Recently (Jaekel 2023; Van Hirtum et al. 2024) the ninth Dedekind number (D(9)) was computed. In fact, the result of two independent computations, confirming each other’s result, were published nearly at the same time. In one of these, the authors of the present paper were involved. D(n) counts the monotone Boolean functions or antichains on subsets of a set of n elements. The number rises doubly exponentially in the number of elements n, and until now no algorithm of a lower combinatorial complexity is known to compute D(n). In our computation, we use coefficients representing the number of solutions of a specific set of equations on antichains over a finite set. We refer to these coefficients as P-coefficients. These can be computed efficiently. In this paper, we generalise this coefficient and apply it to four different systems of equations. Finally we show how the coefficient was used in our computation of D(9), and how its generalisations can be used to compute D(n).

This manuscript makes the claim of having computed the \(9^{th}\) Dedekind number, D(9). This was done by accelerating the core operation of the process with an efficient FPGA design that outperforms an optimized 64-core CPU reference by 95 \(\times\) . The FPGA execution was parallelized on the Noctua 2 supercomputer at Paderborn University. The resulting value for D(9) is \(286386577668298411128469151667598498812366\) . This value can be verified in two steps. We have made the data file containing the 490M results available, each of which can be verified separately on CPU, and the whole file sums to our proposed value. The paper explains the mathematical approach in the first part, before putting the focus on a deep dive into the FPGA accelerator implementation followed by a performance analysis. The FPGA implementation was done in RTL using a dual-clock architecture and shows how we achieved an impressive FMax of 450MHz on the targeted Stratix 10 GX 2800 FPGAs. The total compute time used was 47’000 FPGA Hours.

Predicting and comparing algorithm performance on graph instances is challenging for multiple reasons. First, there is not always a standard set of instances to benchmark performance. Second, using existing graph generators results in a restricted spectrum of difficulty and the resulting graphs are not always diverse enough to draw sound conclusions. That is why recent work proposes a new methodology to generate a diverse set of instances by using evolutionary algorithms. We can then analyze the resulting graphs and get key insights into which attributes are most related to algorithm performance. We can also fill observed gaps in the instance space in order to generate graphs with previously unseen combinations of features. We apply this methodology to the instance space of the Hamiltonian completion problem using two different solvers, namely the Concorde TSP Solver and a multi-start local search algorithm.

This preprint makes the claim of having computed the $9^{th}$ Dedekind Number. This was done by building an efficient FPGA Accelerator for the core operation of the process, and parallelizing it on the Noctua 2 Supercluster at Paderborn University. The resulting value is 286386577668298411128469151667598498812366. This value can be verified in two steps. We have made the data file containing the 490M results available, each of which can be verified separately on CPU, and the whole file sums to our proposed value.

In this article we propose a heuristic algorithm to explore search space trees associated with instances of combinatorial optimization problems. The algorithm is based on Monte Carlo tree search, a popular algorithm in game playing that is used to explore game trees and represents the state-of-the-art algorithm for a number of games. Several enhancements to Monte Carlo tree search are proposed that make the algorithm more suitable in a combinatorial optimization context. These enhancements exploit the combinatorial structure of the problem and aim to efficiently explore the search space tree by pruning subtrees, using a heuristic simulation policy, reducing the domains of variables by eliminating dominated value assignments and using a beam width. The algorithm was implemented with its components specifically tailored to two combinatorial optimization problems: the quay crane scheduling problem with non-crossing constraints and the 0-1 knapsack problem. For the first problem our algorithm surpasses the state-of-the-art results and several new best solutions are found for a benchmark set of instances. For the second problem our algorithm typically produces near-optimal solutions that are slightly worse than the state-of-the-art results, but it needs only a small fraction of the time to do so. These results indicate that the algorithm is competitive with the state-of-the-art for two entirely different combinatorial optimization problems.

