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arXiv2026

Networked Admissibility-Preserving Control for Directed Safe Coordination

This paper addresses safety-critical coordination for scalar agents whose distributed commands are implemented through constrained physical-input dynamics. Agents communicate over a fixed weighted digraph with a directed spanning tree, while their outputs must remain inside a common moving safety corridor and their realized inputs must satisfy heterogeneous asymmetric bounds. We propose a networked Admissibility-Preserving Control (APC) architecture in which an Admissibility-Preserving Input Realization (APIR) governs physical inputs and a logarithmic barrier coordinate represents the safety corridor. The synthesis yields an exact cascade in which exponentially decaying realization errors drive nonsymmetric consensus dynamics. For every compatible compact initial set, the closed-loop system admits a unique complete solution, renders the moving corridor and actuator intervals forward invariant with uniform margins, keeps commands bounded, and achieves exponential consensus. We derive direction-specific sufficient conditions under which positive and negative control demands remain within their corresponding actuator limits. The analysis yields a closed-form barrier-coordinate limit determined by the left Perron vector and initial APIR mismatch. Under strong connectivity and the stated gain and compatibility conditions, partial pinning propagates a constant barrier reference from a nonempty informed subset and assigns the induced safety corridor trajectory. A non-weight-balanced example illustrates the directional certificate and predicted collective motion.

eess.SY↗
arXiv2026

Near-Feasible Stable Matchings: Incentives and Optimality

Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where agent capacities are slightly modified to ensure the existence of desirable outcomes. While useful when no stable matching exists, or some agents are left unmatched, it has not previously been investigated whether near-feasible stable matchings satisfy desirable properties with regard to their stability in the original instance. Furthermore, prior works leave open deviation incentive issues that arise when the centralised authority modifies agents' capacities. We consider these issues in the Stable Fixtures problem model, which generalises many classical models through non-bipartite preferences and capacitated agents. We develop a formal framework to analyse and quantify agent incentives to adhere to computed matchings. Then, we embed near-feasible stable matchings in this framework and study the trade-offs between instability, capacity modifications, and computational complexity. We prove that capacity modifications can be simultaneously optimal at individual and aggregate levels, and provide efficient algorithms to compute them. We show that different modification strategies significantly affect stability, and establish that minimal modifications and minimal deviation incentives are compatible and efficiently computable under general conditions. Finally, we provide exact algorithms and experimental results for tractable and intractable versions of these problems.

cs.GT↗
arXiv2026

Tight Bounds for Memory Allocation With and Without Request Fragmentation

The classical memory-allocation problem captures the task of placing objects of different sizes in memory, while minimizing the so-called memory high-water mark. It has been known since the early 1970s that the optimal competitive ratio for any deterministic online allocator is $Θ(\log M)$, where $M$ is the volume high-water mark of the underlying request sequence. This paper begins with a simple observation: many real-world allocators seem to bypass the 1971 lower bound by adopting a slightly different model for memory allocation. These allocators use what we call $k$-aggregate request fragmentation, meaning that the memory allocator is permitted to break requests into multiple fragments, so long as the all-time maximum number of simultaneous fragments is at most $k$ times the all-time maximum number of simultaneous requests. We consider the following basic question: Does request fragmentation fundamentally change the problem of memory allocation, and if so, how? Our results come with several surprises. Among these, we find that even using $k = 1 + o(1)$ request fragmentation, the optimal competitive ratio---which was $Θ(\log M)$ in the classical setting---collapses to $Θ(\log \log M)$. This result is shown to be tight with matching upper and lower bounds, applying to both deterministic and randomized algorithms.

