Uniform Capacitated Clustering with Few Extra Centers
Kangke Cheng, Guanlin Mo, Hu Ding
AI 速览：研究在允许开放 k+⌊εk⌋ 个中心且严格满足容量约束下，均匀容量 (k,z)-聚类的近似算法。
In uniform capacitated $(k,z)$-clustering, we open $k$ centers and assign every client to a center so that no center serves more than $U$ clients, minimizing the sum of the $z$-th powers of the assignment distances; $k$-median and $k$-means are the cases $z=1$ and $z=2$. We study algorithms that may open $k+\lfloor\varepsilon k\rfloor$ centers for a fixed $\varepsilon\in(0,1]$ and must respect the capacities exactly. In low dimensions, we give deterministic algorithms that are, except for $k$-median in the plane, the first polynomial-time algorithms with $(1+\varepsilon)k$ centers whose ratio is bounded independently of $\varepsilon$. In Euclidean space of any dimension, with clients of bounded doubling dimension, the ratio is $\mathcal{B}_z-(\mathcal{B}_z-1)\varepsilon+δ$ for every fixed $δ>0$, where $\mathcal{B}_z<3^z$ for $z>1$; this is $3-2\varepsilon+δ$ for $k$-median and $5-4\varepsilon+δ$ for $k$-means. In doubling metrics, the ratio is $3^z-(3^z-1)\varepsilon+δ$ when centers may share a location, and a constant otherwise, such as $\min\{7-4\varepsilon,\,3+2\ln(1/\varepsilon)\}+δ$ for $k$-median. In general metrics, deleting centers greedily gives ratio $3+2\ln(1/\varepsilon)+δ$ for $k$-median when centers may share a location and $7+2\ln(1/\varepsilon)+δ$ otherwise, improving the previous $O(\varepsilon^{-2}\log(1/\varepsilon))$. Our main tool is a rescue lemma. Splitting a near-optimal uncapacitated solution gives at most $2k-1$ blocks of at most $U$ clients, and giving a block a private center saves its share of the extra cost of following an optimal solution. Since a center of the optimal solution can be closed once all its blocks are rescued, $\lfloor\varepsilon k\rfloor$ extra centers save almost an $\varepsilon$-fraction of this extra cost instead of an $\varepsilon/2$-fraction.
On Zeros and Algorithms for Disordered Systems II: The Antiferromagnetic Ising Model up to Kesten-Stigum
Ferenc Bencs, Brice Huang, Daniel Z. Lee, Kuikui Liu, Guus Regts
AI 速览：作者提出首个在均匀随机 d-正则图上、对反铁磁 Ising 模型在直至 Kesten-Stigum 阈值的全部逆温度范围内以任意高精度估计配分函数的多项式时间算法。
We give the first polynomial-time algorithm to estimate the partition function of the antiferromagnetic Ising model on the uniformly random $d$-regular graph to arbitrarily high accuracy for all inverse temperatures up to the Kesten-Stigum threshold. This is the entire regime in which the model is "replica symmetric," and is conjectured to be optimal for polynomial-time counting and sampling. To achieve our algorithmic result, we establish the absence of Fisher zeros for the partition function in a constant-width strip around the interval from $0$ up to the Kesten-Stigum threshold, with high probability over the randomness of the input graph. AI Disclosure: The overarching vision and proof architecture were developed solely by the authors without AI input. GPT 5.5 Pro was used, with extensive guidance from the authors, for only a few minor technical aspects of the proof. The authors independently verified correctness and wrote everything by hand. We take full responsibility for its contents. A more detailed statement can be found in Section 1.2.
Approximating Minimum Spanning Trees in Metrics under Semi-Random Corruptions
Aditya Anand, Anupam Gupta, Rajesh Jayaram, Euiwoong Lee, Ali Vakilian
AI 速览：研究在半随机损坏的度量空间中近似最小生成树，针对距离只增、只减和任意改变三种情形给出近似算法与下界。
We study the problem of finding minimum spanning trees in semi-random corrupted metrics. In this problem proposed by Bateni, Dharangutte, Jayaram, and Wang (COLT'24), we start with a metric space $\mathcal{M} = (X, d)$. The adversary independently corrupts each pairwise distance with probability $p$. The algorithm only observes these possibly corrupted distances and must output a spanning tree whose expected cost (in the true metric) is multiplicatively close to the minimum spanning tree cost. Prior work showed that losing a factor of $Ω(\log n)$ is necessary, even if the algorithm may adaptively select $O(n/\log n)$ points and probe the true distances between them. However, no upper bounds were known. We study three regimes based on whether the adversary can only increase, decrease, or arbitrarily change the corrupted distances. We show the following: (1) In the increase-only setting, we give a polynomial-time $O(1)$-approximation algorithm for any constant $0 \le p < 1$. (2) In the decrease-only setting, we give an $O(\log n)$-approximation algorithm for any constant $0 \le p < 1$. Additionally, we show an $O(\log^4 n)$-approximation algorithm that runs in quasi-polynomial time $n^{O(\log n)}$, for any constant $p \leq 0.1$. Moreover, we show that a polynomial-time algorithm computing an $n^{1/3-\varepsilon}$-approximation is unlikely to exist. (3) In the general setting, where the adversary can both increase and decrease distances, we show an $O(\log n)$-approximation algorithm for every $0 \leq p < \frac{1}{4}$. Further, we show that no $o(n)$-approximation is possible for $\frac{1}{2} \leq p < 1$. Our algorithms answer the main open question in Bateni et al.'s work; moreover, the $O(\log n)$-approximations in the latter two settings are information-theoretically optimal.
