We propose UTSP, an Unsupervised Learning (UL) framework for solving the Travelling Salesman Problem (TSP). We train a Graph Neural Network (GNN) using a surrogate loss. The GNN outputs a heat map representing the probability for each edge to be part of the optimal path. We then apply local search to generate our final prediction based on the heat map. Our loss function consists of two parts: one pushes the model to find the shortest path and the other serves as a surrogate for the constraint that the route should form a Hamiltonian Cycle. Experimental results show that UTSP outperforms the existing data-driven TSP heuristics. Our approach is parameter efficient as well as data efficient: the model takes 10% of the number of parameters and 0.2% of training samples compared with Reinforcement Learning or Supervised Learning methods.

Recent work [Ma et al., 2019] shows that in the non-convex case, sampling

We propose a non-autoregressive framework for the Travelling Salesman Problem where solutions emerge directly from learned permutations, without requiring explicit search. By applying a similarity transformation to Hamiltonian cycles, the model learns to approximate permutation matrices via continuous relaxations. Our unsupervised approach achieves competitive performance against classical heuristics, demonstrating that the inherent structure of the problem can effectively guide combinatorial optimization without sequential decision-making. Our method offers concrete evidence that neural networks can directly capture and exploit combinatorial structure.

Recent work [Ma et al., 2019] shows that in the non-convex case, sampling

Recent advancements in multimodal large language models have driven breakthroughs in visual question answering. Yet, a critical gap persists, `conceptualization'-the ability to recognize and reason about the same concept despite variations in visual form, a basic ability of human reasoning. To address this challenge, we introduce the Visual Graph Arena (VGA), a dataset featuring six graph-based tasks designed to evaluate and improve AI systems' capacity for visual abstraction. VGA uses diverse graph layouts (e.g., Kamada-Kawai vs. planar) to test reasoning independent of visual form. Experiments with state-of-the-art vision models and multimodal LLMs reveal a striking divide: humans achieved near-perfect accuracy across tasks, while models totally failed on isomorphism detection and showed limited success in path/cycle tasks. We further identify behavioral anomalies suggesting pseudo-intelligent pattern matching rather than genuine understanding. These findings underscore fundamental limitations in current AI models for visual understanding. By isolating the challenge of representation-invariant reasoning, the VGA provides a framework to drive progress toward human-like conceptualization in AI visual models. The Visual Graph Arena is available at: \href{https://vga.csail.mit.edu/}{vga.csail.mit.edu}

Online learning algorithms are widely used in strategic multi-agent settings, including repeated auctions, contract design, and pricing competitions, where agents adapt their strategies over time. A key question in such environments is how an optimizing agent can best respond to a learning agent to improve its own long-term outcomes. While prior work has developed efficient algorithms for the optimizer in special cases - such as structured auction settings or contract design - no general efficient algorithm is known. In this paper, we establish a strong computational hardness result: unless $\mathsf{P} = \mathsf{NP}$, no polynomial-time optimizer can compute a near-optimal strategy against a learner using a standard no-regret algorithm, specifically Multiplicative Weights Update (MWU). Our result proves an $\Omega(T)$ hardness bound, significantly strengthening previous work that only showed an additive $\Theta(1)$ impossibility result. Furthermore, while the prior hardness result focused on learners using fictitious play - an algorithm that is not no-regret - we prove intractability for a widely used no-regret learning algorithm. This establishes a fundamental computational barrier to finding optimal strategies in general game-theoretic settings.

Multirotor UAVs have been typically considered for aerial manipulation, but their scarce endurance prevents long-lasting manipulation tasks. This work demonstrates that the non-stop flights of three or more carriers are compatible with holding a constant pose of a cable-suspended load, thus potentially enabling aerial manipulation with energy-efficient non-stop carriers. It also presents an algorithm for generating the coordinated non-stop trajectories. The proposed method builds upon two pillars: (1)~the choice of $n$ special linearly independent directions of internal forces within the $3n-6$-dimensional nullspace of the grasp matrix of the load, chosen as the edges of a Hamiltonian cycle on the graph that connects the cable attachment points on the load. Adjacent pairs of directions are used to generate $n$ forces evolving on distinct 2D affine subspaces, despite the attachment points being generically in 3D; (2)~the construction of elliptical trajectories within these subspaces by mapping, through appropriate graph coloring, each edge of the Hamiltonian cycle to a periodic coordinate while ensuring that no adjacent coordinates exhibit simultaneous zero derivatives. Combined with conditions for load statics and attachment point positions, these choices ensure that each of the $n$ force trajectories projects onto the corresponding cable constraint sphere with non-zero tangential velocity, enabling perpetual motion of the carriers while the load is still. The theoretical findings are validated through simulations and laboratory experiments with non-stopping multirotor UAVs.

Abstract: The Vehicle Routing Problem is about optimizing the routes of vehicles to meet the needs of customers at specific locations. The route graph consists of depots on several levels and customer positions. Several optimization methods have been developed over the years, most of which are based on some type of classic heuristic: genetic algorithm, simulated annealing, tabu search, ant colony optimization, firefly algorithm. Recent developments in machine learning provide a new toolset, the rich family of neural networks, for tackling complex problems. The main area of application of neural networks is the area of classification and regression. Route optimization can be viewed as a new challenge for neural networks. The article first presents an analysis of the applicability of neural network tools, then a novel graphical neural network model is presented in detail. The efficiency analysis based on test experiments shows the applicability of the proposed NN architecture.

This paper aims to develop a learning method for a special class of traveling salesman problems (TSP), namely, the pickup-and-delivery TSP (PDTSP), which finds the shortest tour along a sequence of one-to-one pickup-and-delivery nodes. One-to-one here means that the transported people or goods are associated with designated pairs of pickup and delivery nodes, in contrast to that indistinguishable goods can be delivered to any nodes. In PDTSP, precedence constraints need to be satisfied that each pickup node must be visited before its corresponding delivery node. Classic operations research (OR) algorithms for PDTSP are difficult to scale to large-sized problems. Recently, reinforcement learning (RL) has been applied to TSPs. The basic idea is to explore and evaluate visiting sequences in a solution space. However, this approach could be less computationally efficient, as it has to potentially evaluate many infeasible solutions of which precedence constraints are violated. To restrict solution search within a feasible space, we utilize operators that always map one feasible solution to another, without spending time exploring the infeasible solution space. Such operators are evaluated and selected as policies to solve PDTSPs in an RL framework. We make a comparison of our method and baselines, including classic OR algorithms and existing learning methods. Results show that our approach can find tours shorter than baselines.