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arXiv2026

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↗

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↗

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

Improved Upper Bounds for Dynamic Bin Packing of General, Unit-Fraction, and Power-Fraction Squares

This paper presents significant upper-bound improvements for dynamic 2D square bin packing, where square items arrive and depart over time and the objective is to minimize the peak number of concurrent active unit bins. In our model, repacking is permitted only within a destination bin upon item arrival; migration between active bins is strictly forbidden. By introducing a streamlined two-list algorithm and proving a tight $5/16$ occupied-area bound for Next-Fit Decreasing Height, we reduce the upper bound on the asymptotic competitive ratio for arbitrary squares from 4.2154 down to 3.918, breaking a longstanding theoretical ceiling. For restricted variants, we establish asymptotic competitive ratios of at most 3.356 for unit-fraction side lengths and 2.211 for power-fraction side lengths.

cs.DS↗

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

Tight bounds on the number of non-equivalent parameterized squares in a word

Two words $x,y$ of the same length are said to be \emph{parameterized equivalent} if there exists a character bijection that transforms $x$ into $y$. A word $w$ is called a parameterized square if $w$ is a concatenation of two parameterized equivalent words. Kociumaka et al. [TCS 2016] showed that in a word of length $n$ that contains $σ$ distinct characters, the number of \emph{parameterized squares} that are non-equivalent with respect to parameterized equivalence is at most $2 σ! n$. In this paper, we show that the maximum number of non-equivalent parameterized squares is less than $σn$, which significantly improves the best-known upper bound by Kociumaka et al. Moreover, we construct a family of words containing $Ω(σn)$ non-equivalent parameterized squares, which demonstrates that the upper bound is asymptotically tight.

cs.DS↗

arXiv2026

Efficient Algorithms to Compute Closed Substrings

A closed string $u$ is either of length one or contains a border that occurs only as a prefix and as a suffix in $u$ and nowhere else within $u$. In this paper, we present fast $\mathcal{O}(n\log n)$ time algorithms to compute all $\mathcal{O}(n^2)$ closed substrings by introducing a compact representation for all closed substrings of a string $ w[1..n]$, using only $\mathcal{O}(n \log n)$ space. These simple and space-efficient algorithms also compute maximal closed strings. Furthermore, we compare the performance of these algorithms and identify classes of strings where each performs best. Finally, we show that the exact number of MCSs ($M(f_n)$) in a Fibonacci word $ f_n $, for $n \geq 5$, is $\approx \left(1 + \frac{1}{ϕ^2}\right) F_n \approx 1.382 F_n$, where $ ϕ$ is the golden ratio.

cs.DS↗

arXiv2026

Kernelization of 2-Club Cluster Edge Deletion on Interval Graphs

The \emph{$s$-Club Cluster Edge Deletion} problem asks whether, given a graph $G$ and an integer $k$, one can delete at most $k$ edges so that every remaining connected component has diameter at most~$s$. This generalizes the classical \emph{Cluster Edge Deletion} problem by permitting components of bounded diameter instead of requiring cliques. On general graphs, $2$-Club Cluster Edge Deletion is known to be fixed-parameter tractable when parameterized by $k$, but it remains open whether it admits a polynomial kernel, as posed in~\cite{ABUKHZAM2023113864}. Motivated by this question, we study the problem on interval graphs and obtain a polynomial vertex kernel of size $\mathcal{O}(k^{5})$. As a complementary result, we also show that the \emph{$s$-Club Cluster Edge Deletion} problem is polynomial time solvable on unit interval graphs. We also show that $2$-Club Cluster Edge Deletion is NP-hard even on split graphs.

cs.DS↗

arXiv2026

Machine Unlearning as Private Retroactive Algorithms

Machine unlearning typically aims to emulate retraining from scratch: upon a deletion request, the unlearning algorithm should produce an outcome that would have been obtained had the deleted point never been included. Recent work has shown that this emulation requirement carries no meaningful privacy semantics against an adversary who observes a sequence of releases. Machine unlearning is thus not a privacy question per se, but rather a data maintenance question, which is precisely the subject of retroactive algorithms. These are algorithms supporting modifications of past operations, guaranteeing that all subsequent answers reflect the revised history as if it had always been in force. We put forward a definition of private retroactive algorithms, combining the retroactivity requirement with differential privacy under continual observation. We present constructions achieving both privacy and retroactivity at no asymptotic cost over privacy alone for linear statistics, clustering, and histograms, alongside impossibility results.

