This paper solves three NP-hard routing problems, traveling salesman problem (TSP), prize collecting TSP (PCTSP), and capacitated vehicle routing problem (CVRP). This section provides detailed descriptions of PCTSP and CVRP (for TSP, see section 3). The PCTSP is similar to TSP, while there are differences in that we do not have to visit all the nodes and that the destination is not the first node but the depot node, i.e., a tour is not a cycle. Let N be the number of nodes. The problem instance of PCTSP is s = {(xi,λi,µi)}N+1i=1, where the xi R2 is in 2D euclidean coordinates, λi R is the penalty of unvisited node, and µi R is the prize of visited node. The L(π|s) is the tour length, and λ(π|s) is the total penalty of the unvisited nodes.

Foundation models are becoming valuable tools in medicine. Yet despite their promise, the best way to leverage Large Language Models (LLMs) in complex medical tasks remains an open question. We introduce a novel multi-agent framework, named **M**edical **D**ecision-making **Agents** (**MDAgents**) that helps to address this gap by automatically assigning a collaboration structure to a team of LLMs. The assigned solo or group collaboration structure is tailored to the medical task at hand, a simple emulation inspired by the way real-world medical decision-making processes are adapted to tasks of different complexities. We evaluate our framework and baseline methods using state-of-the-art LLMs across a suite of real-world medical knowledge and clinical diagnosis benchmarks, including a comparison ofLLMs' medical complexity classification against human physicians. MDAgents achieved the **best performance in seven out of ten** benchmarks on tasks requiring an understanding of medical knowledge and multi-modal reasoning, showing a significant **improvement of up to 4.2\%** ($p$ < 0.05) compared to previous methods' best performances. Ablation studies reveal that MDAgents effectively determines medical complexity to optimize for efficiency and accuracy across diverse medical tasks. Notably, the combination of moderator review and external medical knowledge in group collaboration resulted in an average accuracy **improvement of 11.8\%**.

We utilize the sparsity operation proposed in Section 3.1 for ResNet-50. We generalize section 4.2 in the main text to ResNet-50 and ViT-B architectures (Figure 1). We apply the Sparsity layer in a subset of the network. It is based on the intuition that the brain utilizes sparsity for long range communication butcan allowlocal dense computation. Wedivide thenetworks into chunks where within each chunk theneuron'sactivities areallowed tobedense (keep original) but the communication across different chunks is set to be sparse.

In this section, we show the proofs of the results in the main body. Eq. (1) satisfies the triangle inequality, i.e., for any scoring functions For the second inequality, we prove it similarly. Before we present the proof of the theorem, we first provide some lemmas. By applying Lemma A.2, the following holds with probability at least 1 α: null R F). Thus we have: null R A.1, we can get that the margin loss satisfies the triangle inequality. By Lemma A.4, we have R By Theorem 4.4, the following holds for any Based on Theorem A.6, the following standard error bound for gradual AST can be derived similarly to Corollary 4.6.

The color has been normalized to be between 0 and 1 which does not affect the clustering or visualization. We can see that output representation from later layers yield more patterned kernel matrices with more erratic clustering.