AI 速览：作者提出一个在一般图上以 n^{2+o(1)} 时间求解最大基数匹配的算法，思路是扩展近期组合最大流技术到斜对称流问题。
We consider the maximum cardinality matching problem in general graphs, and give an $n^{2+o(1)}$-time algorithm for the problem. Our work builds on recent work on combinatorial algorithms for maximum flow and extends the techniques to skew-symmetric flow problems which capture general matchings as a special case.
When Imputation Changes the Nearest Neighbor: Exact Certification of Similarity Retrieval for Incomplete Sensor Time Series
Jianru Shen
AI 速览：针对缺失值插补会改变最近邻的问题，提出对窗口最小值距离族在任意允许补全下保持检索结论的精确认证方法。
Sensor time series routinely reach similarity search with missing values, and the retrieved neighbours then depend on how the gaps were filled. We ask which retrieval conclusions hold for every admissible completion of an incomplete query. For a family of window-minimum distances that contains the $L_1$ distance as its separable end we give a pairwise certificate that one candidate is strictly closer than another, a guaranteed candidate set that contains every possible nearest neighbour, and a certificate that a nearest-neighbour label cannot change. The guarantees hold uniformly over the specified box and apply to the true query whenever it lies inside. At the separable end a closed form is known, but beyond it the worst-case gap need not be attained at a corner of the box. We prove a finite reduction to critical values, give an exact interval dynamic programme for the worst-case ranking gap, and a quadratic-time fast layer of valid bounds and feasible witnesses, supported by an exact criterion for when the feature-box relaxation is tight. On three traffic-sensor collections with real gap patterns, linear interpolation changes the nearest neighbour in 15 to 33 percent of the queries. At the full band the feature-box relaxation misses up to 14 percent of the exactly stable boundary pairs, whereas fixed-completion and Monte Carlo checks issue false certificates. The fast layer resolves about 90 percent of all tested boundary pairs without dynamic programming, and on paired network-scale queries the exact method is 82 to 110 times faster than the programme alone. Label certificates transfer to three public benchmarks, and the band width trades forecasting accuracy and certification coverage for lower imputation sensitivity, with dataset-dependent crossovers.
Minimum Label $k$-Cut on Planar Graphs: Complexity and Color Structure
Yuanhao Wang, Wei Wang
AI 速览：研究底层图结构与颜色类结构如何影响最小标签 k-割问题的参数化复杂性与可解性边界。
Minimum Label $k$-Cut asks for a minimum-size set of colors such that deleting all edges of these colors leaves at least $k$ connected components. We study how the structure of the underlying graph and its color classes affects the complexity of this problem. Regarding fixed-parameter tractability with respect to $k$ on simple planar graphs, we prove that the problem is W[1]-hard, even when the underlying graph is a cactus graph and each color appears at most twice. Furthermore, for cactus graphs in which every color class is connected, we establish a complexity boundary in terms of the maximum monochromatic degree: when this degree is at most two, an optimal solution can be found in $O(Nk^2)$ time, where $N$ is the input size, with no restriction on color frequency or the number of blocks spanned by a color; allowing the maximum monochromatic degree to increase to three already makes the decision version NP-complete. Finally, we prove that the problem remains W[1]-hard parameterized by $k$ on fan graphs, even when every color class is an induced path. These results show that bounding color frequency does not suffice to eliminate parameterized hardness. They also establish a monochromatic-degree boundary for connected colors on cactus graphs and identify complexity barriers to extending this tractable case to outerplanar graphs.
