Machine Learning-based heuristics have recently shown impressive performance in solving a variety of hard combinatorial optimization problems (COPs). However they generally rely on a separate neural model, specialized and trained for each single problem. Any variation of a problem requires adjustment of its model and re-training from scratch. In this paper, we propose GOAL (for Generalist combinatorial Optimization Agent Learning), a generalist model capable of efficiently solving multiple COPs and which can be fine-tuned to solve new COPs. GOAL consists of a single backbone plus light-weight problem-specific adapters, mostly for input and output processing. The backbone is based on a new form of mixed-attention blocks which allows to handle problems defined on graphs with arbitrary combinations of node, edge and instance-level features. Additionally, problems which involve heterogeneous nodes or edges, such as in multi-partite graphs, are handled through a novel multi-type transformer architecture, where the attention blocks are duplicated to attend only the relevant combination of types while relying on the same shared parameters. We train GOAL on a set of routing, scheduling and classic graph problems and show that it is only slightly inferior to the specialized baselines while being the first multi-task model that solves a variety of COPs. Finally, we showcase the strong transfer learning capacity of GOAL by fine-tuning or learning the adapters for new problems, with only few shots and little data.

Over the last decades, new mobility offers have emerged to enlarge the coverage and the accessibility of public transportation systems. In many areas, public transit now incorporates on-demand transport lines, that can be activated at user need. In this paper, we propose to integrate lines without predefined schedules but with predefined stop sequences into a state-of-the-art trip planning algorithm for public transit, the Trip-Based Public Transit Routing algorithm [33]. We extend this algorithm to non-scheduled lines and explain how to model other modes of transportation, such as bike sharing, with this approach. The resulting algorithm is exact and optimizes two criteria: the earliest arrival time and the minimal number of transfers. Experiments on two large datasets show the interest of the proposed method over a baseline modelling.

Despite the success of neural-based combinatorial optimization methods for end-to-end heuristic learning, out-of-distribution generalization remains a challenge. In this paper, we present a novel formulation of Combinatorial Optimization Problems (COPs) as Markov Decision Processes (MDPs) that effectively leverages common symmetries of COPs to improve out-of-distribution robustness. Starting from a direct MDP formulation of a constructive method, we introduce a generic way to reduce the state space, based on Bisimulation Quotienting (BQ) in MDPs. Then, for COPs with a recursive nature, we specialize the bisimulation and show how the reduced state exploits the symmetries of these problems and facilitates MDP solving. Our approach is principled and we prove that an optimal policy for the proposed BQ-MDP actually solves the associated COPs. We illustrate our approach on five classical problems: the Euclidean and Asymmetric Traveling Salesman, Capacitated Vehicle Routing, Orienteering and Knapsack Problems. Furthermore, for each problem, we introduce a simple attention-based policy network for the BQ-MDPs, which we train by imitation of (near) optimal solutions of small instances from a single distribution. We obtain new state-of-the-art results for the five COPs on both synthetic and realistic benchmarks. Notably, in contrast to most existing neural approaches, our learned policies show excellent generalization performance to much larger instances than seen during training, without any additional search procedure.

Despite the success of Neural Combinatorial Optimization methods for end-to-end heuristic learning

Neural Combinatorial Optimization approaches have recently leveraged the expressiveness and flexibility of deep neural networks to learn efficient heuristics for hard Combinatorial Optimization (CO) problems. However, most of the current methods lack generalization: for a given CO problem, heuristics which are trained on instances with certain characteristics underperform when tested on instances with different characteristics. While some previous works have focused on varying the training instances properties, we postulate that a one-size-fit-all model is out of reach. Instead, we formalize solving a CO problem over a given instance distribution as a separate learning task and investigate meta-learning techniques to learn a model on a variety of tasks, in order to optimize its capacity to adapt to new tasks. Through extensive experiments, on two CO problems, using both synthetic and realistic instances, we show that our proposed meta-learning approach significantly improves the generalization of two state-of-the-art models.

Time series prediction is a widespread and well studied problem with applications in many domains (medical, geoscience, network analysis, finance, econometry etc.). In the case of multivariate time series, the key to good performances is to properly capture the dependencies between the variates. Often, these variates are structured, i.e. they are localised in an abstract space, usually representing an aspect of the physical world, and prediction amounts to a form of diffusion of the information across that space over time. Several neural network models of diffusion have been proposed in the literature. However, most of the existing proposals rely on some a priori knowledge on the structure of the space, usually in the form of a graph weighing the pairwise diffusion capacity of its points. We argue that this piece of information can often be dispensed with, since data already contains the diffusion capacity information, and in a more reliable form than that obtained from the usually largely hand-crafted graphs. We propose instead a fully data-driven model which does not rely on such a graph, nor any other prior structural information. We conduct a first set of experiments to measure the impact on performance of a structural prior, as used in baseline models, and show that, except at very low data levels, it remains negligible, and beyond a threshold, it may even become detrimental. We then investigate, through a second set of experiments, the capacity of our model in two respects: treatment of missing data and domain adaptation.

We study the problem of finding bi-criteria Pareto optimal journeys in public transit networks. We extend the Trip-Based Public Transit Routing (TB) approach [18] to allow for users to select modes of interest at query time. As a first step, we modify the preprocessing of the TB method for it to be correct for any set of selected modes. Then, we change the bi-criteria earliest arrival time queries, and propose a similar algorithm for latest departure time queries, that can handle the definition of the allowed mode set at query time. Experiments are run on 3 networks of different sizes to evaluate the cost of allowing for mode personalization. They show that although preprocessing times are increased, query times are similar when all modes are allowed and lower when some part of the network is removed by mode selection. 2012 ACM Subject Classification Mathematics of computing → Graph theory

Covering location problems is well-known and very important class of combinatorial optimization problems. Standard models for covering location problems cannot encompass real-life problems, because they contain some degree of uncertainty. The use of fuzzy sets in modeling covering location problems allows the implementation of these conditions. Depending on the type of problems, it is necessary to use different aggregation operators in calculating solution’s quality. The aim of this study is introducing of fuzzy sets with different corresponding conorms in modeling most known types of covering location problems.

The objective of Maximal Covering Location Problem is locating facilities such that they cover the maximal number of locations in a given radius or travel time. MCLP is applied in many different real-world problems with several modifications. In this paper a new model of MCLP with fuzzy conditions is presented. It uses two types of fuzzy numbers for describing two main parameters of MCLP - coverage radius and distances between locations. First, the model is defined, then Particle Swarm Optimization method for solving the problem is described and tested.