost required by sublinear external range partitioning algorithms are proved, which indicate that at least a full scan of the whole input is needed in the worst case and a general sublinear external partitioning algorithm does not exist. Motivated by the hard instances utilized in the proof of lower bounds, a model for describing the input distributions of the range partitioning problem in practical applications is proposed. Finally, for the special cases described by the model, a sublinear external partitioning algorithm with O(klog⁡(N/δ)wBϵ2) I/O cost is designed.

Sequence-Based Selection Hyper-Heuristic Model via MAP-Elites

Although the number of solutions in combinatorial optimization problems (COPs) is finite, some problems grow exponentially and render exact approaches unfeasible. So, approximate methods, such as heuristics, are customary. Each heuristic usually specializes in specific kinds of problems. Hence, other approaches seek to merge their strengths. One of them is selection hyper-heuristics. However, they usually provide scarce information about their sensitivity. Illumination algorithms may fix this issue since they focus on exploration rather than exploitation while preserving the best solutions under different criteria. Still, literature falls short when merging both approaches, representing a knowledge gap. This work tests the feasibility of using an illumination algorithm, MAP-Elites (ME), for tuning a sequence-based selection hyper-heuristic model for Balanced Partition problems. We choose ME since other researchers have successfully applied it to a different COP. So, we may achieve a hyper-heuristic that represents the best combination of heuristics while simultaneously gaining intel on the performance of diverse alternatives. Our approach operates by creating a multi-dimensional map, where each design variable represents the application of a heuristic. Afterward, ME generates mutated sequences and tests them to determine if they represent a better-performing solution. We consider 1500 instances that include easy and hard instances, analyzed under different scenarios to test our approach. We also include limit instances that are neither easy nor hard. Our resulting data support the proposed approach, as it performs toe-to-toe with a synthetic oracle and may even outperform it. This represents an outstanding result, since a brute-force approach is needed to achieve such an oracle. So, merging ME and hyper-heuristics is a path worth pursuing. We also present how each parameter affects the model performance and identify the critical and virtually irrelevant ones. This serves as the groundwork for future works that focus on exploiting the most relevant parameters.

Integer Linear Programming Formulations for the RCPSP considering Multi-Skill, Multi-Mode, and Minimum and Maximum Time Lags

The project scheduling problem is essential both in the theoretical part, as in the field of operational research, and practice, with the project management in corporate environments. Integer linear programming formulations indexed on time are studied for the Resource-Constrained Project Scheduling Problem (RCPSP). Moreover, the multi-skill, multiple modes, and time lag constraints are taken into consideration. The objective of the RCPSP is to minimize the makespan. The formulations are solved with the default branch-and-cut algorithm of the solver Gurobi Optimizer. The formulations and solver are analyzed concerning the runtime, the number of optimal solutions, and the gap on the resolution of more than 2000 instances. Results indicate the solver can have better performance when instances with up to 50 activities are solved. Then, to develop models to handle hard instances of this problem is a challenge. Moreover, it can bring significant advantages to the corporate environment, helping managers to make accurate decisions and reduce costs.

A Novel Local Search-Based Approximation Algorithm to Optimize Virtual Machine Placement With Resource Constraints

Many problems in cloud computing are not solvable in polynomial time and only option left is to choose approximate solution instead of optimum. Virtual Machine placement is one of such problem with resource constraints in which overall objective is to optimize multiple resources of hosts during placement process. In this paper we have addressed this problem with large size NP-Hard instances and proposed novel local search-based approximation algorithm. This problem is not yet studied in the research community with NP hard instances. A new proposed algorithm is empirically evaluated with state-of-the-art techniques. and our algorithm has improved placement result by 18% in CPU utilization, 21% in resource contention and 26% in overall resource utilization for benchmark instances collected from azure private cloud data center.