cs.CR↗

arXiv2026

Fine-Grained Visual Preprocessing and Dual-Stream Temporal Modeling for Multimodal Sentiment Analysis on Social Media

Multimodal sentiment analysis often remains text-dominant due to raw-video noise and insufficient temporal modeling. Using CH-SIMS v2.0S, this study proposes three improvements: the NAPS pipeline---a seven-stage system integrating face tracking,identity embedding, and normalized lip-motion analysis to reduce visual noise;DS-TANet, combining an EfficientNetB2 static stream, RAFT optical-flow motion stream, motion-guided attention, and Bi-GRU temporal modeling; and DS-TAFNet, fusing visual and MacBERT-Base textual representations via concatenation fusion. With NAPS, the static visual baseline achieves 80.98\% Macro F1, comparable to the text baseline of 80.55\%; DS-TANet improves visual Macro F1 to 82.58\%;and DS-TAFNet achieves 87.49\% accuracy and 87.48\% Macro F1. These results demonstrate that improving visual input quality and temporal representation is more effective than increasing fusion complexity under limited-data conditions.

cs.CV↗

arXiv2026

Twelve Simple Algorithms to Compute Fibonacci Numbers

The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known, and the algorithms to compute them are simple enough that they are often used in introductory algorithms courses. In this paper, we present twelve such algorithm together with their time and space complexity analyses. Though very simple, these algorithms illustrate eleven concepts from the algorithms field, ranging from top-down vs. bottom-up dynamic programming to recursion depth, and we say which algorithms illustrate which concept. We also present the results of a small-scale experimental comparison of their runtimes on a personal laptop, where the slowest algorithm takes about four orders of magnitude longer than the fastest. Finally, we provide a list of homework questions for students. We hope that this paper can serve as a useful resource for students learning the basics of algorithms.

cs.DS↗

arXiv2026

Gate-Efficient Implementation of the Query-Optimal Time-Dependent Hamiltonian Simulation

The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

quant-ph↗

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

Batched Pandora's Box

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

cs.DS↗

arXiv2026

Don't Be Afraid to Die: Black Hole Search in Dynamic Graphs with Fewer Agents

We consider a team of synchronous mobile agents operating in a port-labeled network. There is one node in the network, called a black hole, that permanently destroys any agent that visits the node. The team of agents must safely locate the black hole, i.e., at least one agent must survive, terminate its algorithm at a node adjacent to the black hole, and output the port number that leads to the black hole from its current position. In the setting where the network is a 1-bounded 1-interval connected dynamic graph, Kaur et al. (SSS 2025) showed that a team consisting of $2δ_{BH}+17$ agents is sufficient to solve the task from a scattered configuration, where $δ_{BH}$ denotes the degree of the black hole node. We show that $2δ_{BH}+3$ agents are sufficient, nearly matching the $2δ_{BH}+1$ lower bound provided in Kaur et al. (ICDCN 2025).

cs.DC↗

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

Two-State Max-Plus Comparison Is Decidable

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

cs.FL↗

arXiv2026

Generalized Graph Search Trees

Graph search algorithms and their corresponding graph search trees are commonly used in algorithmic graph theory. In recent years, the recognition problem of these graph search trees has received significant attention. So far, the research has focused on two types of search trees: first-in trees that behave like BFS-trees and last-in trees that behave like DFS-trees. The search tree paradigms differ from each other by the parent a vertex is connected to. In first-in trees, it is the first visited neighbor, while in last-in trees it is the last neighbor visited before that vertex. Here, we will generalize these concepts of graph search trees by allowing every preceding neighbor of a vertex to be the parent. We study the complexity of the recognition problem of these generalized graph search trees. We present NP-completeness proofs for most searches. We also show that the problem is trivial for Generic Search and polynomial-time solvable for several searches on bipartite graphs and chordal graphs. We also study the question how fixing the start vertex influences the complexity of the problem.

cs.DM↗