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arXiv2026

Batched Pandora's Box

Motivated by numerous parallelizable stochastic search problems, most notable and timely among them being LLM inference-time scaling, we propose and study batched versions of the Pandora's Box problem of Weitzman. In particular, boxes are opened in capacity-constrained batches, each batch has a setup cost, and all rewards in a batch are revealed together. We consider two different variants, motivated by different application environments: one where boxes are reusable (i.e., can provide multiple i.i.d.~samples) and another where they are not. For both variants we rule out most ``simple'' natural heuristics, and also formally prove NP-hardness of approximation in the traditional sense. We then relax the problem to allow bi-criteria approximations, with respect to both rewards and setup costs, where we exhibit constant approximation algorithms for both the reusable and non-reusable settings. This is obtained through a linear-programming relaxation of Pandora's Box problem, followed by randomized or Pipage rounding.

cs.DS↗
arXiv2026

Data-efficient Kernel Methods for Learning Hamiltonian Systems

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.

math.NA↗
arXiv2026

Online Differentially Private Consistent Clustering

We study differentially private (DP) $k$-means and $k$-median clustering in the online streaming setting. In this model, points arrive sequentially, and at each time step, we need to output a set of $k$ centers that optimizes the clustering objective for all points seen so far. We give a generic reduction that transforms the (sensitive) input stream into a private stream, which is a semi-coreset of the input stream. This implies that any (non-private) online clustering algorithm, run as a post-processing step, can achieve good utility for the original clustering objective. Our algorithm matches or improves upon the approximation ratio, space usage, and running time of existing algorithms [Epasto et al., 2026, Dupré la Tour et al., 2024]. A key aspect of our reduction is that it inherits desirable properties of the underlying non-private clustering algorithm, such as consistency [Lattanzi and Vassilvitskii, 2017]--a property not satisfied by previous DP algorithms.

cs.DS↗
arXiv2025

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification

The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

math.NA↗
arXiv2026

On two proofs of $d^2$ mixing of weighted Dikin walks

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

cs.DS↗
arXiv2026

Structural Corrections to the Bethe Approximation of the Permanent

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

cs.DS↗
arXiv2025

A-PINN: Auxiliary Physics-informed Neural Networks for Structural Vibration Analysis in Continuous Euler-Bernoulli Beam

Recent advancements in physics-informed neural networks (PINNs) and their variants have garnered substantial focus from researchers due to their effectiveness in solving both forward and inverse problems governed by differential equations. In this research, a modified Auxiliary physics-informed neural network (A-PINN) framework with balanced adaptive optimizers is proposed for the analysis of structural vibration problems. In order to accurately represent structural systems, it is critical for capturing vibration phenomena and ensuring reliable predictive analysis. So, our investigations are crucial for gaining deeper insight into the robustness of scientific machine learning models for solving vibration problems. Further, to rigorously evaluate the performance of A-PINN, we conducted different numerical simulations to approximate the Euler-Bernoulli beam equations under the various scenarios. The numerical results substantiate the enhanced performance of our model in terms of both numerical stability and predictive accuracy. Our model shows improvement of at least 40% over the baselines.

cs.LG↗
arXiv2026

FirstFit online coloring in the random order model

The average performance of FirstFit online coloring on trees in the random order model is completely determined in recent works of Frei et al. and Bosek et al., showing $Θ(\log n /\log\log n)$ number of colors, improving the $Θ(\log n)$ colors in the adversarial model. We provide a few further results on slightly more general graph classes. Firstly, we extend their method to obtain a simple path-counting principle for sparse graph classes, which immediately yields for example that cactus graphs and uniform hypertrees exhibit a similar improvement. We then show that FirstFit uses only $O(1)$ colors on crown graphs, a standard example where adversarial arrival forces $Θ(n)$ colors. We further show that density alone (even linear minimum degree) is insufficient to guarantee $O(1)$ colors even on bipartite graphs. Finally, we identify graph classes, including unit interval graphs and some graphs of high chromatic number, for which random arrival provides only limited improvement. We end with some open problems.

cs.DS↗
arXiv2026

Two-State Max-Plus Comparison Is Decidable

Daviaud, Guillon, and Merlet proved that comparison of max-plus automata is undecidable under a fixed state bound of 553 and explicitly left the range from 2 to 552 states open. We resolve the two-state endpoint. More strongly, given an arbitrary finite max-plus automaton $A$ and a max-plus automaton $B$ with at most two states, it is decidable whether $[\![A]\!](w)\leq [\![B]\!](w)$ for every word $w$. The structural reason is a one-dimensional projective normal form for two-state dynamics. Outside an effective bounded region, a transition has one of three tail behaviors: it propagates the unbounded projective gap with gap-independent height increment, forgets the gap with gap-independent height increment, or reads the gap magnitude into the height increment and then forgets it. In particular, any transition whose output depends on the unbounded gap necessarily destroys that gap. This yields an exact one-counter realization of $B$. Effective semilinearity of context-free Parikh images then reduces comparison to Presburger arithmetic. As a consequence, two-state max-plus comparison, equivalence, and positivity are decidable.

