A.1 Omitted Proofs (Details for Lemma 3) Clustering Nets Next, we give a detailed proof of Lemma 4. Proof of Lemma 4. Our objective is to generate a small set of cost vectors that satisfy the desired guarantee. We first define the cost vectors (the reader familiar with the proof sketch from the main body of the submission may skip this and the next paragraph). For each subset U of size O(min(α 22i,α 2+ki), we consider the the subspace ΠU spanned by U. In this subspace we consider (α/2i) p cost(p,A)nets of every ball centered around ΠUpwith radius 60 2i/2 p cost(p,A)for all p P. Such a net has size exp(γ rank(U)ilogα), for some constant γ and there exist at most P 0 exp(γ |U|ilogα) Furthermore, there are at most P 0|U| = P |U|0 such subsets. Now, for every point p, define an exponential sequence α2(1 + α/2i)j for j {0,...log102i}. There exist at most P 0 such sequences and every such sequence consists of at most O(α 1 2i i) many values. We combine every net point in ever ball of every subspace with all values in the exponential sequence to obtain the evaluation for a single candidate center.

It imposes lower and upper bound constraints on the number of facilities opened for each label, ensuring fair representation of all demographic groups by the selected facilities.

When such outliers occur at the end of the data stream, this can cause the final solution to have unexpectedly low accuracy.

Selecting interpretable feature sets in underdetermined ($n \ll p$) and highly correlated regimes constitutes a fundamental challenge in data science, particularly when analyzing physical measurements. In such settings, multiple distinct sparse subsets may explain the response equally well. Identifying these alternatives is crucial for generating domain-specific insights into the underlying mechanisms, yet conventional methods typically isolate a single solution, obscuring the full spectrum of plausible explanations. We present GEMSS (Gaussian Ensemble for Multiple Sparse Solutions), a variational Bayesian framework specifically designed to simultaneously discover multiple, diverse sparse feature combinations. The method employs a structured spike-and-slab prior for sparsity, a mixture of Gaussians to approximate the intractable multimodal posterior, and a Jaccard-based penalty to further control solution diversity. Unlike sequential greedy approaches, GEMSS optimizes the entire ensemble of solutions within a single objective function via stochastic gradient descent. The method is validated on a comprehensive benchmark comprising 128 synthetic experiments across classification and regression tasks. Results demonstrate that GEMSS scales effectively to high-dimensional settings ($p=5000$) with sample size as small as $n = 50$, generalizes seamlessly to continuous targets, handles missing data natively, and exhibits remarkable robustness to class imbalance and Gaussian noise. GEMSS is available as a Python package 'gemss' at PyPI. The full GitHub repository at https://github.com/kat-er-ina/gemss/ also includes a free, easy-to-use application suitable for non-coders.

The seeder generates as diversified candidate solutions as possible (seeds) while being dedicated to exploring over the full combinatorial action space (i.e.,sequence ofassignment action).

We further evaluate SGBS+EAS on nine real-world based instance sets from [15]. Each instance set consists of 20 instances that have similar characteristics (i.e., they have been sampled from the same underlying distribution). To account for this new evaluation setting, we always perform 10 runs in parallel for EAS and SGBS+EAS. This improves the solution quality, while leading only to a slight increase of the requiredruntime. For SGBS+EAS we set (β, γ) = (35,5), the learning rate α = 0.005 and λ = 0.05.

Recently, deep reinforcement learning (DRL) frameworks have shown potential for solving NP-hard routing problems such as the traveling salesman problem (TSP) without problem-specific expert knowledge. Although DRL can be used to solve complex problems, DRL frameworks still struggle to compete with state-of-the-art heuristics showing a substantial performance gap. This paper proposes a novel hierarchical problem-solving strategy, termed learning collaborative policies (LCP), which can effectively find the near-optimum solution using two iterative DRL policies: the seeder and reviser. The seeder generates as diversified candidate solutions as possible (seeds) while being dedicated to exploring over the full combinatorial action space (i.e., sequence of assignment action). To this end, we train the seeder's policy using a simple yet effective entropy regularization reward to encourage the seeder to find diverse solutions. On the other hand, the reviser modifies each candidate solution generated by the seeder; it partitions the full trajectory into sub-tours and simultaneously revises each sub-tour to minimize its traveling distance. Thus, the reviser is trained to improve the candidate solution's quality, focusing on the reduced solution space (which is beneficial for exploitation). Extensive experiments demonstrate that the proposed two-policies collaboration scheme improves over single-policy DRL framework on various NP-hard routing problems, including TSP, prize collecting TSP (PCTSP), and capacitated vehicle routing problem (CVRP).