cs.DS↗
arXiv2026

High-dimensional Linear Bandits with Knapsacks

We investigate the contextual bandits with knapsack (CBwK) problem in a high-dimensional linear setting, where the feature dimension can be very large. Our goal is to harness sparsity to obtain sharper regret guarantees. To this end, we first develop an online variant of the hard thresholding algorithm that performs the sparse estimation in an online manner. We then embed this estimator in a primal-dual scheme: every knapsack constraint is paired with a dual variable, which is updated by an online learning rule to keep the cumulative resource consumption within budget. This integrated approach achieves a two-phase sub-linear regret that scales only logarithmically with the feature dimension, improving on the polynomial dependency reported in prior work. Furthermore, we show that either of the following structural assumptions is sufficient for a sharper regret bound of $\tilde{O}(s_{0} \sqrt{T})$: (i) a diverse-covariate condition; and (ii) a margin condition. When both conditions hold simultaneously, we can further control the regret to $O(s_{0}^{2} \log(dT)\log T)$ by a dual resolving scheme. As a by-product, applying our framework to high-dimensional contextual bandits without knapsack constraints recovers the optimal regret rates in both the data-poor and data-rich regimes. Finally, numerical experiments confirm the empirical efficiency of our algorithms in high-dimensional settings.

cs.LG↗
arXiv2026

Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minima

An important quantity in the theory of gradient descent (GD) is the \emph{sharpness}, defined as the largest eigenvalue of the objective Hessian. Classical analyses typically require the step size to be uniformly smaller than twice the reciprocal of the sharpness, but this condition is frequently violated in the training of deep neural networks. Recent work bridges this gap in the setting of overparametrised least-squares with a \emph{single scalar output}, providing a normal form for large-step GD in a neighbourhood of an \emph{isolated} flat minimum and establishing three corresponding convergence results. In this paper, we extend this theory in two directions: (1) to overparametrised least-squares with \emph{vector-valued outputs} (including regression with arbitrarily many observations), and (2) to a neighbourhood of a \emph{manifold} of flat minima (which we show is essential for applications such as matrix factorisation). We generalise both the normal form and all three convergence theorems of \cite{macdonaldeos} to this broader setting, overcoming several technical challenges. We further show that our framework applies to deep matrix factorisation under mild assumptions, yielding several new structural results. In particular, we prove that the set of flat minima forms a fibre bundle over a product of spheres, and that the sharpness is Morse-Bott along this manifold.

cs.LG↗
arXiv2026

Structure-Preserving Data-Driven Identification of Port-Hamiltonian Differential-Algebraic Systems

We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.

math.NA↗
arXiv2026

Optimality of Random Regular Graphs in Sparse Network Designs

The problems of designing sparse networks arise frequently in resource allocation and operations research. In production systems, for example, sparse process flexibility designs are used to handle uncertain demand effectively: the goal is to construct the sparsest bipartite graph between supply and demand that still achieves an expected fulfilled demand comparable to that of a fully flexible system. In middle-mile transportation, sparse delivery-route subgraphs that sustain large matchings after random node deletions help reduce delivery costs; here, the goal is to design the sparsest graph whose maximum matching size remains comparable to that of the fully connected graph under node deletions. The design of sparse networks has been studied extensively, with state-of-the-art results providing order-wise optimal designs for both bipartite and unipartite networks (Chen et al., 2015; Feng et al., 2024). However, identifying designs that achieve the sharp theoretical limit -- where the average degree asymptotically matches the lower bound of any graph to achieve a given loss level, has remained open. In this paper, we prove that the random regular graph achieves this sharp optimal condition in both bipartite and unipartite settings. Numerical experiments further validate this optimality. Our results highlight a practical guideline for sparse flexibility networks: designs that combine degree regularity with dispersed edge placement can achieve optimal performance under uncertainty.

