Combinatorial Optimization (CO) has been a long-standing challenging research topic featured by its NP-hard nature. Traditionally such problems are approximately solved with heuristic algorithms which are usually fast but may sacrifice the solution quality. Currently, machine learning for combinatorial optimization (MLCO) has become a trending research topic, but most existing MLCO methods treat CO as a single-level optimization by directly learning the end-to-end solutions, which are hard to scale up and mostly limited by the capacity of ML models given the high complexity of CO. In this paper, we propose a hybrid approach to combine the best of the two worlds, in which a bi-level framework is developed with an upper-level learning method to optimize the graph (e.g.

Given a network, the critical node detection problem finds a subset of nodes whose removal disrupts the network connectivity. Since many real-world systems are naturally modeled as graphs, assessing the vulnerability of the network is essential, with applications in transportation systems, traffic forecasting, epidemic control, and biological networks. In this paper, we consider a stochastic version of the critical node detection problem, where the existence of edges is given by certain probabilities. We propose heuristics and learning-based methods for the problem and compare them with existing algorithms. Experimental results performed on random graphs from small to larger scales, with edge-survival probabilities drawn from different distributions, demonstrate the effectiveness of the methods. Heuristic methods often illustrate the strongest results with high scalability, while learning-based methods maintain nearly constant inference time as the network size and density grow.

Combinatorial optimization problems are traditionally tackled with handcrafted heuristic algorithms, which demand extensive domain expertise and significant implementation effort. Recent progress has highlighted the potential of automatic heuristics design powered by large language models (LLMs), enabling the automatic generation and refinement of heuristics. These approaches typically maintain a population of heuristics and employ LLMs as mutation operators to evolve them across generations. While effective, such methods often risk stagnating in local optima. To address this issue, we propose the Experience-Guided Reflective Co-Evolution of Prompt and Heuristics (EvoPH) for automatic algorithm design, a novel framework that integrates the island migration model with the elites selection algorithm to simulate diverse heuristics populations. In EvoPH, prompts are co-evolved with heuristic algorithms, guided by performance feedback. We evaluate our framework on two problems, i.e., Traveling Salesman Problem and Bin Packing Problem. Experimental results demonstrate that EvoPH achieves the lowest relative error against optimal solutions across both datasets, advancing the field of automatic algorithm design with LLMs.

-- In swarm robotics, confrontation scenarios, including strategic confrontations, require efficient decision-making that integrates discrete commands and continuous actions. Traditional task and motion planning methods separate decision-making into two layers, but their unidirectional structure fails to capture the interdependence between these layers, limiting adaptability in dynamic environments. Here, we propose a novel bidirectional approach based on hierarchical reinforcement learning, enabling dynamic interaction between the layers. This method effectively maps commands to task allocation and actions to path planning, while leveraging cross-training techniques to enhance learning across the hierarchical framework. Furthermore, we introduce a trajectory prediction model that bridges abstract task representations with actionable planning goals. In our experiments, it achieves over 80% in confrontation win rate and under 0.01 seconds in decision time, outperforming existing approaches. Demonstrations through large-scale tests and real-world robot experiments further emphasize the generalization capabilities and practical applicability of our method. I. INTRODUCTION Recent advances in artificial intelligence lead to significant progress in robotics [1], [2], with particular attention given to robotic swarm confrontations [3], [4].

We would also like to discuss the limitations of the approaches including ours. As shown in Tab. 4, the PPO-Single that serves as a baseline in our paper is designed following As shown in Tab. 4, NerRewritter is most general because it can be viewed as a learning-based local It is also worth noting that there are some problems that are beyond our knowledge to tackle, e.g. the expression simplify problem, and it may requires experts with specific domain We have discussed the model details of PPO-BiHyb in Sec. 4, and in this section, we discuss the DAG. Considering the structure of DAG, we design two GCNs: the first GCN processes the original DAG, and the second GCN processes the DAG with all edges reversed. The predicted doubly-stochastic matrix by SK is processed by considering the partial matching matrix. Graph-level features are obtained via attention pooling, which are fed to the critic net.

Intercepting a criminal using limited police resources presents a significant challenge in dynamic crime environments, where the criminal's location continuously changes over time. The complexity is further heightened by the vastness of the transportation network. To tackle this problem, we propose a layered graph representation, in which each time step is associated with a duplicate of the transportation network. For any given set of attacker strategies, a near-optimal defender strategy is computed using the A-Star heuristic algorithm applied to the layered graph. The defender's goal is to maximize the probability of successful interdiction. We evaluate the performance of the proposed method by comparing it with a Mixed-Integer Linear Programming (MILP) approach used for the defender. The comparison considers both computational efficiency and solution quality. The results demonstrate that our approach effectively addresses the complexity of the problem and delivers high-quality solutions within a short computation time.

We consider the problem of propagating the uncertainty from a possibly large number of random inputs through a computationally expensive model. Stratified sampling is a well-known variance reduction strategy, but its application, thus far, has focused on models with a limited number of inputs due to the challenges of creating uniform partitions in high dimensions. To overcome these challenges, we perform stratification with respect to the uniform distribution defined over the unit interval, and then derive the corresponding strata in the original space using nonlinear dimensionality reduction. We show that our approach is effective in high dimensions and can be used to further reduce the variance of multifidelity Monte Carlo estimators.

The graph coloring problem (GCP) is a classic combinatorial optimization problem that aims to find the minimum number of colors assigned to vertices of a graph such that no two adjacent vertices receive the same color. GCP has been extensively studied by researchers from various fields, including mathematics, computer science, and biological science. Due to the NP-hard nature, many heuristic algorithms have been proposed to solve GCP. However, existing GCP algorithms focus on either small hard graphs or large-scale sparse graphs (with up to 10^7 vertices). This paper presents an efficient hybrid heuristic algorithm for GCP, named HyColor, which excels in handling large-scale sparse graphs while achieving impressive results on small dense graphs. The efficiency of HyColor comes from the following three aspects: a local decision strategy to improve the lower bound on the chromatic number; a graph-reduction strategy to reduce the working graph; and a k-core and mixed degree-based greedy heuristic for efficiently coloring graphs. HyColor is evaluated against three state-of-the-art GCP algorithms across four benchmarks, comprising three large-scale sparse graph benchmarks and one small dense graph benchmark, totaling 209 instances. The results demonstrate that HyColor consistently outperforms existing heuristic algorithms in both solution accuracy and computational efficiency for the majority of instances. Notably, HyColor achieved the best solutions in 194 instances (over 93%), with 34 of these solutions significantly surpassing those of other algorithms. Furthermore, HyColor successfully determined the chromatic number and achieved optimal coloring in 128 instances.