cs.FL↗
arXiv2026

SurgeGen: A Hybrid Generative Diffusion Framework for Storm Surge Scenario Synthesis

Predicting storm surge induced by landfalling tropical cyclones is crucial for flood mitigation and coastal risk management. Traditionally, physics-based numerical models simulate storm surge by solving the Navier--Stokes equations using numerical methods, but these simulations are computationally expensive. Generative models are promising for storm surge emulation because they can generate diverse realizations rather than producing a single deterministic prediction. However, their use for storm surge emulation remains largely unexplored. In this paper, we leverage diffusion models for storm surge surrogate modeling, combining a baseline prediction stage with conditional generation to provide a more interpretable modeling framework. We develop SurgeGen, a two-stage generative framework for generating storm surge scenarios conditioned on hypothetical storms with parameters defined in a continuous space. First, a baseline model produces a coarse estimate of the storm surge height. This estimate then conditions a diffusion model, which generates refined storm surge scenarios that better capture spatial patterns and variability. We demonstrate that our approach can generate realistic and diverse storm surge scenarios under conditions both within and outside the training distribution.

math.DS↗
arXiv2026

Visualizing Treewidth

A witness drawing of a graph is a visualization that clearly shows a given property of a graph. We study and implement various drawing paradigms for witness drawings to clearly show that graphs have bounded pathwidth or treewidth. Our approach draws the tree decomposition or path decomposition as a tree of bags, with induced subgraphs shown in each bag, and with ''tracks'' for each vertex of the graph connecting its copies in multiple bags. Within bags, we optimize the vertex layout to avoid crossings of edges and tracks. We implement a visualization prototype for crossing minimization using dynamic programming for graphs of small width and heuristic approaches for graphs of larger width. We explore the design space for width-witness drawings and investigate drawing styles that render the subgraph for each bag as an arc diagram with one or two pages or as a circular layout with straight-line edges, and we render tracks either with straight lines or with orbital-radial paths. Finally, we report results from an expert evaluation assessing different witness drawing styles.

cs.CG↗
arXiv2026

Online Matching in Convex Bipartite Graphs

Online resource-allocation systems, like outpatient scheduling and spectrum allocation, often assign sequentially arriving requests to an ordered pool of scarce resources, where each request accepts a contiguous interval of feasible options. We study the resulting online matching problem on convex bipartite graphs under irrevocable decisions and adversarial arrivals. We first show that convexity alone does not improve the classic worst-case guarantee of 1-1/e, achieved by Ranking. We then consider the uniform-length model, in which every online request has exactly d consecutive offline neighbors. We propose Flip, which uses one random bit to commit ex-ante to either earliest-feasible assignment or latest-feasible assignment. Although either natural deterministic policy can waste capacity and be asymptotically only 1/2-competitive, we show that their randomized mixture is 2/3-competitive. This guarantee is tight for Flip and remains valid against a semi-adaptive adversary that observes the selected policy before choosing the arrival order. We also prove that no randomized online algorithm can achieve a competitive ratio strictly larger than 3/4 in the uniform-length model.

cs.DS↗
arXiv2026

Automated Lower Bounds for Bilinear Complexity over Finite Fields

We present a general, automated framework for proving lower bounds on the bilinear complexity (tensor rank) of multiplication problems over a finite field $\mathbb{F}_q$. The framework is parameterized only by the multiplication tensor and by a group of rank-preserving symmetries acting on one argument: it classifies the subspaces of the argument into orbits under the group, runs a dynamic program over the orbits combining four lower-bound techniques, and emits a proof certificate that a verifier rechecks, typically faster than the search. Instantiating the framework for matrix multiplication, we improve the lower bounds for three small formats over $\mathbb{F}_2$, most notably showing that the bilinear complexity of multiplying two $3 \times 3$ matrices over $\mathbb{F}_2$ is at least $20$, raising the bound of $19$ that had stood since Bläser (2003). Instantiating it for polynomial multiplication, we obtain eighteen new lower bounds over $\mathbb{F}_2$ and $\mathbb{F}_3$, for the full product, cyclic convolution, and the truncated (modulo $x^N$) and negacyclic (modulo $x^N+1$) products. Every bound is backed by a machine-checkable certificate.

cs.CC↗
arXiv2026

Fuzzy Pattern Matching in Ordered Structures

The problem of pattern matching, that is, finding all occurrences of a given pattern in a string, is one of the fundamental problems in computer science that has applications in many areas. In this paper, we consider fuzzy patterns, defined as sequences of fuzzy properties over the basic alphabet. We first consider fuzzy pattern matching for sequences of elements of the basic alphabet and then extend the problem to partially ordered sets of nodes labeled by elements of the basic alphabet. For sequences, we seek segments that match the pattern, whereas for partially ordered structures, we seek saturated chains of nodes that match the pattern. The key concept underlying the solutions to these problems is the notion of a trajectory, which generalizes the concept of the prefix function used in the Knuth--Morris--Pratt (KMP) algorithm. A trajectory is processed together with the corresponding data structure, allowing the proposed algorithms to be represented as transition systems whose states are trajectories for sequences and trajectories associated with nodes for partially ordered structures. The trajectory-based approach provides a unified framework for fuzzy pattern matching in various data structures.