The Tight Competitive Ratio of Greedy for Online Steiner Forest
Iman Gholami, Ilya Hajiaghayi, MohammadTaghi Hajiaghayi, Peyman Jabbarzade
AI 速览：该论文证明在线Steiner森林问题中自然贪心算法的竞争比紧确为Θ(log k log log k)，从而否定了Awerbuch-Azar-Bartal的O(log k)猜想。
The online Steiner Forest problem is a fundamental network design problem in which terminal pairs arrive one at a time and an algorithm must irrevocably purchase edges connecting each pair. The natural greedy algorithm connects each terminal pair by a shortest path after making all previously purchased edges free. Awerbuch, Azar, and Bartal (SODA 1996) proved that this algorithm is $O(log^2 k)$-competitive and conjectured an $O(log k)$ bound, where $k$ is the number of terminal pairs. Bamas, Drygala, and Maggiori (SODA 2022) obtained an $O(\log \NumPairs\log\log \NumPairs)$ bound for greedy on a restricted class of inputs, leaving the general case open. In this paper, we resolve the competitive ratio of the greedy algorithm for online Steiner Forest, proving that it is tightly $Θ(log(k) log log(k))$. Specifically, we establish an $O(log(k) log log(k))$ upper bound for arbitrary instances and construct a matching $Ω(log(k) log log(k))$ lower bound. Our results refute the $O(log k)$ conjecture of Awerbuch, Azar, and Bartal and establish a tight $Θ(log log k)$ separation between the greedy algorithm and the best possible online algorithm for Steiner Forest.
Nearly-Optimal Time-Sample Trade-off in Consensus Dynamics
Francesco d'Amore, Joel Rybicki, Nobutaka Shimizu, Takeharu Shiraga, Robin Vacus
AI 速览：该论文研究同步 h-Majority 共识动力学中样本数 h 与达成共识所需时间之间的权衡，并给出近乎最优的上界与匹配下界。
How does the number of sampled opinions affect the time required to reach consensus in distributed consensus dynamics? We study the synchronous $h$-\textsf{Majority}, in which every node independently samples $h$ nodes uniformly at random with replacement in each round and adopts a most frequent sampled opinion, breaking ties uniformly at random. We show that $h$-\textsf{Majority} achieves the nearly-optimal trade-off between the sample size and the consensus time, among a broad class of consensus dynamics. In particular, we prove the following results. For every sample size $h \ge 3$ and for every initial configuration, we prove that the process reaches consensus with high probability within $\widetilde{O}\left(1+\min\left\{\frac{1}{\min\{h^2p_{\max},1\}},\frac{\sqrt{n}}{h}\right\}\right)$ rounds, where $n$ is the number of nodes and $p_{\max}$ is the largest initial fraction of nodes supporting any single opinion. This is the first bound on the consensus time for $h$-\textsf{Majority} that holds for every $h\ge3$ and every initial configuration. When the initially most popular opinion has a sufficiently large lead over every other opinion, we show that all nodes adopt this opinion with high probability within $O(\log n/\min\{h^2p_{\max},1\})$ rounds. We establish a matching lower bound up to polylogarithmic factors for any fixed update rule that treats all opinions symmetrically and that selects either a sampled opinion or the node's current opinion.
The Value of Experiments in Assortment Optimization
Mohammadreza Ahmadnejadsaein, Amine Bennouna, Omar El Housni
AI 速览：研究在多项 logit 模型下，利用少量固定实验 assortment 的顾客选择数据，如何设计实验与最终 assortment 选择规则以最小化最坏情况遗憾。
Retailers can learn customer preferences by experimenting with different assortments. However, the cost of each experiment and the operational burden of repeatedly changing product offerings can make extensive experimentation impractical. We study how much a retailer can improve its final assortment decision using customer observations from one (or a very small number of) fixed experimental assortments. We consider the multinomial logit model, where product revenues are known but customer preferences are unknown. In this setting, we jointly design the experimental assortment and the rule for selecting a final assortment from observed customer choices to minimize worst-case regret: the expected revenue loss of the final assortment relative to an optimal assortment under the true preferences, maximized over possible customer preferences. We first characterize exactly the minimax regret and an optimal assortment when no experimentation is possible. We then show, surprisingly, that offering a single experimental assortment to just one customer can already substantially reduce regret, depending on how the products' revenues compare. We derive the exact minimax regret and an optimal experimental policy in this setting. When a single fixed experimental assortment is offered to many customers, we give a geometric characterization of the limiting minimax regret and construct an asymptotically optimal policy. Depending on product revenues, this single experiment can yield no improvement over no experimentation or nearly complete regret reduction. Finally, we extend our analysis to multiple experimental assortments chosen in advance, characterizing the limiting minimax regret, constructing asymptotically optimal policies, and identifying which collections of assortments allow worst-case regret to vanish as the number of customer observations grows.