cs.DS↗
arXiv2026

The Popular Dimension of Matchings

We study popular matchings in three classical settings: the house allocation problem, the marriage problem, and the roommates problem. In the popular matching problem, (a subset of) the vertices in a graph have preference orderings over their potential matches. A matching is popular if it gets a plurality of votes in a pairwise election against any other matching. Unfortunately, popular matchings typically do not exist. So we study a natural relaxation, namely popular winning sets which are a set of matchings that collectively get a plurality of votes in a pairwise election against any other matching. The $\textit{popular dimension}$ is the minimum cardinality of a popular winning set, in the worst case over the problem class. We prove that the popular dimension is exactly $2$ in the house allocation problem, even if the voters are weighted and ties are allowed in their preference lists. For the marriage problem and the roommates problem, we prove that the popular dimension is between $2$ and $3$, when the agents are weighted and/or their preferences orderings allow ties. In the special case where the agents are unweighted and have strict preference orderings, the popular dimension of the marriage problem is known to be exactly $1$ and we prove the popular dimension of the roommates problem is exactly $2$.

cs.GT↗
arXiv2026

Quadratization of Autonomous Partial Differential Equations: Algorithmic Solutions

Quadratization for partial differential equations (PDEs) is a process that formally transforms a PDE with a nonquadratic right-hand side into a quadratic form by introducing auxiliary variables. Even though the existence and uniqueness of the solution of this quadratic form are, as of yet, unknown in the general case, this symbolic transformation has been used in diverse fields to simplify the analysis, simulation, and control of PDE models. This paper presents a rigorous definition of PDE quadratization, a sample case study on the solutions of quadratic representations, and theoretical contributions for the PDE quadratization problem of spatially one-dimensional PDEs, including results on existence and complexity. Its main focus, however, is introducing and analyzing QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE. This algorithm is the first computational tool to find quadratizations for PDEs to date. We demonstrate QuPDE's performance by applying it to fourteen nonquadratic PDEs in diverse areas such as fluid mechanics, space physics, chemical engineering, and biological processes. QuPDE delivers a low-order quadratization in each case, uncovering quadratic transformations with fewer auxiliary variables than those previously discovered in the literature for some examples, and finding quadratizations for systems that had not been transformed to quadratic form before.

cs.SC↗
arXiv2026

Scalable Neighborhood Local Search for Single-Machine Scheduling with Family Setup Times

In this work, we study the task of scheduling jobs on a single machine with sequence dependent family setup times under the goal of minimizing the makespan, that is, the completion time of the last job in the schedule. This notoriously NP-hard problem is highly relevant in practical productions and requires heuristics that provide good solutions quickly in order to deal with large instances. In this paper, we present a heuristic based on the approach of parameterized local search. That is, we aim to replace a given solution by a better solution having distance at most $k$ in a pre-defined distance measure. This is done multiple times in a hill-climbing manner, until a locally optimal solution is reached. We analyze the trade-off between the allowed distance $k$ and the algorithm's running time for four natural distance measures. Example of allowed operations for our considered distance measures are: swapping $k$ pairs of jobs in the sequence, or rearranging $k$ consecutive jobs. For two distance measures, we show that finding an improvement for given $k$ can be done in $f(k) \cdot n^{\mathcal{O}(1)}$ time, while such a running time for the other two distance measures is unlikely. We provide a preliminary experimental evaluation of our local search approaches.

cs.DS↗
arXiv2026

Fast Evaluation of Polynomials with Rational Preprocessing

Horner's rule evaluates a monic degree-$n$ polynomial using $n-1$ multiplications. We show that with rational preprocessing of the coefficients, any such polynomial can be evaluated using only $\lfloor n/2 \rfloor + 1$ multiplications over fields of characteristic zero or of characteristic $p>n$. This resolves the multiplication side of a conjecture of Rabin and Winograd (Comm. Pure Appl. Math 1972), who achieved $n/2 + 2\lceil\log_2 n\rceil$ multiplications and conjectured the logarithmic overhead was necessary. We show that this multiplication count can't be beaten in general, proving that three multiplications do not suffice for degree~$6$. This strengthens the lower bound of Pan (STOC 1978), who proved a tight bound for general, complex preprocessing. In characteristic~2, for every $n>1$ and every finite field of size at least $2n$, we prove that an $n$-multiplication chain cannot parametrize all value vectors at $2n$ distinct evaluation points, even with arbitrary preprocessing. We give $\lfloor n/2 \rfloor + 1$ multiplication schedules over characteristic~2, each with an explicit inverse, for every odd degree $n\le 25$ and conjecture that this is possible for all $n$. We also give an injective polynomial construction for universal hashing that uses $N$ multiplications to hash $2N$ values with a single random key. This improves the best previous construction by Daniel J. Bernstein (cryp.to).

