Recent advances in neural models have shown considerable promise in solving Traveling Salesman Problems (TSPs) without relying on much hand-crafted engineering. However, while non-autoregressive (NAR) approaches benefit from faster inference through parallelism, they typically deliver solutions of inferior quality compared to autoregressive ones. To enhance the solution quality while maintaining fast inference, we propose DEITSP, a diffusion model with efficient iterations tailored for TSP that operates in a NAR manner. Firstly, we introduce a one-step diffusion model that integrates the controlled discrete noise addition process with self-consistency enhancement, enabling optimal solution prediction through simultaneous denoising of multiple solutions. Secondly, we design a dual-modality graph transformer to bolster the extraction and fusion of features from node and edge modalities, while further accelerating the inference with fewer layers. Thirdly, we develop an efficient iterative strategy that alternates between adding and removing noise to improve exploration compared to previous diffusion methods. Additionally, we devise a scheduling framework to progressively refine the solution space by adjusting noise levels, facilitating a smooth search for optimal solutions. Extensive experiments on real-world and large-scale TSP instances demonstrate that DEITSP performs favorably against existing neural approaches in terms of solution quality, inference latency, and generalization ability. Our code is available at $\href{https://github.com/DEITSP/DEITSP}{https://github.com/DEITSP/DEITSP}$.

Distributed Constraint Optimization Problems (DCOPs) offer a powerful framework for multi-agent coordination but often rely on labor-intensive, manual problem construction. To address this, we introduce VL-DCOPs, a framework that takes advantage of large multimodal foundation models (LFMs) to automatically generate constraints from both visual and linguistic instructions. We then introduce a spectrum of agent archetypes for solving VL-DCOPs: from a neuro-symbolic agent that delegates some of the algorithmic decisions to an LFM, to a fully neural agent that depends entirely on an LFM for coordination. We evaluate these agent archetypes using state-of-the-art LLMs (large language models) and VLMs (vision language models) on three novel VL-DCOP tasks and compare their respective advantages and drawbacks. Lastly, we discuss how this work extends to broader frontier challenges in the DCOP literature.

A common characteristic in integer linear programs (ILPs) is symmetry, allowing variables to be permuted without altering the underlying problem structure. Recently, GNNs have emerged as a promising approach for solving ILPs. However, a significant challenge arises when applying GNNs to ILPs with symmetry: classic GNN architectures struggle to differentiate between symmetric variables, which limits their predictive accuracy. In this work, we investigate the properties of permutation equivariance and invariance in GNNs, particularly in relation to the inherent symmetry of ILP formulations. We reveal that the interaction between these two factors contributes to the difficulty of distinguishing between symmetric variables. To address this challenge, we explore the potential of feature augmentation and propose several guiding principles for constructing augmented features. Building on these principles, we develop an orbit-based augmentation scheme that first groups symmetric variables and then samples augmented features for each group from a discrete uniform distribution. Empirical results demonstrate that our proposed approach significantly enhances both training efficiency and predictive performance. Integer Linear Programs (ILPs) are fundamental optimization problems characterized by a linear objective function and linear constraints, where the decision variables are restricted to integer values. These problems play a critical role in various fields, including operations research, computer science, and engineering (Pochet & Wolsey, 2006; Liu & Fan, 2018; Watson & Woodruff, 2011; Luathep et al., 2011; Schöbel, 2001).

Additional Feedback: Do you have any comments on how your hard shape constraint formulation affects overlapping quantiles versus the soft constraint (PDCD)? It would be more informative to display Figure 1 alongside plots from alternative methods. Is it reasonable to attribute the comments on l304 regarding non-crossing to the additional regularizing properties of the concavity constraint? Regarding Table 1, it seems SOC performs _worse_ than PDCD on 5/9 datasets. This opens the question of when and where are hard constraints more beneficial over soft constraints.

General comments: (i) The authors only solve a new special kind of higher order consistency constraints, generalizing soft PN-potentials, but not a truly general class of constraints, as indicated in the title or in the abstract. In case of MAP-inference, which is normally desired, the goal is to obtain a single assignment which satisfies all given linear constraints. The relaxed model the authors optimize is simply a byproduct of looking for marginals instead of MAP-assignments (the added entropy is responsible for this). In case of vanishing entropy one gets the same model. Hence there certainly remains the disadvantage of a parameter in the PN-potential, but now hidden in the entropy.

Many practical settings call for the reconstruction of temporal signals from corrupted or missing data. Classic examples include decoding, tracking, signal enhancement and denoising. Since the reconstructed signals are ultimately viewed by humans, it is desirable to achieve reconstructions that are pleasing to human perception.Mathematically, perfect perceptual-quality is achieved when the distribution of restored signals is the same as that of natural signals, a requirement which has been heavily researched in static estimation settings (i.e. when a whole signal is processed at once). Here, we study the problem of optimal causal filtering under a perfect perceptual-quality constraint, which is a task of fundamentally different nature. Specifically, we analyze a Gaussian Markov signal observed through a linear noisy transformation.

In many applications, learning systems are required to process continuous non-stationary data streams.We study this problem in an online learning framework and propose an algorithm that can deal with adversarial time-varying and nonlinear constraints.As we show in our work, the algorithm called Constraint Violation Velocity Projection (CVV-Pro) achieves \sqrt{T} regret and converges to the feasible set at a rate of 1/\sqrt{T}, despite the fact that the feasible set is slowly time-varying and a priori unknown to the learner. CVV-Pro only relies on local sparse linear approximations of the feasible set and therefore avoids optimizing over the entire set at each iteration, which is in sharp contrast to projected gradients or Frank-Wolfe methods. We also empirically evaluate our algorithm on two-player games, where the players are subjected to a shared constraint.

Recent advances in constrained reinforcement learning (RL) have endowed reinforcement learning with certain safety guarantees. However, deploying existing constrained RL algorithms in continuous control tasks with general hard constraints remains challenging, particularly in those situations with non-convex hard constraints. Inspired by the generalized reduced gradient (GRG) algorithm, a classical constrained optimization technique, we propose a reduced policy optimization (RPO) algorithm that combines RL with GRG to address general hard constraints. Subsequently, RPO calculates the nonbasic actions by solving equations based on equality constraints using the obtained basic actions. The policy network is then updated by implicitly differentiating nonbasic actions with respect to basic actions.

This paper considers online convex optimization with hard constraints and analyzes achievable regret and cumulative hard constraint violation (violation for short). The problem distinguishes itself from online convex optimization with soft constraints, where a violation at one round can be compensated/cancelled by a conservative decision at a different round. We propose a RECtified Online Optimization algorithm (RECOO) and consider two settings: fixed constraints and adversarial constraints. Both settings have been considered in the literature. Compared with existing results, {\em RECOO achieves the best of two worlds and beyond.}

This paper considers stochastic linear bandits with general nonlinear constraints. The objective is to maximize the expected cumulative reward over horizon T subject to a set of constraints in each round \tau\leq T . We propose a pessimistic-optimistic algorithm for this problem, which is efficient in two aspects. First, the algorithm yields \tilde{\cal O}\left(\left(\frac{K {0.75}}{\delta} d\right)\sqrt{\tau}\right) (pseudo) regret in round \tau\leq T, where K is the number of constraints, d is the dimension of the reward feature space, and \delta is a Slater's constant; and {\em zero} constraint violation in any round \tau \tau', where \tau' is {\em independent} of horizon T. Second, the algorithm is computationally efficient. Our algorithm is based on the primal-dual approach in optimization and includes two components.