Combinatorial optimization (CO) problems, central to decision-making scenarios like logistics and manufacturing, are traditionally solved using problem-specific algorithms requiring significant domain expertise. While large language models (LLMs) have shown promise in automating CO problem solving, existing approaches rely on intermediate steps such as code generation or solver invocation, limiting their generality and accessibility. This paper introduces a novel framework that empowers LLMs to serve as end-to-end CO solvers by directly mapping natural language problem descriptions to solutions.

Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem. Existing approaches focus on finding a single optimal solution. We introduce FOSC-X, a framework for extracting the top-M globally optimal flat clusterings from local, non-horizontal cuts of a hierarchical cluster tree, while optionally enforcing constraints on the number of clusters. This enables automatic identification of multiple high-quality alternative clusterings that capture different aspects of the hierarchical structure. Without constraints, the top-M problem can be solved in polynomial time using dynamic programming, exploiting the property that locally optimal partial candidates within subtrees can be combined to form globally optimal solutions while automatically determining the number of clusters. However, this can lead to solutions with numbers of clusters that are ultimately undesirable -- e.g., too large to be meaningful or practically analysed within a particular application domain. Imposing cluster-count constraints breaks the optimality property underlying the unconstrained dynamic programming approach, since locally optimal partial candidates may no longer combine into feasible globally optimal solutions. FOSC-X addresses this challenge through a dynamic programming strategy that maintains compact sets of feasible candidates using lower and upper feasibility bounds while pruning infeasible or dominated combinations. The resulting method guarantees optimal rankings of the top-M solutions with linear-time complexity in the number of cluster nodes and dataset size, both with and without cluster-count constraints. Experiments show that FOSC-X efficiently reveals alternative clustering structures overlooked by single-solution extraction methods.

We study the Quality of Service Degradation (QoSD) problem, in which an adversary perturbs edge weights to degrade network performance. This setting arises in both network infrastructures and distributed ML systems, where communication quality, not just connectivity, determines functionality. While classical methods rely on combinatorial optimization, and recent ML approaches address only restricted linear variants with small-size networks, no prior model directly tackles the QoSD problem under nonlinear edge-weight functions. This work proposes Hephaestus, a self-reinforcing generative framework that synthesizes feasible solutions in latent space, to fill this gap. Our method includes three phases: (1) Forge: a Predictive Path-Stressing (PPS) algorithm that uses graph learning and approximation to produce feasible solutions with performance guarantee, (2) Morph: a new theoretically grounded training paradigm for Mixture of Conditional VAEs guided by an energy-based model to capture solution feature distributions, and (3) Refine: a reinforcement learning agent that explores this space to generate progressively near-optimal solutions using our designed differentiable reward function. Experiments on both synthetic and real-world networks show that our approach consistently outperforms classical and ML baselines, particularly in scenarios with nonlinear cost functions where traditional methods fail to generalize.

Biases with respect to socially-salient attributes of individuals have been well documented in evaluation processes used in settings such as admissions and hiring. We view such an evaluation process as a transformation of a distribution of the true utility of an individual for a task to an observed distribution and model it as a solution to a loss minimization problem subject to an information constraint. Our model has two parameters that have been identified as factors leading to biases: the resource-information trade-off parameter in the information constraint and the risk-averseness parameter in the loss function. We characterize the distributions that arise from our model and study the effect of the parameters on the observed distribution. The outputs of our model enrich the class of distributions that can be used to capture variation across groups in the observed evaluations. We empirically validate our model by fitting real-world datasets and use it to study the effect of interventions in a downstream selection task. These results contribute to an understanding of the emergence of bias in evaluation processes and provide tools to guide the deployment of interventions to mitigate biases.

Let ϑ Rd1 and µ Rd2 be the dual variables corresponding to the d1 equality constraints and the d2 inequality constraints respectively. Let X? be an optimal solution to (SDP) and let X?FW be an optimal solution to (SDP-LSE). For ease of notation, let u= A(1)(X) b(1) andv = b(2) A(2)(X), (1) and define (bu,bv), (uFW,vFW) and (u?,v?) by substituting bX, XFW and X? respectively in (1). Upper bound on the objective. Rearranging the terms, using the duality of the `1 and ` norms, and the fact that µ? 0, gives hC, bX i hC,X?i+

A.1 For LEHD model467 In Table 5, we explore the effects of eliminating normalization from the attention layer in our LEHD468 model. We train three LEHD models with the same training scheme and training budget, differing469 solely in the attention layer: one with batch normalization (BN), one with instance normalization470 (IN), and one without normalization (w/o). We train all three POMO models with the same reinforcement learning method477 with POMO strategy and training budget (1000 epochs). The results show that different types of478 normalization have few effects on the POMO model.479 The results in Table 6 show that removing normalization from attention layer has little impact on the480 model with a heavy encoder and a light decoder.

Constrained optimization in high-dimensional black-box settings is difficult due to expensive evaluations, the lack of gradient information, and complex feasibility regions. In this work, we propose a Bayesian optimization method that combines a penalty formulation, a surrogate model, and a trust region strategy. The constrained problem is converted to an unconstrained form by penalizing constraint violations, which provides a unified modeling framework. A trust region restricts the search to a local region around the current best solution, which improves stability and efficiency in high dimensions. Within this region, we use the Expected Improvement acquisition function to select evaluation points by balancing improvement and uncertainty. The proposed Trust Region method integrates penalty-based constraint handling with local surrogate modeling. This combination enables efficient exploration of feasible regions while maintaining sample efficiency. We compare the proposed method with state-of-the-art methods on synthetic and real-world high-dimensional constrained optimization problems. The results show that the method identifies high-quality feasible solutions with fewer evaluations and maintains stable performance across different settings.

We study the cost function for hierarchical clusterings introduced by [16] where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown in [16] that a top-down algorithm returns a hierarchical clustering of cost at most O(αnlog n) times the cost of the optimal hierarchical clustering, where αn is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao-Vazirani, the top-down algorithm returns a hierarchical clustering of cost at most O log3/2 ntimes the cost of the optimal solution. We improve this by giving an O(log n)-approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of sphere growing which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an O(log n)-approximate hierarchical clustering for a generalization of this cost function also studied in [16]. We also give constant factor inapproximability results for this problem.