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cs.DS: explore 187 source-linked works published from 2022 to 2026, with original documents and citations.

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Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

Breaking the Exponential Barrier: The First Polynomial-Time Algorithm for the Győri-Lovász Theorem

We give the first polynomial-time algorithm, after half a century, for the celebrated Győri-Lovász theorem, which resolved a conjecture of Frank (1975). The theorem, one of the simplest existential theorems to explain, states that every $k$-connected graph can be partitioned into $k$ disjoint connected subgraphs of arbitrary prescribed positive sizes. This is a fundamental structural result with broad applications, such as flexible allocation of connected subnetworks of prescribed sizes in sufficiently connected cloud infrastructures. While Lovász (1977) gave a highly non-constructive proof for a stronger directed version using algebraic topology, Győri's original constructive proof (1976) requires exponential time. Despite more than 50 years of effort, no polynomial-time algorithm was known even for $k>4$. Determining the computational complexity of the Győri-Lovász theorem---whether it admits even a sub-exponential-time algorithm or is computationally hard (in particular, PLS-complete or PPAD)---has remained one of the central open problems in algorithmic graph theory. In this paper, we finally resolve this long-standing problem by a fundamentally new proof of the existential theorem via introducing the novel concept of \emph{flow-essential assignment}, which genuinely marries matching and cut structures and yields the first polynomial-time constructive algorithm for the Győri-Lovász theorem. In fact, we obtain a polynomial-time algorithm for Lovász's stronger directed version, whose proof was non-constructive even for DAGs; for DAGs, we further obtain a near-linear-time algorithm. We also develop polynomial-time algorithms for weighted generalizations where the seminal work of Chen, Kleinberg, Lovász, Rajaraman, Sundaram, and Vetta (JACM'07) on confluent flows established only existential non-constructive results.

cs.DS↗

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↗

Diversity-Fair Online Selection

Online selection problems arise in applications such as crowdsourcing and recruitment, where decision makers may seek representation across multiple, potentially overlapping demographic or skill dimensions. We study diversity-fair online selection under adversarial arrivals. A recruiter must immediately and irrevocably decide whether to accept each candidate while selecting at most \(K\) candidates. Before arrivals begin, the recruiter observes aggregate marginal information: the total number of candidates contributing to each of the \(d\) diversity dimensions. When the candidate pool is large, this information may be estimated from demographic statistics of the applicant population. We evaluate the expected utilities across dimensions using the generalized mean \(M_p=(d^{-1}\sum_{k=1}^d U_k^p)^{1/p}, -\infty\le p\le 1,\) where \(U_k\) denotes the expected utility of dimension \(k\). We first study max-min fairness, corresponding to \(p=-\infty\). We prove that no online policy can achieve a competitive ratio better than \(O(1/\sqrt d)\) and develop a policy with a competitive ratio \(1/[4(2+\sqrt2)\sqrt d]\), establishing the optimal dependence on \(d\) up to a constant factor. Without exact marginal information, the optimal worst-case rate falls to \(Θ(1/d)\), demonstrating the value of this information. We also extend the max-min analysis to nonbinary attributes and characterize the optimal dependence on their value range. Finally, we study generalized-mean objectives. For \(0\le p\le1\), we establish an optimal competitive ratio of \(Θ(1/\log d)\). For each fixed finite negative mean \(p=-q\), where \(q>0\), our policy achieves \(d^{-q/(2q+1)}\) up to polylogarithmic factors, matching the exponent of the corresponding impossibility bound.

econ.TH↗

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↗

Sublinear Edge Fault-Tolerant Hyperspanners for Hypergraphs

In this paper, we initiate the study on fault-tolerant (FT) graph spanners for hypergraphs and show the generalization to hypergraphs in the FT setting is non-trivial. An FT spanner approximates shortest distances under network failures, widely used in applications such as routing and distributed computing. We first provide a systematic study on extending spanners to hyperspanners in both non-faulty and FT settings and reveal that the latter case is more interesting: simple methods can only produce a linear size in the number of allowed faults $f$, while all known optimal sizes of FT graph spanners are sublinear in $f$. Inspired by the FT clustering technique in Parter's paper \cite{partervft}, we propose a hypergraph clustering based algorithm with an improved sublinear size bound. Specifically, for an $n$-node $m$-edge hypergraph with rank $r$ and a stretch parameter $k$, our algorithm constructs edge FT (EFT) hyperspanners of stretch $2k-1$ and size $O(k(k+r)f^{1-1/(rk)}n^{1+1/k}\log n)$ with high probability in time $\widetilde{O}(mr^3+nrf)$ ($\widetilde{O}$ hides polylogarithmic factors). We also establish size lower bounds, $Ω((f/r)^{r-1-1/k+o(1)}n^{1+1/k-o(1)})$ for vertex FT (VFT) hyperspanners and $Ω(f^{1-1/r-1/(rk)+o(1)}n^{1+1/k-o(1)}+fn)$ for EFT hyperspanners, leaving a gap of $k(k+r)f^{1/r}$ yet to close. We believe that this work will spark interest in developing optimal-sized FT hyperspanners for hypergraphs.