Cross-Epoch Learning for Weakly Supervised Anomaly Detection in Surveillance Videos

Weakly Supervised Anomaly Detection (WSAD) in surveillance videos is a complex task since usually only video-level annotations are available. Previous work treated it as a regression problem by giving different scores on normal and anomaly events. However, the widely used mini-batch training strategy may suffer from the data imbalance between these two types of events, which limits the model's performance. In this work, a cross-epoch learning (XEL) strategy associated with a hard instance bank (HIB) is proposed to introduce additional information from previous training epochs. Two new losses are proposed for XEL to achieve a higher detection rate as well as a lower false alarm rate of anomaly events. Moreover, the proposed XEL can be directly integrated into any existing WSAD framework. Experimental results of three XEL embedded models have shown promising AUC improvement (3%∼7%) on two public datasets, surpassing the state-of-the-art methods. Our code is available at: https://github.com/sdjsngs/XEL-WSAD .

Revisiting where are the hard knapsack problems? via Instance Space Analysis

In 2005, David Pisinger asked the question “where are the hard knapsack problems?”. Noting that the classical benchmark test instances were limited in difficulty due to their selected structure, he proposed a set of new test instances for the 0–1 knapsack problem with characteristics that made them more challenging for dynamic programming and branch-and-bound algorithms. This important work highlighted the influence of diversity in test instances to draw reliable conclusions about algorithm performance. In this paper, we revisit the question in light of recent methodological advances – in the form of Instance Space Analysis – enabling the strengths and weaknesses of algorithms to be visualised and assessed across the broadest possible space of test instances. We show where the hard instances lie, and objectively assess algorithm performance across the instance space to articulate the strengths and weaknesses of algorithms. Furthermore, we propose a method to fill the instance space with diverse and challenging new test instances with controllable properties to support greater insights into algorithm selection, and drive future algorithmic innovations.

A multi-objective linear threshold influence spread model solved by swarm intelligence-based methods

The influence maximization problem (IMP) is one of the most important topics in social network analysis. It consists of finding the smallest seed of users that maximizes the influence spread in a social network. The main influence spread models are the linear threshold model (LT-model) and the independent cascade model (IC-model). These models have mainly been treated by using the single-objective paradigm which covers just one perspective: maximize the influence spread starting by given seed size, or minimize the seed set to reach a given number of influenced nodes. Sometimes, this minimization problem has been called the least cost influence problem (LCI). In this work, we propose a new optimization model for both perspectives under conflict, through the LT-model, by applying a binary multi-objective approach. Swarm intelligence methods are implemented to solve our proposal on real networks. Results are promising and suggest that the new multi-objective solution proposed can be properly solved in harder instances.

Quantum annealing using vacuum states as effective excited states of driven systems

Quantum annealing, which is particularly useful for combinatorial optimization, becomes more powerful by using excited states, in addition to ground states. However, such excited-state quantum annealing is prone to errors due to dissipation. Here we propose excited-state quantum annealing started with the most stable state, i.e., vacuum states. This counterintuitive approach becomes possible by using effective energy eigenstates of driven quantum systems. To demonstrate this concept, we use a network of Kerr-nonlinear parametric oscillators, where we can start excited-state quantum annealing with the vacuum state of the network by appropriately setting initial detuning frequencies for the oscillators. By numerical simulations of four oscillators, we show that the present approach can solve some hard instances whose optimal solutions cannot be obtained by standard ground-state quantum annealing because of energy-gap closing. In this approach, a nonadiabatic transition at an energy-gap closing point is rather utilized. We also show that this approach is robust against errors due to dissipation, as expected, compared to quantum annealing started with physical excited (i.e., nonvacuum) states. These results open new possibilities for quantum computation and driven quantum systems.

