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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↗

arXiv2026

Random attractors and almost-sure stability under discretization of a stochastic autoparametric system

For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

math.DS↗

arXiv2026

Sparse Koopman Autoencoders Identify Local Dynamical Regimes in Multibasin Systems

Koopman autoencoders (KAEs) seek a higher-dimensional latent representation in which nonlinear dynamics evolve linearly. However, many interesting systems have multiple basins of attraction, and both theoretical and empirical work has shown these multibasin systems cannot generally admit a single finite-dimensional global Koopman embedding under standard assumptions. We posit that encoders with a sparsity-inducing objective encouraging few active latent coefficients will provide latent supports as an inspectable basin-modeling principle for Koopman autoencoders. We use these encoders producing sparse latents in training Sparse Koopman Autoencoders (SKAEs) without basin labels or other regime annotations, and treat the learned latent supports as model-produced regime variables after training. Across a range of procedurally generated multibasin systems and chaotic flows, we show that SKAEs have superior forecasting performance compared to dense-latent KAEs. We also perform a mechanistic study that shows latent supports produced by SKAEs are both essential for the quality of the representation and useful for identifying basins on held-out basin interior states, whereas dense-latent KAEs collapse to an uninformative single family. These results identify sparse latents and their corresponding supports as label-free, interpretable regime variables for Koopman learning in nonlinear systems with multiple local dynamical laws.

cs.LG↗

arXiv2026

Constructive Characterization and Recognition Algorithm for Grafts with a Connected Minimum Join

Minimum joins in a graft $(G, T)$, also known as minimum $T$-joins of a graph $G$, are said to be connected if they determine a connected subgraph of $G$. Grafts with a connected minimum join have gained interest ever since Middendorf and Pfeiffer showed that they satisfy Seymour's min-max formula for joins and $T$-cut packings; that is, in such grafts, the size of a minimum join is equal to the size of a maximum packing of $T$-cuts. In this paper, we provide a constructive characterization of grafts with a connected minimum join. We also obtain a polynomial time algorithm that decides whether a given graft has a connected minimum join and, if so, outputs one. Our algorithm has two bottlenecks; one is the time required to compute a minimum join of a graft, and the other is the time required to solve the single-source all-sink shortest path problem in a graph with conservative $\pm 1$-valued edge weights. Thus, our algorithm runs in $O(n(m + n\log n) )$ time. In the nondense case, it improves upon the time bound for this problem due to Sebő and Tannier that was introduced as an application of their results on metrics on graphs.

cs.DM↗

arXiv2026

DAG Covers for Structured Graphs: The Steiner Point Effect

Given a weighted digraph $G$, a $(t,g,μ)$-DAG cover is a collection of $g$ dominating DAGs $D_1,\dots,D_g$ such that all distances are approximately preserved: for every pair $(u,v)$ of vertices, $\min_id_{D_i}(u,v)\le t\cdot d_{G}(u,v)$, and the total number of non-$G$ edges is bounded by $|(\cup_i D_i)\setminus G|\le μ$. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of \emph{Steiner} DAG cover, where the DAGs are allowed to contain Steiner points. We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a $(1,2,\tilde{O}(n\cdot tw))$-Steiner DAG cover. For planar digraphs we provide a $(1+\varepsilon,2,\tilde{O}_\varepsilon(n))$-Steiner DAG cover. We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth $1$ with stretch $t<2$ and sub-quadratic number of extra edges requires $Ω(\log n)$ DAGs.

cs.DS↗

arXiv2026

Scalable Reachability Analysis of Linear Continuous Systems with Property-Driven Time-Step Adaptation

We study safety verification for linear time-invariant systems with bounded inputs in continuous time. The standard approach reduces to a reachability analysis in two steps: first discretize time and then apply a forward analysis in the discretized system. Existing algorithms use either a fixed time step or an adaptive time step that changes based on the approximation error compared to the underlying continuous system. In this paper, we present an efficient reachability algorithm that adapts the time step based on a given safety property. Essentially, our algorithm makes the largest possible time step such that it can still prove safety. For this approach to be scalable in practice, we discuss several optimizations such as avoiding the repeated expensive calculation of the matrix exponential during discretization and a careful balance how we tame the approximation error stemming from the states and the inputs. This allows our algorithm to yield a moderate approximation error even when using a large time step, thus requiring much fewer steps than prior algorithms. We demonstrate the effectiveness and scalability on the large-scale SLICOT benchmark suite, where our algorithm consistently outperforms other state-of-the-art approaches.