cs.DS↗

Sensitivity and Size Relationships of the Lempel-Ziv Factorization

The Lempel-Ziv (LZ) factorization is a fundamental method for compressing repetitive strings. Its multiplicative sensitivity to an operation measures how much that operation can increase the number of phrases, as the maximum ratio of the phrase counts after and before the operation. While its sensitivity to single-character edits is known, its sensitivity to more global operations such as prefix deletion and string reversal remains unknown. Further open questions concern its size relationships with other compressed representations and whether it can be converted into an LZ-like encoding of polylogarithmic height without asymptotically increasing its size. In this paper, We determine the multiplicative sensitivity of the LZ factorization to prefix deletion, cyclic rotation, and string reversal by establishing an $Ω(\log n)$ lower bound for each operation that matches the known upper bound, where $n$ is the input length. Using the result for prefix deletion, we show that the combined size of the relative LZ factorization and the LZ encoding of its reference string can be asymptotically smaller than that of the LZ factorization. We also obtain asymptotically tight worst-case ratios of $Θ(\log n)$ for the size of the LZ factorization divided by the minimum size of a collage system and for the size of the lex-parse divided by that of the LZ factorization. We give lower and upper bounds on the worst-case ratio of the minimum size of an LZ-like encoding of height at most $h$ to the size of the LZ factorization, with matching bounds for $h \geq \lg n$. In particular, we show that the minimum size of an LZ-like encoding with height at most $h \in O({\rm poly}\,\log n)$ can be larger than the size of the LZ factorization by a factor of $Ω(\log n / \log\log n)$. All our lower bounds follow from a common string construction based on the bit-reversal permutation.

cs.DS↗

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↗

Adversarial Online Classification with a Preview

Worst-case online classification is governed by sequential complexity, such as Littlestone dimension, and can be impossible even for statistically simple classes, such as thresholds of VC dimension one. We study a preview model in which an oblivious adversary fixes an entire labeled sequence of length $T$, a uniformly random subset of size $pT$ is revealed before prediction begins, and the remaining $(1-p)T$ examples are then presented in their original adversarial order. Against the best full-sequence hypothesis evaluated on the unrevealed examples, we characterize the dependence on the preview rate $p$: for binary classes of VC dimension $d$, the optimal excess loss is $Θ(d/p+\sqrt{dT})$, up to the trivial cap at $T$; for multiclass classes we obtain the corresponding $\widetilde O(d_{\rm DS}/p+\sqrt{d_{\rm Nat}T})$ bound with no dependence on the number of labels. Thus a random preview can replace worst-case sequential complexity by classical statistical dimensions without randomizing the online order. To achieve the sharp binary bound, our ChainedPrediction algorithm uses an online analogue of chaining, implemented as a multiscale aggregation algorithm rather than only as an analytic argument.

cs.LG↗

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↗

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↗

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↗

Spectral and combinatorial methods for efficiently computing the rank of unambiguous finite automata

A zero-one matrix is a matrix with entries from $\{0, 1\}$. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite automaton, an important generalisation of deterministic finite automata which shares many of their good properties. Let $\mathcal{A}$ be a finite set of $n \times n$ zero-one matrices generating a monoid of zero-one matrices, and $m$ be the cardinality of $\mathcal{A}$. We study the computational complexity of computing the minimum rank of a matrix in the monoid generated by $\mathcal{A}$. By using linear-algebraic techniques, we show that this problem is in $\textsf{NC}$ and can be solved in $\mathcal{O}(mn^4)$ time and $\mathcal{O}(n^2)$ space. We also provide a combinatorial algorithm finding a matrix of minimum rank in $\mathcal{O}(mn^4)$ time and $\mathcal{O}(n^3)$ space. As a byproduct, we show a very weak version of a generalisation of the Černý conjecture: there always exists a straight line program of size $\mathcal{O}(n^2)$ describing a product resulting in a matrix of minimum rank. For the special case corresponding to total DFAs (that is, for the case where all matrices have exactly one 1 in each row), the minimum rank is the size of the smallest image of the set of all states under the action of a word. Our combinatorial algorithm finds a matrix of minimum rank in time $\mathcal{O}(n^3 + mn^2)$ in this case.

cs.FL↗

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↗

Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams

Let $S$ be a set of $n$ points in $\mathbb{R}^2$. Our goal is to preprocess $S$ to efficiently compute the smallest enclosing disk of the points in $S$ that lie inside an axis-aligned query rectangle. Previous data structures for this problem achieve a query time of $O(\log^6 n)$ with $O(n \log^2 n)$ preprocessing time and space by lifting the points to 3D, dualizing them into polyhedra, and searching through their intersections. We present a significantly simpler approach, solely based on 2D geometric structures, specifically 2D farthest-point Voronoi diagrams. Our approach achieves a deterministic query time of $O(\log^4 n)$ and, via randomization, an expected query time of $O(\log^{5/2} n \log\log n)$ with the same preprocessing bounds.