Exact Competitive Ratios of Index Rules for Scheduling with Obligatory Testing
Felix Buld
AI 速览：作者提出一种保持竞争比的实例归约方法，精确计算了1-SORT等索引规则在带强制测试调度中的竞争比，证明1-SORT的竞争比恰为黄金比例φ≈1.618。
We revisit scheduling with obligatory testing, where each job consists of two operations: a test with an initially known time and an operation whose processing time is revealed upon completion of the job's test. We develop an iterative competitive-ratio-preserving instance-reduction method that enables us to compute exact competitive ratios directly on highly structured instances for various index rules. Our main result is a clean analysis showing that the competitive ratio of the algorithm $1$-SORT is exactly the Golden Ratio $\varphi\approx 1.618$, improving the recent best-known upper bound of $1.861$ for arbitrary testing times. We further leverage this method to obtain near-tight bounds for a simple randomized algorithm on instances with uniform testing times.
Zero-freeness for Random Regular Graphs up to the Replica Symmetry Breaking Threshold
Zejia Chen, Chunyang Wang, Xinyuan Zhang
AI 速览：作者在随机 d-正则图上证明硬核模型与反铁磁 Potts 模型（含正常 q-染色）在副本对称区内具有零点自由区域，并由此得到多项式时间近似配分函数的算法。
We study antiferromagnetic spin systems on random regular graphs through the lens of the phase transitions characterized by the complex zeros of the partition function. Our motivating examples include the hardcore model and the antiferromagnetic Potts model (including proper $q$-colorings at $β=0$). For both models on random $d$-regular graphs, we establish, with high probability over the graph, zero-free regions throughout the interior of the replica-symmetric regime. Asymptotically, this regime extends up to tree occupation density $$α< α_{\mathrm{RSB}}(d) = (2\log d - 2\log\log d + 2 - 2\log 2 + o_d(1))/d$$ for the hardcore model and up to degree $$d < d_{\mathrm{RSB}}(q) = (2q-1)\log q - 2\log 2 + o_q(1)$$ for proper $q$-colorings. Leveraging the standard algorithmic implications of zero-freeness, we obtain deterministic polynomial-time algorithms that multiplicatively approximate the corresponding partition functions at fixed rational parameters throughout this regime, with high probability over the graph. Our algorithmic guarantees significantly improve upon prior work and extend beyond the parameter ranges reached by previously known search algorithms. To establish our results, we first develop reinforced small subgraph conditioning, which resolves short-cycle fluctuations in the complex plane by lifting classical small subgraph conditioning on random regular graphs to random holomorphic functions. Second, we overcome the standard second-moment barrier caused by the exponential blowup of the bare second moment inside the replica-symmetric phase, by showing that replica symmetry implies a nonpositive Bethe functional and upgrading planted overlap concentration to an exponentially small bound, which enables a typical-graph truncation that controls complex second moments up to the replica symmetry breaking threshold.
We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a simulation working directly with the nonlinear complementarity equations. Using a two-stage quantum eigenpath traversal schedule, this approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\widetilde{O}\!\left((m+n)^{3/2}\,\varepsilon^{-1}+(m+n)^{1/2}\,\varepsilon^{-3/2}\right)$ queries to an oracle that evaluates a potential function. In the standard gate model (i.e., without access to quantum RAM), our algorithm can obtain $\varepsilon$-precise solutions to LO problems using at most $$\widetilde{O}\!\left(\left[\frac{(m+n)^{3/2}}{\varepsilon} + \frac{(m+n)^{1/2}}{\varepsilon^{3/2}}\right] \cdot \mathrm{nnz}(A)\right)$$ elementary gates, where $\mathrm{nnz}(A)$ is the total number of non-zero elements found in the constraint matrix.
Sampling Sphere Packings with Continuum Glauber Dynamics
Aiya Kuchukova, Santosh S. Vempala, Daniel J. Zhang
AI 速览：作者为排斥对势的Gibbs点过程建立连续Glauber动力学的谱隙，并据此给出任意程对势的高效采样算法，改进了固定尺寸/密度堆积的可采样阈值。
Continuum Glauber dynamics is a spatial birth-death process whose stationary distribution is a Gibbs distribution. We establish a spectral gap for Continuum Glauber dynamics applied to Gibbs point processes with repulsive pair potentials, a well-known special case of which is the hard sphere model. For arbitrary-range repulsive pair potentials, we show that a continuous version of Spectral Independence suffices to establish a spectral gap. This extends the regime of activity for which Continuum Glauber dynamics is known to mix, yielding a simple efficient sampling algorithm for arbitrary-range pair potentials that matches the known efficient sampling regime for finite-range pair potentials currently based on specialized algorithms. As a consequence, we also improve the threshold up to which packings of fixed size/density can be efficiently sampled from a bounded domain, the first improvement since Kannan, Mahoney and Montenegro (2003). To prove these results, we develop continuous analogs of Spectral Independence and negative fields localization. We show that a stronger variant of zero-freeness implies Spectral Independence, which in turn allows us to run the localization scheme to boost the spectral gap of Continuum Glauber dynamics from smaller activity to larger activity. While this follows the high-level blueprint of Chen and Eldan (2022) for the discrete setting, we have to address several novel difficulties due to the continuous setting. Notably, we avoid discretization in the algorithm and the analysis and work directly in the continuous setting.