The objective of the nurse rostering problem (NRP) is to obtain a scheduling plan that optimizes the allocation of human resources, effectively reducing work pressure on nurses and improving work efficiency and quality. Because various constraints must be considered during scheduling, the NRP is complicated and known to be NP-hard. Existing research has not combined learning mechanisms with NRP. This study constructively explores the possibility of combining an optimization method and a learning mechanism to automatically produce feasible solutions and proposes a feature vector and a reconstruction mechanism to assist in this exploration. We aim to learn a policy that is generalizable for NRPs of various sizes and design a hybrid algorithm with learning and optimization methods to solve the general NRP. The algorithm has two main parts: a deep neural network (DNN) improvement part and a reconstruction part. In the DNN improvement part, a feature vector is used to describe heterogeneous NRP solutions and normalizes these solutions to the same dimension. Then, the DNN model determines the best heuristic for approximating the local optimal solution. The method reconstructs the structure of the current solution with embedded mixed integer programming (MIP), quickly escaping the local optimum and enhancing the diversity of the search process, increasing the likelihood of determining an optimal solution. Different experiments and statistical tests were conducted by comparing various configurations and approaches. The detailed computational and statistical results demonstrate the competitive performance of the proposed method.

Patient scheduling is a difficult task involving stochastic factors such as the unknown arrival times of patients. Similarly, the scheduling of radiotherapy for cancer treatments needs to handle patients with different urgency levels when allocating resources. High priority patients may arrive at any time, and there must be resources available to accommodate them. A common solution is to reserve a flat percentage of treatment capacity for emergency patients. However, this solution can result in overdue treatments for urgent patients, a failure to fully exploit treatment capacity, and delayed treatments for low-priority patients. This problem is especially severe in large and crowded hospitals. In this paper, we propose a prediction-based approach for online dynamic radiotherapy scheduling that dynamically adapts the present scheduling decision based on each incoming patient and the current allocation of resources. Our approach is based on a regression model trained to recognize the links between patients' arrival patterns, and their ideal waiting time in optimal offline solutions where all future arrivals are known in advance. When our prediction-based approach is compared to flat-reservation policies, it does a better job of preventing overdue treatments for emergency patients, while also maintaining comparable waiting times for the other patients. We also demonstrate how our proposed approach supports explainability and interpretability in scheduling decisions using SHAP values.

Decades of research on the 0-1 knapsack problem led to very efficient algorithms that are able to quickly solve large problem instances to optimality. This prompted researchers to also investigate whether relatively small problem instances exist that are hard for existing solvers and investigate which features characterize their hardness. Previously the authors proposed a new class of hard 0-1 knapsack problem instances and demonstrated that the properties of so-called inclusionwise maximal solutions (IMSs) can be important hardness indicators for this class. In the current paper, we formulate several new computationally challenging problems related to the IMSs of arbitrary 0-1 knapsack problem instances. Based on generalizations of previous work and new structural results about IMSs, we formulate polynomial and pseudopolynomial time algorithms for solving these problems. From this we derive a set of 14 computationally expensive features, which we calculate for two large datasets on a supercomputer in approximately 540 CPU-hours. We show that the proposed features contain important information related to the empirical hardness of a problem instance that was missing in earlier features from the literature by training machine learning models that can accurately predict the empirical hardness of a wide variety of 0-1 knapsack problem instances. Using the instance space analysis methodology, we also show that hard 0-1 knapsack problem instances are clustered together around a relatively dense region of the instance space and several features behave differently in the easy and hard parts of the instance space.

In the literature, much effort has been devoted to providing taxi drivers with recommendations on how to find passengers efficiently. However, the load balancing problem is not taken seriously, leading to a situation where taxis gather and compete at some points while passengers wait a long time in other places due to low taxi supply. Besides, there are multiple passenger-finding strategies, and recommending appropriate strategies according to time and location helps to improve drivers' performance. This paper proposes a global load balancing approach that can be applied to different passenger-finding strategies; specifically, a new measure dealing with load balance between taxis and passengers is introduced, and an adaptive passenger-finding recommendation method is proposed. These two problems are integrated into a single coherent framework using a multi-objective optimisation approach to maximise drivers' potential benefits while minimising supply-demand imbalance. Extensive experiments on real-world taxi data from New York City and San Francisco verify that the proposed method improves upon the performance of existing methods in terms of both drivers' net profit and passengers' waiting time.

