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arXiv2026

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↗

arXiv2026

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↗

arXiv2026

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↗

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

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↗

arXiv2026

GPU-Native Approximate Nearest Neighbor Search with IVF-RaBitQ: Fast Index Build and Search

Approximate nearest neighbor search (ANNS) on GPUs is gaining increasing popularity for modern retrieval and recommendation workloads that operate over massive high-dimensional vectors. Graph-based indexes deliver high recall and throughput but incur heavy build-time and storage costs. In contrast, cluster-based methods build and scale efficiently yet often need many probes for high recall, straining memory bandwidth and compute. Aiming to simultaneously achieve fast index build, high-throughput search, high recall, and low storage requirement for GPUs, we present IVF-RaBitQ (GPU), a GPU-native ANNS solution that integrates the cluster-based method IVF with RaBitQ quantization into an efficient GPU index build/search pipeline. Specifically, for index build, we develop a scalable GPU-native RaBitQ quantization method that enables fast and accurate low-bit encoding at scale. For search, we develop GPU-native distance computation schemes for RaBitQ codes and a fused search kernel to achieve high throughput with high recall. With IVF-RaBitQ implemented and integrated into the NVIDIA cuVS Library, experiments on cuVS Bench across multiple datasets show that IVF-RaBitQ offers a strong performance frontier in recall, throughput, index build time, and storage footprint. For Recall approximately equal 0.95, IVF-RaBitQ achieves 3.0x higher QPS than the state-of-the-art graph-based method CAGRA, while also constructing indices 14.7x faster on average. Compared to the cluster-based method IVF-PQ, IVF-RaBitQ delivers on average over 4.5x higher throughput while avoiding accessing the raw vectors for reranking.

cs.DB↗

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

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

Dynamic Edge Coloring of Forests

In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $Δ$ with at most $Δ+c$ colors, under edge updates. A prominent objective is to minimize the \emph{recourse}, namely the number of edges that are recolored. We study this problem on forests, arguably the simplest graph class that already captures much of the complexity of the problem. We consider both the \emph{incremental} model, where edges are only inserted and the \emph{fully dynamic} model where edges may also be deleted. In the deterministic setting, we focus on the natural greedy algorithm. We show that it achieves $O(\frac{1}{c + \sqrtΔ})$ amortized recourse in the incremental model, and that this is tight up to tie-breaking. In contrast, in a fully dynamic forest, greedy can be forced to have $Ω(\log_Δn)$ amortized recourse. To partially overcome this limitation of greedy within the deterministic setting, we give an optimal non-greedy algorithm with $O(1)$ amortized recourse for \emph{rooted} fully dynamic forests and $c=Δ-2$. In the randomized setting, we give a natural distribution-maintaining algorithm. In the incremental model, it achieves $Θ(\frac{1}Δ)$ expected amortized recourse, and we show that this is optimal for every constant $c$. In the fully dynamic model, the same algorithm achieves $Θ(\min \{ \fracΔ{c}, \log_Δ n \})$ expected recourse for $c > 0$, and $Θ(\log_Δ n)$ for $c = 0$. We show that this is optimal for $c = 0$, and prove an $Ω(1)$ lower bound for every constant $c$.

cs.DS↗

arXiv2026

Optimal Mixing of Glauber Dynamics for the Sherrington-Kirkpatrick Model at $β< 1/2$

We prove that for every fixed inverse temperature $β< 1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_β\left(n \log\left(n / \varepsilon\right)\right)$ steps. The bound holds uniformly over all external fields and is optimal up to constants depending only on $β$. The main ingredient is a deterministic criterion for optimal-order Poincaré inequalities in general Ising models, established via the integrated Bakry-Émery criterion together with a new two-spin estimate. A standard application of the localization-scheme framework of Chen and Eldan then upgrades the Poincaré inequality to a modified log-Sobolev inequality, yielding the optimal mixing-time bound. The main ideas underlying the proof of the Poincaré inequality were generated by GPT-5.6 Sol Ultra.

math.PR↗

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↗

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

An Exposition of the $\widetilde{O}(\log^{1/4} n)$ Bound for the Komlós Problem

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $Ω(\sqrt{\log n})$ should hold.

math.CO↗

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

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

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

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

Quantum Computing: Lecture Notes

This is a set of lecture notes suitable for a Master's course on quantum computation and information from the perspective of theoretical computer science. The first version was written in 2011, with many extensions and improvements in subsequent years. The first 10 chapters cover the circuit model and the main quantum algorithms (Deutsch-Jozsa, Simon, Shor, Hidden Subgroup Problem, Grover, quantum walks, Hamiltonian simulation and HHL). They are followed by 4 chapters about complexity, 4 chapters about distributed ("Alice and Bob") settings, a chapter about quantum machine learning, one about stabilizer states and Clifford circuits, and a final chapter about quantum error correction. Appendices A and B give a brief introduction to the required linear algebra and some other mathematical and computer science background. All chapters come with exercises, with some hints provided in Appendix C.

quant-ph↗