Albert Gong, Annabelle Michael Carrell, Raaz Dwivedi, Lester Mackey
AI 速览：提出 Express 工具，将非因果注意力近似转化为因果近似，并结合 Thinformer 实现更优的因果注意力保证与高效实现。
We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.
Half-Optimal Induced Matchings in Random Graphs: Online and Low-Degree Algorithms
Abhishek Dhawan, Eren C. Kızıldağ, Neeladri Maitra, Cheng Mao, Timothy L. H. Wee
AI 速览：作者证明在 Erdős-Rényi 随机图中，寻找大诱导匹配存在尖锐的因子二间隙：简单在线贪心可达约一半最优，而在线算法与低阶多项式算法均无法突破该阈值。
For many optimization problems on random graphs, most prominently the largest independent set problem, natural algorithms reach only half the optimum, and such factor-two gaps are known to be sharp within broad algorithmic classes. We prove such a gap for finding large induced matchings in Erdős-Rényi random graphs. In the worst case, requiring a matching to be induced turns a polynomial-time solvable problem into an NP-hard one; our results reveal an average-case counterpart. In $G(n,p)$ with $C/n\le p\le 1-κ$, the largest induced matching has size $(1+o(1))L$ with high probability, where $L=\log_{1/(1-p)}(np)$. A simple online greedy algorithm finds an induced matching of size $(1/2-ε)L$, while no online algorithm, even with adaptive inspection order and unbounded computation, finds one of size $(1/2+ε)L$ except with probability $\exp(-Ω(L\log(np)))$. In the sparse regime $p=d/n$, low-degree polynomial algorithms exhibit the same sharp threshold. Our guarantees rest on a maximality principle: with high probability, every maximal induced matching is half-optimal. Greedy outputs a maximal induced matching and our low-degree algorithm a near-maximal one. We bypass the delicate coupling arguments classically used to analyze greedy since Karp. For low-degree hardness, we adapt Wein's overlap gap argument, with forbidden structures that track overlaps between endpoint sets rather than edge sets, since a new edge may join two previously used endpoints and thus encounter no fresh randomness. For online hardness, we give a new and significantly simpler argument that avoids the correlated instances of prior OGP-based proofs: at a suitable stopping time, all pairs involving uninspected vertices remain unrevealed, so a direct first-moment bound suffices.
A Deterministic Near-Linear Time Approximation Algorithm for Restricted Shortest Paths in Directed Graphs
Vikrant Ashvinkumar
AI 速览：针对有向图受限最短路径问题，作者提出确定性双准则 (1+ε,1+ε) 近似算法，运行时间近线性，并可推广到多资源情形。
In the restricted shortest paths problem, we are given an $n$-vertex $m$-edge directed graph $G$ whose edges are assigned two non-negative weights (or resources) called lengths and costs, a source vertex $s$, and a cost threshold $C$. The goal is to find, for each target vertex $t$, the length of the shortest $(s,t)$-path whose total cost is at most $C$. Ashvinkumar, Bernstein, and Karczmarz (SODA 2025) gave the first randomized bicriteria $(1+\varepsilon,1+\varepsilon)$-approximate algorithm on directed graphs running in sub-$mn$ time, answering a problem posed by Bernstein (SODA 2012). We substantially improve this on two fronts, supplying a deterministic bicriteria $(1+\varepsilon,1+\varepsilon)$-approximate algorithm running in near-linear time. Our approach straightforwardly generalizes to more than two resources. For $k$ resources, a deterministic $(1+\varepsilon_1,\ldots,1+\varepsilon_k)$-approximate algorithm running in near-linear time is possible, with $\varepsilon_1=\ldots=\varepsilon_k=\varepsilon$.