eess.SY↗

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

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

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

On the Parameterized Complexity of $s$-Club Cluster Edge Deletion

We study the parameterized and kernelization complexity of the \emph{\textsc{$s$-Club Cluster Edge Deletion}} problem, a distance-bounded generalization of \emph{\textsc{Cluster Edge Deletion}}. Given a graph $G=(V,E)$ and integers $k,s$, the goal is to delete at most $k$ edges so that every resulting connected component has diameter at most $s$. On the structural side, we settle an open question of Montecchiani, Ortali, Piselli, and Tappini (\emph{Theoretical Computer Science}, 2023) by proving W[1]-hardness parameterized by pathwidth plus the maximum number of allowed $s$-clubs, and consequently by treewidth plus this parameter. Thus, the diameter bound $s$ is inecessary for tractability under these parameters. In contrast, we show that dependence on $s$ is unnecessary for several structural parameters: the problem is fixed-parameter tractable when parameterized by treedepth, neighborhood diversity, or cluster vertex deletion number, generalizing known results for $s=1.$ We further prove that no polynomial kernel exists when parameterized by vertex cover, even for $s=2$. On the positive side, we present an FPT bicriteria approximation scheme for graphs excluding long induced cycles, running in time $f(k,1/ε)\cdot n^{\mathcal{O}(1)}$ and producing a solution of size at most $k$ whose components have diameter at most $(1+ε)s$. Finally, we initiate the study of the directed variant, \textsc{$s$-Club Cluster Arc Deletion}, and prove that it is W[1]-hard parameterized by $k$, even on directed acyclic graphs.

cs.DM↗

arXiv2026

Multi-tier Flexible Graph Connectivity

Motivated by non-uniform edge failures in network design, we introduce a multi-tier model of flexible graph connectivity. In k-tier Flexible Graph Connectivity (k-tier FGC), the input is an undirected graph G=(V, E) with non-negative edge costs, along with a classification of the edges into nested tiers T_1 subseteq T_2 subseteq ... subseteq T_k = E and non-negative integral tier requirements q_1 <= q_2 <= ... <= q_k. A non-empty proper subset R of vertices is safe if it is safe along one of the tiers, i.e., there exists i in [k] such that |delta(R) cap T_i| >= q_i. The goal is to find a minimum cost subset F subseteq E of edges such that the subgraph (V, F) has no unsafe cuts. The case of k=1 corresponds to the min-cost p-edge-connected spanning subgraph problem which is APX-hard. We design approximation algorithms for every fixed constant k for three variants of k-tier FGC: (i) for k-tier FGC, we design an LP-based logarithmic approximation, (ii) for min-cardinality k-tier FGC, we design a combinatorial approximation whose factor depends only on the tier requirements q_1 and q_k, and (iii) for k-tier Flexible Multi-Graph Connectivity, where we are allowed to use multiple copies of each edge while paying the cost of the edge for each chosen copy of the edge, we design an LP-based 2-approximation.

cs.DS↗

arXiv2026

Scheduling to Maximize Weighted Throughput with an Active-Time Budget

We study the active-time scheduling problem with weighted throughput maximization. In this setting, a set of $n$ jobs $J$ arrive at integer release times, each with an integer processing time and integer deadline. Jobs may be preempted at integer time slot boundaries. A schedule assigns jobs to time slots, with at most $m$ jobs assigned to the same time slot. A slot is called \emph{active} if at least one job is scheduled in it. Instead of scheduling all jobs to minimize the number of active time slots, we consider the more general variant of \emph{weighted throughput} with an active-time budget $K$, where each job $j\in J$ has a weight $w_j$. The objective is to maximize the total weight of \emph{completed} jobs using at most $K$ active time slots. This means that partially scheduled jobs do not count towards the objective. The classical active-time minimization problem is recovered by asking whether all jobs can be completed within a given active-time budget. We give hardness, approximation, and exact algorithmic results. For general intervals with unbounded parallelism, we prove NP-hardness, rule out an FPTAS unless $\mathrm{P}=\mathrm{NP}$, and give a pseudo-polynomial time $Ω(1/\log K)$-approximation. For proper intervals, we prove a canonical structural lemma and obtain an exact $(nK)^{O(m)}$-time algorithm. For laminar intervals, we give an exact $f(K,m)\cdot n^{O(1)}$-time algorithm.

cs.DS↗

arXiv2026

Parameterized Complexity of Connected Network Microaggregation: The Role of Cluster Size

Network microaggregation is a fundamental technique in statistical disclosure control, where vertices of a graph are partitioned into clusters satisfying size constraints and admitting a center within bounded distance. We study the parameterized complexity of the \emph{unweighted Connected Network Microaggregation} problem, focusing on structural parameters and natural clustering parameters such as the distance bound $d$ and cluster size gap $u-\ell$. We show that, unlike the weighted variant, the unweighted connected problem is fixed-parameter tractable when parameterized by neighborhood diversity, and hence by vertex cover. In contrast, it remains $\mathrm{W[1]}$-hard for more general structural parameters, including vertex deletion to paths, stars, and cliques. These hardness results hold even for every $d\ge 2$ and any fixed gap $u-\ell$, showing that these clustering parameters do not overcome the structural hardness. We further show that adding the cluster size bound $u$ restores tractability for structural parameters such as treewidth and cluster vertex deletion. Moreover, $u$ is essential: the problem remains $\mathrm{W[1]}$-hard when these structural parameters are considered alone. For kernelization, we prove that the problem has no polynomial kernel parameterized by vertex cover unless $\mathrm{coNP}\subseteq\mathrm{NP/poly}$, even when the distance constraint is vacuous. Adding $u$ yields a polynomial kernel for vertex cover, while kernelization remains unlikely for more general structural parameters even when combined with $u$. Finally, we show that the problem is NP-hard on graphs of bounded clique-width.