To formulate a systematic and scientific nurse scheduling plan based on the nurse rostering problem (NRP), individual preferences and legal and other constraints must be fully considered. The NRP aims to optimize the allocation of human resources and to effectively reduce the workload to improve the efficiency and quality of nurses' work. As various constraints must be considered, the NRP is complicated and is known to be NP-hard. The fast attainment of an efficient schedule is a challenging problem that requires urgent resolution. In this paper, we propose a neural network-assisted method for automating the complex and costly design of the heuristics. The method integrates a generic deep neural network (DNN) model and a generic recurrent neural network (RNN) model. The DNN model is used to guide the heuristic selection; the RNN model is embedded into the reconstruction mechanism to escape the local optimum. By treating schedules as matrices, the neural networks can help decide which heuristic to apply next and generate feasible solutions. Four experiments with different focuses based on benchmark instances are conducted by comparing different parameters as well as other state-of-the-art methods. Detailed computational results revealed the performance of the proposed method, results from the proposed algorithm outperforms the results reported in recent literature and state-of-the-art heuristic approaches in at least half of the instances, the proposed method also shows better performance compared with three reproduced deep reinforcement (DRL) learning methods, it is also competitive as an component of the hybrid method.

The 0-1 knapsack problem is an important optimization problem, because it arises as a special case of a wide variety of optimization problems and has been generalized in several ways. Decades of research have resulted in very powerful algorithms that can solve large knapsack problem instances involving thousands of decision variables in a short amount of time. Current problem instances in the literature no longer challenge these algorithms. However, hard problem instances are important to demonstrate the strengths and weaknesses of algorithms and this knowledge can in turn be used to create better performing algorithms. In this paper, we propose a new class of hard problem instances for the 0-1 knapsack problem and provide theoretical support that helps explain why these problem instances are hard to solve to optimality. A large dataset of 3240 hard problem instances was generated and subsequently solved on a supercomputer, using approximately 810 CPU-hours. The analysis of the obtained results shows to which extent different parameters influence the hardness of the problem instances. This analysis also demonstrates that the proposed problem instances are a lot harder than the previously known hardest instances, despite being much smaller.(c) 2021 Elsevier B.V. All rights reserved.

The Radiotherapy Scheduling Problem (RTSP) focuses on optimizing the planning of radiotherapy treatment sessions for cancer patients. In this paper, we propose a two-phase approach for the RTSP. In the first phase, radiotherapy sessions are assigned to specific linear accelerators (linacs) and days. The second phase then decides the sequence of patients on each day/linac and the specific appointment times. For the first phase, an Integer Linear Programming (IP) model is proposed and solved using CPLEX. For the second phase, a Mixed Integer Linear Programming (MIP) and a Constraint Programming (CP) model are proposed. The test data is generated based on real data from CHUM, a large cancer center in Montréal, Canada, with an average of 3,500 new patients and 40,000 radiotherapy treatments per year. The results show that in the second phase, CP is better at finding good solutions quickly while MIP is better at closing optimality gaps with more run time. Lastly, a simulation is conducted to evaluate the impact of different scheduling strategies on the outcome of the scheduling. Preliminary results show that batch scheduling reduces patients’ waiting time and overdue time.

This paper studies partitions in the space of antimonotonic boolean functions on sets of n elements. The antimonotonic functions are the antichains of the partially ordered set of subsets. We analyse and characterise a natural partial ordering on this set. We study the intervals according to this ordering. We show how intervals of antimonotonic functions, and a fortiori the whole space of antimonotonic functions can be partitioned as disjoint unions of certain classes of intervals. These intervals are uniquely determined by antimonotonic functions on smaller sets. This leads to recursive enumeration algorithms and new recursion relations. Using various decompositions, we derive new recursion formulae for the number of antimonotonic functions and hence for the number of monotonic functions (i.e. the Dedekind number).

Semi-supervised or constrained graph clustering incorporates prior information in order to improve clustering results. Pairwise constraints are often utilized to guide the clustering process. This work addresses a constrained graph clustering problem in biological networks where (1) subgraph connectivity constraints are strictly required to be satisfied and (2) clustering quality is assessed with respect to pairwise constraint violations. Existing constrained graph clustering methods often fail to fully satisfy connectivity constraints. This paper presents an iterated local search algorithm which aims to find the clustering with the highest quality in a short computing time. Experiments demonstrate how the proposed solutions are of good quality, often being optimal. Additionally, the proposed method significantly outperforms an existing branch-and-cut algorithm in terms of computational runtime and produces competitive results with regard to other local search techniques and graph clustering algorithms. Furthermore, a multilevel algorithm for clustering is designed to handle large-scale graphs. The performance of the overall scheme for a variety of coarsening methods from the literature is studied on a large number of biological networks exceeding 10,000 genes.