AI 速览：该论文研究在线无权重背包问题，在允许移除已接受物品的基础上，结合物品规模上界知识与有限追索权（重新插入曾被拒物品），给出竞争比的紧界并提出自适应在线算法。
In the online unweighted knapsack problem, items arrive sequentially and must be accepted or rejected immediately for a bounded-capacity knapsack, aiming to maximize the total size of accepted items. When decisions are irrevocable, no algorithm achieves a non-trivial competitive ratio. An essential relaxation that circumvents this impossibility is to allow previously accepted items to be removed from the knapsack (a.k.a. removal). In this work, we investigate possible further improvements on the competitive ratio with removal by bringing together two other previously proposed model extensions: knowledge of an upper bound on the item sizes (a.k.a. bound awareness) and ability to reinsert previously rejected items(a.k.a.recourse). Our contributions are twofold. First, we establish close upper and lower bounds on the competitive ratio under bound awareness, removal and recourse. We show that significant improvements on the best known competitive ratio are possible when the upper bound on the item sizes is not too restrictive and only few uses of recourse are available. Second, we introduce the notion of adaptive online algorithms which do not require to know any upper bound on the item sizes but still achieve performance guarantees comparable to the ones obtained under the bound awareness model.
Layering and reverse extension for weighted $k$-path vertex cover
Jiajing Wu, An Zhang, Shiming Li, Guohui Lin
AI 速览：作者在分层与反向扩展局部比值框架下设计局部权重函数，为加权 k-路径顶点覆盖问题给出新的近似比。
We study the weighted $k$-path vertex cover problem for any fixed integer $k \ge 2$, in which a vertex non-negatively weighted simple graph $G = (V, E)$ is given and one wants to find a minimum-weight vertex subset $C \subseteq V$ such that every simple path on $k$ vertices in $G$ contains at least one vertex of $C$. The problem generalizes the classic weighted vertex cover problem, that is, $k = 2$, and it admits a $k$-approximation algorithm of Hochbaum~[SIAM J. Comput., 1982] and a $(k - \frac 12)$-approximation algorithm of Brüstle et al.~[IWOCA 2021]. In this paper, we carefully design local weight functions in the Layering and Reverse Extension local ratio framework of Bar-Yehuda et al.~[ACM Comput. Surv., 2004], leading to an approximation algorithm of ratio $\frac {3k-1}4$ for odd $k \ge 3$ and of ratio $\frac {3k}4$ for even $k \ge 4$. When $k = 3$ and $k = 4$, these ratios are $2$ and $3$, respectively, matching the state-of-the-art approximation ratios for the corresponding special cases.
In a compact storage system with side access, bins are arranged in adjacent vertical stacks. In each retrieval cycle, bins blocking access are lifted to create a temporary tunnel through which a robot extracts one requested bin from the side. The lifted bins are then lowered again, and the energy expenditure of the cycle equals the number of lifted bins. Given a pick list known in advance, we seek a retrieval order that minimizes total energy expenditure. For every number of stacks, we prove that the total energy separates into an instance constant and a pairwise penalty, and that every energy-optimal order is laminar within each stack. For two stacks, we prove that the retrieval sequencing problem is solvable in polynomial time. For three stacks, we give an exact polynomial-time reduction to a merge problem and characterize which merge-problem instances arise from the storage model. We define a recursive class of retrieval orders and construct a polynomial-time dynamic program that minimizes energy expenditure over this class. Finally, we quantify the additional energy savings from batching, which allows several requested bins to be retrieved in one cycle.
We study approximation algorithms for constructing sparse navigable graphs in general metrics. Given an $n$-point metric $(P,d)$ and a parameter $α\ge1$, a directed graph on $P$ is $α$-navigable if, for every pair of distinct points $s,t\in P$, the source $s$ has an outgoing neighbor $p$ with $d(p,t)<d(s,t)/α$. The objective is to minimize the number of edges. Recent independent works of Khanna, Padaki, and Waingarten (2026) and Conway et al. (2026) give $O(\log n)$-approximation algorithms and show that an $o(\log n)$ approximation is NP-hard in general. Before our work, the best known running time for such an approximation for general $α\ge1$ was $\widetilde O(\min\{n\operatorname{OPT}_α,n^{2.5}\})$, where $\operatorname{OPT}_α$ is the minimum number of edges in an $α$-navigable graph. In this paper, we give a randomized algorithm that, with high probability, runs in $\widetilde O(n^2)$ time and achieves an $O(\log n)$ approximation for every $α\ge1$. In general, these guarantees are nearly optimal up to constant factors in the approximation and polylogarithmic factors in the running time.
