Combinatorial problems which have been proven to be NP-hard are faced in Higher Education Institutions and researches have extensively investigated some of the well-known combinatorial problems such as the timetabling and student project allocation problems. However, NP-hard problems faced in Higher Education Institutions are not only confined to these categories of combinatorial problems. The majority of NP-hard problems faced in institutions involve grouping students and/or resources, albeit with each problem having its own unique set of constraints. Thus, it can be argued that techniques to solve NP-hard problems in Higher Education Institutions can be transferred across the different problem categories. As no method is guaranteed to outperform all others in all problems, it is necessary to investigate heuristic techniques for solving lesser-known problems in order to guide stakeholders or software developers to the most appropriate algorithm for each unique class of NP-hard problems faced in Higher Education Institutions. To this end, this study described an optimization problem faced in a real university that involved grouping students for the presentation of semester results. Ordering based heuristics, genetic algorithm and the ant colony optimization algorithm implemented in Python programming language were used to find feasible solutions to this problem, with the ant colony optimization algorithm performing better or equal in 75% of the test instances and the genetic algorithm producing better or equal results in 38% of the test instances.

Can deep neural networks learn to solve any task, and in particular problems of high complexity? This question attracts a lot of interest, with recent works tackling computationally hard tasks such as the traveling salesman problem and satisfiability. In this work we offer a different perspective on this question. Given the common assumption that $\textit{NP} \neq \textit{coNP}$ we prove that any polynomial-time sample generator for an $\textit{NP}$-hard problem samples, in fact, from an easier sub-problem. We empirically explore a case study, Conjunctive Query Containment, and show how common data generation techniques generate biased datasets that lead practitioners to over-estimate model accuracy. Our results suggest that machine learning approaches that require training on a dense uniform sampling from the target distribution cannot be used to solve computationally hard problems, the reason being the difficulty of generating sufficiently large and unbiased training sets.

We propose an algorithm based on reinforcement learning for solving NP-hard problems on graphs. We combine Graph Isomorphism Networks and the Monte-Carlo Tree Search, which was originally used for game searches, for solving combinatorial optimization on graphs. Similarly to AlphaGo Zero, our method does not require any problem-specific knowledge or labeled datasets (exact solutions), which are difficult to calculate in principle. We show that our method, which is trained by generated random graphs, successfully finds near-optimal solutions for the Maximum Independent Set problem on citation networks. Experiments illustrate that the performance of our method is comparable to SOTA solvers, but we do not require any problem-specific reduction rules, which is highly desirable in practice since collecting hand-crafted reduction rules is costly and not adaptive for a wide range of problems.

Cluster Hire is defined as a problem of hiring a group of experts to maximize profits with the ability to complete multiple projects simultaneously under a budget. It assumes that we have a set of projects which require skills and experts who possess various skills. The process of hiring a group of experts to complete a set of projects under the given conditions is proven to be the NP-hard problem. Individuals expect financial support (i.e.salary) which can be handled by a specific budget that we get, to work on the projects. Addition to maximizing the total profit, we are interested in hiring productive experts who can work many projects concurrently with effective result. Therefore, this paper examines the problem of hiring a cluster of experts, so that the total salary does not exceed more than a given budget and maximizes the total benefit of the projects that a highly productive team can cover collectively. We propose two greedy algorithms to solve this problem with different strategies. We illustrate the effectiveness of our approach by experimenting with the synthetic data sets. The results from a study of the synthetic dataset were compared with Bruteforce and Random Algorithm. It suggested that our proposed both project greedy and expert greedy algorithms performed well regarding both accuracy and run-time.

We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph convolutional network that is trained to estimate the likelihood, for each vertex in a graph, of whether this vertex is part of the optimal solution. The network is designed and trained to synthesize a diverse set of solutions, which enables rapid exploration of the solution space via tree search. The presented approach is evaluated on four canonical NP-hard problems and five datasets, which include benchmark satisfiability problems and real social network graphs with up to a hundred thousand nodes. Experimental results demonstrate that the presented approach substantially outperforms recent deep learning work, and performs on par with highly optimized state-of-the-art heuristic solvers for some NP-hard problems. Experiments indicate that our approach generalizes across datasets, and scales to graphs that are orders of magnitude larger than those used during training.

We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph convolutional network that is trained to estimate the likelihood, for each vertex in a graph, of whether this vertex is part of the optimal solution. The network is designed and trained to synthesize a diverse set of solutions, which enables rapid exploration of the solution space via tree search. The presented approach is evaluated on four canonical NP-hard problems and five datasets, which include benchmark satisfiability problems and real social network graphs with up to a hundred thousand nodes. Experimental results demonstrate that the presented approach substantially outperforms recent deep learning work, and performs on par with highly optimized state-of-the-art heuristic solvers for some NP-hard problems. Experiments indicate that our approach generalizes across datasets, and scales to graphs that are orders of magnitude larger than those used during training.

We study the application of min-max propagation, a variation of belief propagation, for approximate min-max inference in factor graphs. We show that for “any” high-order function that can be minimized in O(ω), the min-max message update can be obtained using an efficient O(K(ω + log(K)) procedure, where K is the number of variables. We demonstrate how this generic procedure, in combination with efficient updates for a family of high-order constraints, enables the application of min-max propagation to efficiently approximate the NP-hard problem of makespan minimization, which seeks to distribute a set of tasks on machines, such that the worst case load is minimized.

There exist a number of problem classes for which obtaining the exact solution becomes exponentially expensive with increasing problem size. The quadratic assignment problem (QAP) or the travelling salesman problem (TSP) are just two examples of such NP-hard problems. In practice, approximate algorithms are employed to obtain a suboptimal solution, where one must face a trade-off between computational complexity and solution quality. In this paper, we propose to learn to solve these problem from approximate examples, using recurrent neural networks (RNNs). Surprisingly, such architectures are capable of producing highly accurate solutions at minimal computational cost. Moreover, we introduce a simple, yet effective technique for improving the initial (weak) training set by incorporating the objective cost into the training procedure. We demonstrate the functionality of our approach on three exemplar applications: marginal distributions of a joint matching space, feature point matching and the travelling salesman problem. We show encouraging results on synthetic and real data in all three cases.

I provide a personal view of some of the major research challenges in the area of combinatorial search. These include solving and playing games with chance, hidden information, and multiple players, optimally solving larger instances of well-known single-agent toy problems, applying search techniques to more realistic problem domains, analyzing the time complexity of heuristic search algorithms, and capitalizing on advances in computing hardware, such as very large external memories and multi-core processors.

The neural network model is a discrete time system that can be represented by a weighted and undirected graph. There is a weight attached to each edge of the graph and a threshold value attached to each node (neuron) of the graph.