cs.DS↗

arXiv2026

Sensitivity and Size Relationships of the Lempel-Ziv Factorization

The Lempel-Ziv (LZ) factorization is one of the most fundamental methods for compressing highly repetitive strings, and the number of phrases in its factorization is considered a repetitiveness measure. Sensitivity to an edit operation measures the maximum increase in a repetitiveness measure when the operation is applied to a string. While asymptotically tight bounds are known for the sensitivity of the LZ factorization to single-character edits, whether its multiplicative sensitivity is bounded by a constant has remained open for operations that change a large part of the structure of a string, such as prefix deletion, substring deletion, cyclic rotation, and string reversal. We resolve this question. For each of these four operations, we construct a family of strings in which a string of length $n$ has sensitivity $Ω(\log n)$ to that operation. We also determine the size relationships among the LZ factorization, collage systems and the lex-parse. We construct a family of strings whose LZ factorizations are $Ω(\log n)$ times larger than their minimum collage systems, and a family of strings whose lex-parses are $Ω(\log n)$ times larger than their LZ factorizations. Furthermore, we prove that there exists a family of strings for which every LZ encoding of height $O({\rm poly}\,\log n)$ is $Ω(\log n / \log \log n)$ times larger than the standard LZ factorization. Except for the lower bound on height-bounded LZ encodings, all of these lower bounds are asymptotically tight, matching $O(\log n)$ upper bounds.

cs.DS↗

arXiv2026

Parameterized Complexity of Edge-Constrained Graph Partitioning

We study the Edge-Constrained Graph Partitioning Problem (ECGP), which asks whether the vertices of a graph can be partitioned into r parts, each inducing at least gamma edges. We also consider a balanced variant (BECGP), requiring equal-sized parts, and signed variants, where the utility of a part is the difference between its numbers of positive and negative edges. We show that ECGP and BECGP remain NP-hard for fixed gamma, while BECGP is also NP-hard for fixed r. For the natural parameterization r+gamma, both problems admit polynomial kernels. We obtain FPT algorithms for ECGP and BECGP parameterized by maximum leaf number, vertex deletion distance to a clique, cluster vertex deletion number plus gamma, and vertex integrity. Furthermore, ECGP is FPT parameterized by vertex deletion distance to stars plus gamma and vertex deletion distance to paths plus gamma. On the negative side, ECGP and BECGP are W[1]-hard when parameterized by r together with several structural parameters. In particular, hardness holds for feedback edge set, vertex deletion distance to stars or paths, and modular width even when the corresponding parameter is zero. The problems are also W[1]-hard parameterized by cluster vertex deletion number plus r, and by clique-width even when gamma=3. For signed graphs, both variants are NP-hard even when r+gamma=3 and the input is a disjoint union of two cliques. Finally, the balanced signed variant is W[1]-hard parameterized by treedepth plus r, even when gamma=0.

cs.DS↗

arXiv2026

A Configuration-LP Framework for Connected $k$-Median Clustering

We study the \emph{connected $k$-median} clustering problem, a clustering problem that augments the classical $k$-median objective with connectivity constraints. We focus on the \emph{overlapping} variant of the problem, where clusters are allowed to share vertices. In addition to a metric space $(V,d)$, the input contains a connected graph $G$ on the same vertex set $V$ of size $n$. The goal is to select at most $k$ centers $C$ and assign vertices to them so as to minimize the $k$-median cost (i.e., $\sum_{v\in V} d(v,C)$), subject to the constraint that each cluster induces a connected subgraph of $G$. Since the metric space and the connectivity graph are independent, the problem is significantly more challenging than standard clustering. Eube et al.~\cite{eube2025esa} showed that even the assignment version is $Ω(\log n)$-hard to approximate and gave approximation algorithms with guarantees depending polynomially on $k$. We develop a configuration-LP-based framework that combines covering LP techniques with a rooted minimum-density oracle. For the assignment version, we obtain an $O(\log^2 n)$-approximation. For the general version, we develop a bicriteria framework that opens $O(k\log n)$ centers while achieving an $O(\log^2 n)$-approximation in cost. %Our results provide a different LP-based approach for handling connectivity constraints in clustering problems and demonstrate that configuration LPs, covering LPs, and rooted density oracles can be combined effectively to obtain approximation guarantees for clustering objectives under graph-theoretic constraints.

cs.DS↗

arXiv2026

From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity

A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furmańczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balabán [MFCS 2026].

cs.DM↗