Linear size MIP formulation of Max-Cut: new properties, links with cycle inequalities and computational results

We consider the Max-Cut problem on an undirected graph G = (V, E) with |V | = n nodes and |E| = m edges. We investigate a linear size MIP formulation, referred to as (MIP-MaxCut), which can easily be derived via a standard linearization technique. However, the efficiency of the Branchand-Bound procedure applied to this formulation does not seem to have been investigated so far in the literature. Branch-and-bound based approaches for Max-Cut usually use the semi-metric polytope which has either an exponential size formulation consisting of the cycle inequalities or a compact size formulation consisting of O(mn) triangle inequalities [2], [16]. However, optimizing over the semi-metric polytope can be computationally demanding due to the slow convergence of cutting-plane algorithms and the high degeneracy of formulations based on the triangle inequalities. In this paper, we exhibit new structural properties of (MIP-MaxCut) that link the binary variables with the cycle inequalities. In particular, we show that fixing a binary variable at 0 or 1 in (MIP-MaxCut) can result in imposing the integrity of several original variables and the satisfaction of a possibly exponential number of cycle inequalities in the semi-metric formulation. Numerical results show that for sparse instances of Max-Cut, our approach exploiting this capability outperforms the branch-and-cut algorithms based on semi-metric polytope when implemented on the same framework; and even without any extra sophistication, the approach is capable of solving hard instances of Max-Cut within acceptable CPU times. The work is supported by the Programme Gaspard Monge pour Optimisation (PGMO).

Aspect-level sentiment analysis based on gradual machine learning

The state-of-the-art solutions for Aspect-Level Sentiment Analysis (ALSA) were built on a variety of Deep Neural Networks (DNN), whose efficacy depends on large quantities of accurately labeled training data. Unfortunately, high-quality labeled training data usually require expensive manual work, thus may not be readily available in real scenarios. In this paper, we propose a novel approach for aspect-level sentiment analysis based on the recently proposed paradigm of Gradual Machine Learning (GML), which can enable accurate machine labeling without the requirement for manual labeling effort. It begins with some easy instances in a task, which can be automatically labeled by the machine with high accuracy, and then gradually labels the more challenging instances by iterative factor graph inference. In the process of gradual machine learning, the hard instances are gradually labeled in small stages based on the estimated evidential certainty provided by the labeled easier instances. Our extensive experiments on the benchmark datasets have shown that the performance of the proposed solution is considerably better than its unsupervised alternatives, and also highly competitive compared with the state-of-the-art supervised DNN models.

A story of diameter, radius, and (almost) Helly property

AbstractWe present new algorithmic results for the class of Helly graphs, that is, for the discrete analogues of hyperconvex metric spaces. Specifically, an undirected unweighted graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. It is known that every graph isometrically embeds into a Helly graph that makes of the latter an important class of graphs in metric graph theory. We study diameter and radius computations within the Helly graphs, and related graph classes. This is in part motivated by a conjecture on the fine‐grained complexity of these two distance problems within the graph classes of bounded fractional Helly number—that contain as particular cases the proper minor‐closed graph classes and the bounded clique‐width graphs. Note that under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, we first present algorithms which given an n‐vertex m‐edge Helly graph G as input, compute with high probability (w.h.p.) its radius and its diameter in time (i.e., subquadratic in n + m). Our algorithms are based on the Helly property and on the unimodality of the eccentricity function in Helly graphs: every vertex of locally minimum eccentricity is a central vertex. Then, we improve our results for the C4‐free Helly graphs, that are exactly the Helly graphs whose balls are convex. For this subclass, we present linear‐time algorithms for computing the eccentricity of all vertices. Doing so, we generalize previous results on strongly chordal graphs to a much larger subclass, that includes, among others, all the bridged Helly graphs and the hereditary Helly graphs. Lastly, we derive approximate versions of our results for the class of chordal graphs: with the latter satisfying an almost‐Helly‐type property, and a stronger (induced‐path) convexity property than the C4‐free Helly graphs. For the chordal graphs, we can compute in quasi linear time the eccentricity of all vertices with an additive one‐sided error of at most one, which is best possible under the strong exponential‐time hypothesis. This answers an open question of Dragan. In fact, we obtain this last result as a byproduct from a more general reduction: from diameter computation on chordal graphs to the Disjoint Sets problem. Roughly, it implies that the split graphs are the only hard instances for diameter computation on chordal graphs. We also get from our reduction that on any subclass of chordal graphs with constant VC‐dimension (and so, for undirected path graphs), the diameter can be computed in truly subquadratic time.