cs.CG↗

A Deterministic Constant-Competitive Algorithm for Dynamic Mixture-of-Experts Serving

Dynamic Mixture-of-Experts Serving allocates k replica GPUs among m experts as workloads change. At each round, the online algorithm sees the current workload, chooses integral replica counts, and pays bottleneck service cost plus replica movement. It does not know future workloads. Huang, Lou, and Xiao gave an O(sqrt(log k))-competitive randomized algorithm for this problem. We prove a deterministic O(1)-competitive algorithm. For every number of experts and every k>=1, the algorithm satisfies ALG_det <= 10 C_PB OPT + (5 C_PB + 8) k + 16, where C_PB is the absolute constant from Chasing Positive Bodies at resource augmentation one and covering sparsity two. Consequently, CR_det(k)<=10 C_PB for every k>=1, so CR_det(k)=Theta(1). The multiplicative factor does not depend on the number of experts, replica budget, horizon, or workload values. Thus randomization is not needed for the asymptotic guarantee. The proof has two layers. A finite tangent envelope, summable positive resets, and a nonexpansive balanced projection reduce reciprocal-max service costs to a deterministic exact-budget fractional path. A new deterministic rounding theorem converts every such path to integral allocations with service distortion three and movement bounded by the fractional movement plus 6k. The complete reduction, rounding theorem, causal composition, and quantified main theorem are machine-checked in Lean 4 relative to the positive-body result as the sole scientific source premise. The theorem concerns the allocation model above. It does not include network topology, shared-edge congestion, or routing decisions.

cs.DS↗

Hardness of Approximation of Rank Aggregation on Ulam Metric

We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a permutation minimizing the sum of its Ulam distances to the input permutations, while in the \emph{Ulam center} problem the objective is to minimize the maximum such distance. Both problems are known to be NP-hard, but no explicit approximation hardness was previously known. We prove that, for every $\varepsilon>0$, it is NP-hard to approximate either Ulam median or Ulam center within a factor of $51/50-\varepsilon$, even when the input consists of only four permutations. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. The hardness result for Ulam median is established via a reduction from MAX-E3-LIN-2. The corresponding hardness for Ulam center is then obtained through a reduction from Ulam median.

cs.CC↗

A Simplified Analysis of the Good-Bad $3/2$-Approximation Algorithm for Some Minimum-Cost Graph Problems

In this paper, we consider an easy greedy approximation algorithm, the good-bad algorithm, introduced by Couëtoux for finding a minimum-cost set of edges such that every connected component has at least $k$ vertices. Couëtoux proves that the good-bad algorithm achieves a $3/2$-approximation for this problem. Davis and Williamson extend this result to the more general problem of finding a minimum-cost edge set that contains at least one edge from every cut $S\subseteq V$ satisfying $h(S) = 1$ where $h:2^V \rightarrow \{0,1\}$ is downward monotone; that is, $h(S) = 1$ implies $h(T) = 1$ for every nonempty subset $T \subseteq S$. The original problem corresponds to $h(S) =1$ when $|S|<k$. We give a simplified analysis of the good-bad algorithm for downward monotone functions.

cs.DS↗

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↗
Compare source metadata on this page
WorkPublishedSource identifierSource
Breaking the Exponential Barrier: The First Polynomial-Time Algorithm for the Győri-Lovász Theorem2026-08-312608.30945arxiv
Structural Corrections to the Bethe Approximation of the Permanent2026-08-312608.31061arxiv
Diversity-Fair Online Selection2026-08-302504.10389arxiv
On the Parameterized Complexity of $s$-Club Cluster Edge Deletion2026-08-302510.07065arxiv
Sublinear Edge Fault-Tolerant Hyperspanners for Hypergraphs2026-08-302511.22803arxiv
Sensitivity and Size Relationships of the Lempel-Ziv Factorization2026-10-082608.03351arxiv
From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity2026-08-302608.03819arxiv
Adversarial Online Classification with a Preview2026-08-302608.29503arxiv
FirstFit online coloring in the random order model2026-08-302608.29603arxiv
Socially Fair Clustering: Parameterized Approximation and Local Search2026-08-302608.30026arxiv
Constructive Characterization and Recognition Algorithm for Grafts with a Connected Minimum Join2026-08-292510.26975arxiv
Spectral and combinatorial methods for efficiently computing the rank of unambiguous finite automata2026-10-052511.09703arxiv
Automated Lower Bounds for Bilinear Complexity over Finite Fields2026-08-292603.07280arxiv
Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams2026-08-292605.00743arxiv
A Deterministic Constant-Competitive Algorithm for Dynamic Mixture-of-Experts Serving2026-08-292608.16947arxiv
Hardness of Approximation of Rank Aggregation on Ulam Metric2026-08-292608.29180arxiv
A Simplified Analysis of the Good-Bad $3/2$-Approximation Algorithm for Some Minimum-Cost Graph Problems2026-08-292608.29274arxiv
Scheduling to Maximize Weighted Throughput with an Active-Time Budget2026-08-292608.29418arxiv

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