cs.DS↗
arXiv2026

Flow Shop Scheduling with Stochastic Reentry

We study flow shop scheduling with stochastic reentry, where jobs must complete multiple passes through the entire shop, and the number of passes that a job requires for completion is drawn from a discrete probability distribution. The goal is to find policies that minimize performance measures in expectation. Our main contribution is a reduction to a stochastic scheduling problem on identical parallel machines augmented by machine arrivals. This reduction preserves objective values and enables the transfer of structural results and performance guarantees from the auxiliary problems to the reentrant flow shop setting. We demonstrate the usefulness of this reduction by proving the optimality of simple priority policies for minimizing the makespan and the total completion time in expectation under geometric and, more generally, monotone hazard rate distributions. For minimizing the total weighted completion time, we derive an approximation guarantee for a simple priority policy that depends only on the squared coefficient of variation of the underlying distributions. Our results constitute the first optimality and approximation guarantees for flow shops with stochastic reentry and demonstrate that established scheduling policies naturally extend to this setting through the proposed reduction.

cs.DS↗
arXiv2026

Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization

Recent advances at the intersection of control theory, neuroscience, and machine learning have revealed novel mechanisms by which dynamical systems perform computation. These advances encompass a wide range of conceptual, mathematical, and computational ideas, with applications for model learning and training, memory retrieval, data-driven control, and optimization. This tutorial focuses on neuro-inspired approaches to computation that aim to improve scalability, robustness, and energy efficiency across such tasks, bridging the gap between artificial and biological systems. Particular emphasis is placed on energy-based dynamical models that encode information through gradient flows and energy landscapes. We begin by reviewing classical formulations, such as continuous-time Hopfield networks and Boltzmann machines, and then extend the framework to modern developments. These include dense associative memory models for high-capacity storage, oscillator-based networks for large-scale optimization, and proximal-descent dynamics for composite and constrained reconstruction. The tutorial demonstrates how control-theoretic principles can guide the design of next-generation neurocomputing systems, steering the discussion beyond conventional feedforward and backpropagation-based approaches to artificial intelligence.

cs.LG↗
arXiv2026

Algebraic Geometry Codes Approach the Half-Singleton Bound with Constant Field Size

We study linear codes for insertion and deletion (insdel) errors through the lens of evaluation codes. We develop a general framework for analyzing random puncturings of evaluation codes, where the edit distance is controlled by only the size of the evaluation domain and the maximum number of zeros of a nonzero function in the underlying function space. Our proof generalizes the results of Con, Guo, Li, and Zhang (ICALP 2025), and simultaneously simplifies their arguments by avoiding an in-depth analysis of longest common subsequences. We demonstrate the applicability of our core theorem by instantiating it with random puncturings of Reed--Muller codes. We then recover the result that random Reed--Solomon codes approach the half-Singleton bound over linear-sized fields while also improving the dependence on the additive gap $\varepsilon$ from $2^{O(1/\varepsilon^2)}$ to $2^{O(1/\varepsilon)}$. Finally, by applying the framework to algebraic geometry codes arising from asymptotically good towers of function fields, we show that there exist randomized families of structured linear codes over constant-sized fields that approach the half-Singleton bound.

cs.IT↗
arXiv2026

The Power of Local Marginals: An $O(\varepsilon^{-1})$-Aspect-Ratio Reduction for Dynamic Weighted Matching

We study dynamic maximum weight matching (MWM) under edge insertions and deletions in two settings: maintaining a $(1\pm\varepsilon)$-approximation to the optimum weight, and maintaining an explicit $(1-\varepsilon)$-approximate matching. Our main result is a reduction that transforms instances of polynomial aspect ratio into instances of aspect ratio $O(\varepsilon^{-1})$. The reduction applies to general graphs in both settings and is compatible with partially dynamic updates. The reduction is based on a structural property of local marginals. After grouping edges into weight classes, the global marginal contribution of one class relative to all lower classes is approximated by its marginal contribution within a local weight window of aspect ratio $O(\varepsilon^{-1})$. Summing these local marginals yields a value composition lemma that uses only approximate optimum values of the local windows. This improves the value reduction of Gupta and Peng (FOCS 2013), whose local aspect ratio is $\varepsilon^{-Θ(\varepsilon^{-1})}$. The same structural property yields an improved matching composition lemma for explicit matchings, reducing the local aspect ratio of Bernstein--Chen--Dudeja--Langley--Sidford--Tu (SODA 2025) from $O(\varepsilon^{-2})$ to $O(\varepsilon^{-1})$.

cs.DS↗