Pulsed Quantum Annealing

We propose a modified quantum annealing protocol, i. e., pulsed quantum annealing} (PQA), in order to increase the success probability by a pulse application during the quantum annealing process. It is well known that the success probability of the conventional quantum annealing is reduced due to the Landau-Zener transitions. By applying a pulse to the system, we modulate the success probability and increase it, compared to the conventional quantum annealing, by optimizing the pulse parameters. We demonstrate our findings for a single qubit both numerically and analytically. The analytical model is based on the tranfer matrix approach and it is in good agreement with the full numerical results. We also investigate the PQA protocol for multi-qubit cases $i. e.,$ random spin-glass instances, and we present an overall increase of the success probability over the conventional quantum annealing, by optimizing the pulse parameters. Our results indicate that PQA can be used to design future high-performance quantum annealing machines, especially for hard instances that the conventional QA protocol behaves poorly.

New online reinsertion approaches for a dynamic Dial-a-Ride Problem

This study is inspired by a challenging dynamic Dial-a-Ride Problem (DARP) encountered in a mobility service operated by the company Padam.11www.padam.io . Customers ask for a transportation service either in advance or in real-time and get an immediate answer about whether their requests are accepted or rejected. The main goal is to maximize the number of accepted requests during the service while respecting maximum capacity of limited number of vehicles. In this study, we propose novel methods based on reinsertion techniques to improve Padam's online system. The proposed reinsertion techniques aim to exploit the neighborhood of a solution and must be run in real-time whenever a request is rejected by an online system. Three main online reinsertion heuristics based on different neighborhoods are proposed. The first heuristic called HDR uses destruction and repair operators, the second heuristic called GH (graph heuristic) uses the ejection chain concept, and allow modeling the reinsertion problem as a constrained shortest path problem on directed graph built from a current solution, and the third heuristic called IGH (improved graph heuristic) based on GH with intensive exploration of the neighborhood. Our proposed approaches were extensively tested on real and hard instances provided by Padam. These instances contain up to 1011 requests and 14 vehicles and with very tight time windows. The obtained results revealed the performance of the proposed reinsertion methods compared to the online heuristic, where the number of served requests is increased and the number of vehicles is reduced while maintaining operating costs close to those of the online system. The results confirm the interest of using reinsertion techniques in the dynamic DARP service.

Improving Matsui’s Search Algorithm For The Best Differential/Linear Trails And Its Applications For DES, DESL And GIFT

Abstract Automatic search methods have been widely used for cryptanalysis of block ciphers, especially for the most classic cryptanalysis methods—differential and linear cryptanalysis. However, the automatic search methods, no matter based on MILP, SMT/SAT or CP techniques, can be inefficient when the search space is too large. In this paper, we propose three new methods to improve Matsui’s branch-and-bound search algorithm, which is known as the first generic algorithm for finding the best differential and linear trails. The three methods, named reconstructing DDT and LAT according to weight, executing linear layer operations in minimal cost and merging two 4-bit S-boxes into one 8-bit S-box, respectively, can efficiently speed up the search process by reducing the search space as much as possible and reducing the cost of executing linear layer operations. We apply our improved algorithm to DESL and GIFT, which are still the hard instances for the automatic search methods. As a result, we find the best differential trails for DESL (up to 14-round) and GIFT-128 (up to 19-round). The best linear trails for DESL (up to 16-round), GIFT-128 (up to 10-round) and GIFT-64 (up to 15-round) are also found. To the best of our knowledge, these security bounds for DESL and GIFT under single-key scenario are given for the first time. Meanwhile, it is the longest exploitable (differential or linear) trails for DESL and GIFT. Furthermore, benefiting from the efficiency of the improved algorithm, we do experiments to demonstrate that the clustering effect of differential trails for 13-round DES and DESL are both weak.