cs.DS↗
arXiv2026

Strategic Facility Location in Euclidean Spaces

The strategic facility location problem is defined as follows: $n$ agents report their location in a metric space, and the objective is to design a \emph{mechanism} deciding the (possibly randomized) location of a facility such that agents have no incentive to lie about their position. We focus on the egalitarian cost, which means that the goal of the mechanism is to minimize the expected maximal facility-agent distance. Meanwhile, mechanisms must be \emph{truthful} (or \emph{strategyproof}): no agent may decrease their expected distance to the facility via lying on their location. Designing truthful mechanisms minimizing the approximation ratio is a well-studied problem, and the optimal solution is known for the real line. We focus in this paper on higher dimension Euclidean spaces, for which gaps remain between the best known lower and upper bounds. We first show that, maybe counter-intuitively, the problem is easier for two agents on the plane rather than on the line: the mechanism can exploit the additional dimension to prevent more efficiently agent lies. Based on this intuition, we devise lower bounds for $\mathbb R^d$ asymptotically matching the best known approximation factor of $2$ for large $d$. We also provide novel mechanism ideas, improving over the best known algorithms on the plane, and when the agents belong to $\mathbb R^d$ but the facility may use an additional dimension.

cs.GT↗
arXiv2026

Socially Fair Clustering: Parameterized Approximation and Local Search

We study the Socially Fair Clustering problem introduced by Abbasi, Bhaskara, and Venkatasubramanian (2021) and Ghadiri, Samadi, and Vempala (2021), along with its extension, the $(p,q)$-Socially Fair Clustering problem. This problem generalizes $k$-medians and $k$-means to settings where data points are partitioned into $\ell$ groups, and the goal is to find a fair clustering that is simultaneously good for all groups. We present several algorithms for this problem. For $\ell_p$-Socially Fair Clustering, we give the first constant-factor FPT-approximation parameterized by the number of groups $\ell$, resolving the open question raised by Ghadiri, Singh, and Vempala (2022). Our main ingredient is a new algorithm for closing additional centers in parameterized time inspired by local search. We then turn to the more general $(p,q)$-Socially Fair Clustering problem. The known algorithm for this problem, proposed by Chlamtáč, Makarychev, and Vakilian (2022) achieves a very good approximation but is complex, slow and difficult to implement. We analyze the performance of a simple local search algorithm and show that it provides an $O(q)$ approximation in the worst case. Finally, we design approximation algorithms for the facility location variant of the problem, where the number of facilities (centers) is not fixed in advance, and opening each facility incurs an opening cost. Unlike in previous work, we do not assume these opening costs are the same for all groups.

cs.DS↗
arXiv2026

A Tight Bound for Facial Distance Patterns in Planar Graphs

Let $G$ be an undirected unweighted planar graph and let $S=(s_0,\dots,s_{k-1})$ be the vertices of a designated face, listed in cyclic order. Consider a vector that stores the distances from an arbitrary vertex $v$ to all vertices of $S$. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values in this vector. Li and Parter [STOC'19] proved an upper bound of $O(k^3)$ on the number of unique patterns over all vertices of $G$. We improve this to $O(k^2)$, matching a known lower bound and settling a conjecture in [ISAAC'22]. The simple proof was found by OpenAI's GPT 5.6-Sol model. Plugging this new bound into known results has the following three immediate implications for undirected unweighted planar graphs: (1) it gives an improved compression of the Okamura-Seymour metric (2) it improves the space required by constant-time exact distance oracles, and (3) it improves the fastest distributed algorithm for computing the diameter. We further present a previously unknown and nontrivial implication: a (centralized) $\tilde{O}(n^{8/5})$-time algorithm for computing the diameter, improving over the $\tilde{O}(n^{5/3})$ algorithm of [SODA'18] which works for weighted directed planar graphs. Thus, there is currently a gap between the time for computing the diameter between weighted and unweighted planar graphs.