Unsigned linear unichromosomal genomes with genes labeled 1, . . . , n are represented by permutations of these labels, whose adjacencies are the unordered pairs of genes occurring consecutively. The breakpoint distance counts the adjacencies of one genome that are absent from the other, and a breakpoint median minimizes the total breakpoint distance to a given set of genomes. Deciding whether a permutation is a median often requires a minimization over the whole symmetric group, which makes the median set resistant to exact counting. We replace this global condition by a local one: a c-approximate median is a genome having at most c adjacencies absent from all the input genomes. For a fixed genome we introduce its approximate median inverse, the set of ordered k-tuples of genomes ad- mitting it as a c-approximate median, and we construct and count this set explicitly. For k genomes chosen uniformly and independently at random, this gives the probability that a given genome is a c-approximate median and the expected number of such approximate medians. The relaxation is not only technical: approximate medians and near-optimal genomes co- incide up to a shift of the tolerance that remains bounded in probability as the number of genes grows. Finally, we derive an explicit upper bound on the probability that a given genome is an exact median, and on the expected number of exact breakpoint medians.
We present deterministic polynomial-space algorithms that solve Max 2-SAT on $n$ variables exactly in time $\Ostar(2^{(1-γ)n})$, for a fixed $γ>0$, on several classes of instances whose clause density may grow with $n$. Our main tool extends the classic DPLL branching algorithm. When the search gives a variable its less promising value, comparing the two values of the variable yields a pruning constraint that is satisfied by some optimal assignment. Accumulating these constraints along each branch prunes all but an exponentially small fraction of the assignments. The classes are formulas with many biased variables, semirandom 2-CNF formulas, and Max-Cut on regular expander graphs and on graphs generated by a stochastic block model. In a semirandom formula, the pairs of variables that may share a clause are arbitrary, and the clauses on these pairs are random. For semirandom formulas and block-model graphs, the algorithms are still exact on every input, and the time bound holds with high probability over the input. To our knowledge, these are the first polynomial-space algorithms with such a running time for nontrivial classes of dense Max 2-SAT instances. Beyond these classes, the pruning constraints can be added to any branching algorithm for Max 2-SAT, and we expect them to be of practical relevance for general formulas, in particular for sparse ones, where each constraint involves only a few variables.
Mugurel-Ionuţ Andreica, Pantelimon George Popescu, Eliana-Dina Tîrşa
The Gauss-Seidel iterative method is commonly used for solving systems of linear equations because of its good convergence properties. However, the Gauss-Seidel algorithm cannot be easily parallelized because of the data dependencies existing at each iteration. The best parallel execution time for dense systems of equations is $O(n)$ given that enough processors are available and no preprocessing is performed. In this paper we propose a new parallel Gauss-Seidel algorithm which is capable of achieving better parallel runtimes at the expense of more preprocessing time. The algorithm can be tuned in order to provide convenient trade-offs between the preprocessing time and the best parallel runtime achievable by the algorithm.
By creating some new concepts and methods: checking tree, long unit path, direct contradiction unit pair, indirect contradiction unit pair, additional contradiction unit pair, 2-unit layer and 3-unit layer, redundant units, and destroying parallel pairs , we successfully transform solving a 3SAT problem to solving 2SAT problems in polynomial time. Thus we proved that NP=P.
With the help of secure hardware such as trusted execution environments (TEEs), private data can be uploaded and computed in untrusted cloud clusters, so that an adversary controlling the cloud can only see the data in encrypted form. However, the communication pattern between the processors may still reveal sensitive information about the data, so it is important to develop parallel algorithms in the BSP model whose communication pattern is independent of the input data, a property known as \textit{communication obliviousness}. In this paper, we study such algorithms for three fundamental problems: permutation, sorting, and 0-1 sorting. For the first two problems, we prove a tight (up to lower-order terms) communication complexity of $(2-2/p)n\pm o(n)$. Here, $p$ is the number of processors and $n$ is the input size, assuming $2\le p$ and $p=o(n^{1/3})$. Our lower bound holds regardless of the number of rounds and local memory size, while for balanced permutation and sorting the upper bound is achieved in a constant number of rounds with load at most $n/p+o(n/p)$. For 0-1 sorting, we also derive a tight communication complexity of $C_{\mathrm{cut}}(p) \cdot n \pm o(n)$, where $C_{\mathrm{cut}}(p)$ is the optimal value of a linear program parameterized by $p$. Although there appears to be no closed-form for $C_{\mathrm{cut}}(p)$, we show that it is strictly smaller than $2-2/p$ for all $p\ge 3$, thus separating 0-1 sorting from general sorting in terms of their communication complexity.