cs.DS↗
arXiv2026

Random Schreier graphs as expanders

Expander graphs, due to their mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph (e.g., the union of $d$ random bijections for a bipartite graph of degree $d$). This construction is much simpler than all known explicit constructions of expanders and gives graphs with good mixing properties (small second largest eigenvalue) with high probability. However, from the practical viewpoint, it uses too many random bits. The natural idea is to restrict the class of the bijections that we use. For example, if both sides are vector spaces $\mathbb{F}_q^k$ over a finite field $\mathbb{F}_q$, we may consider only \emph{linear} bijections, making the number of random bits polynomial in $k$ (and not $q^k$). In this paper we provide some experimental data that shows that this approach conserves the mixing properties (the second eigenvalue) for several types of graphs (undirected regular and biregular bipartite graphs). We also prove some upper bounds for the second eigenvalue (though they are quite weak compared with the experimental results) and a lower bound for a special class of graphs. Finally, we discuss the possibility to decrease the number of random bits further by using Toeplitz matrices; our experiments show that this change makes the mixing properties only marginally worse while the number of random bits decreases significantly.

math.CO↗
arXiv2026

Beating Quadratic Time--Message Trade-off in Distributed Minimum Spanning Tree Construction

We present a new distributed algorithm for computing a minimum spanning tree (MST) in the \textsf{CONGEST-KT$_{1}$} model, where messages are limited to $O(\log n)$ bits and each vertex initially knows the identifiers of its neighbors. Our algorithm exposes a two-parameter time--message trade-off: for any $0 \leq λ\leq κ\leq 1/2$, it runs in $\tilde{O}(n^λD_G + n^{1 - κ- λ} + n^{1 - 2κ+ λ} + n^{1/2})$ rounds and uses $\tilde{O}(\min\{m, n^{1 + κ}\})$ messages, where $n$, $m$, and $D_G$ are the number of vertices, edges, and thenetwork diameter, respectively. In particular, setting $(κ, λ) = (1/3, 1/6)$ yields an MST algorithm running in $\tilde{O}(n^{1/2} + n^{1/6}D_G)$ rounds with only $\tilde{O}(n^{4/3})$ messages. Under the mild assumption $D_G = O(n^{1/3})$, this is round-optimal while improving the best known message bound of $\tilde{O}(n^{3/2})$. More broadly, our algorithm breaks the quadratic time--message trade-off barrier $\mathrm{\# rounds} \cdot \mathrm{\# messages} = \tildeΩ(n^2)$, which no previous MST algorithm in the \textsf{CONGEST-KT$_{1}$} model has been able to overcome, and it does so for almost the entire range of the diameter $D_G$. As a byproduct, we also obtain new low-message broadcast, spanning-tree, and leader-election algorithms.

cs.DS↗
arXiv2026

Infrequent Resolving Algorithm for Online Linear Programming

Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications such as online auctions, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithms only require first-order computations but induce a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only $O(\log\log T)$ times over the time horizon $T$. Moreover, when we are allowed to solve LPs only $M$ times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly $O\left(T^{(1/2)^{M-1}}\right)$ regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms.

cs.DS↗
arXiv2026

Finding Shortest Reconfiguration Sequences on Independent Set Polytopes

We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the $1$-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in $\mathbb{R}^n$ described by $O(n)$ linear inequalities is hard to approximate within a factor of $(1-\varepsilon)\ln n$ for any constant $ε>0$, unless $P=NP$. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.

cs.DS↗