Kellerhals, Krebs, Nichterlein, and Schmid introduced the optimization problem Tree in Binary Tree (TBT), where, given a tree $T$, the task is to find a binary tree $S^*$ with $V(T)=V(S^*)$ that minimizes $\sum_{uv\in E(T)} \operatorname{dist}_{S^*}(u,v)$. They also introduced the variant Tree in Binary Tree With Steiner Nodes (TBT-SN), where the condition $V(T)=V(S^*)$ is relaxed to $V(T)\subseteq V(S^*)$. We give a simple linear-time $\frac{8}{3}$-approximation algorithm for TBT and a polynomial-time approximation scheme (PTAS) for TBT-SN. Our main building idea in both algorithms is to iteratively consider subtrees of the tree $T$ which contain all vertices below some vertex $v$ and construct a solution where $v$ has degree 1. For the PTAS, we cut $T$ into smaller trees, solve these instances, modify the obtained solutions, and combine them to achieve a solution for the original instance.
We study Stable Marriage with Ties and Incomplete lists (SMTI) under weak stability. For an instance with $n$ agents on each side, we give an exact algorithm that returns a maximum-cardinality weakly stable matching in $O^*(22.74^n)$ time. To the best of our knowledge, this is the first exact algorithm for this problem running in single-exponential time.
Tom-Lukas Breitkopf, Vincent Froese, Anton Herrmann, André Nichterlein, Camille Richer
We consider the well-studied problem of finding a spanning tree with minimum average distance between vertex pairs (called a MAD tree). This is a classic network design problem which is known to be NP-hard. While approximation algorithms and polynomial-time algorithms for some graph classes are known, the parameterized complexity of the problem has not been investigated so far. We start a parameterized complexity analysis with the goal of determining the border of algorithmic tractability for the MAD tree problem. To this end, we provide a linear-time algorithm for graphs of constant modular width and a polynomial-time algorithm for graphs of bounded treewidth; the degree of the polynomial depends on the treewidth. That is, the problem is in FPT with respect to modular width and in XP with respect to treewidth. Moreover, we show that it is in FPT when parameterized by vertex integrity, by maximum leaf number, or by an above-guarantee parameter. We complement these algorithms with NP-hardness on split graphs.
Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen, Mikkel Thorup
The famous Four Color Theorem (4CT) states that any planar graph can be colored with four colors, but how do we find a 4-coloring? In our previous paper [FOCS'26], we presented a simple combinatorial algorithm to 4-color planar graphs in $O(n\log n)$ time. We improve it here to $O(n)$ time, which is optimal. The algorithm from our previous paper is based on proving that every planar triangulation has a linear number of reducible configurations or obstructing cycles. The bottleneck for the algorithm is that we need $O(\log n)$ Kempe changes to extend a recursive 4-coloring to the reducible configurations. Each Kempe change can involve a constant fraction of the graph and takes linear time. Here we show that all $O(\log n)$ Kempe changes can be identified and implemented in $O(n)$ time. The idea is to spend $O(n)$ time to preprocess the graph so that we can identify and simulate each good Kempe change in $O(n/\log n)$ amortized time. Thus, we implement all $O(\log n)$ Kempe changes in $O(n)$ total time. Based on this, we implement the whole 4-coloring in linear time. We note that creating such a representation for fast identification and simulation of the right Kempe changes is a challenge of data structures, unlike the previous paper that builds on a new graph theoretic understanding of reducible configurations.
We study \textsc{Max $k$-Weight CSP}: maximize the satisfied weight of Boolean constraints with nonnegative rational weights by setting at most $k$ variables to $1$. Each constraint uses at most $d$ distinct variables and a relation from a finite Boolean language $Γ$. This includes \textsc{Partial Vertex Cover}, \textsc{Densest $k$-Subgraph}, bounded-frequency \textsc{Maximum Coverage}, and cardinality-constrained \textsc{Max-SAT}. We completely classify the fixed finite languages admitting an efficient parameterized approximation scheme (EPAS) with parameter $k+d$. Only nonempty relations excluding the all-zero tuple matter. For such a relation, a coordinate is a \emph{unit witness} if the tuple with a single $1$ in that coordinate is accepted, and a \emph{control} otherwise. A relation is \emph{Direct} if setting any selected control to $0$, with all other coordinates unchanged, preserves satisfaction. It is \emph{Rooted} if every satisfying tuple has a selected unit witness such that keeping it, setting all other unit witnesses to $0$, and setting any subset of selected controls to $0$ preserves satisfaction. If all such relations are Direct, or all are Rooted, we give a deterministic $(1-\varepsilon)$-approximation for every rational $\varepsilon\in(0,1)$ in time $2^{O(d+k\log(1+dk/\varepsilon))}\ell_{\mathrm{in}}^{O(1)}$, where $\ell_{\mathrm{in}}$ is the total input length. The algorithms are uniform in $d$ and give an EPAS parameterized by $k+d$ whenever relation membership is polynomial-time decidable. If neither condition holds, no EPAS exists unless $\mathsf{FPT}=\mathsf{W[1]}$, even for unweighted instances where no two constraints involve the same set of variables.