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Revisiting Garg's 2-Approximation Algorithm for the k-MST Problem in Graphs 图中k-MST问题的Garg 2-逼近算法重述
Pub Date : 2023-06-02 DOI: 10.1137/1.9781611977585.ch6
Emmett Breen, Renee Mirka, Zichen Wang, David P. Williamson
This paper revisits the 2-approximation algorithm for $k$-MST presented by Garg in light of a recent paper of Paul et al.. In the $k$-MST problem, the goal is to return a tree spanning $k$ vertices of minimum total edge cost. Paul et al. extend Garg's primal-dual subroutine to improve the approximation ratios for the budgeted prize-collecting traveling salesman and minimum spanning tree problems. We follow their algorithm and analysis to provide a cleaner version of Garg's result. Additionally, we introduce the novel concept of a kernel which allows an easier visualization of the stages of the algorithm and a clearer understanding of the pruning phase. Other notable updates include presenting a linear programming formulation of the $k$-MST problem, including pseudocode, replacing the coloring scheme used by Garg with the simpler concept of neutral sets, and providing an explicit potential function.
本文根据Paul等人最近的一篇论文,重新审视了Garg提出的$k$-MST的2逼近算法。在$k$-MST问题中,目标是返回一棵树,它生成$k$个顶点,并且总边代价最小。Paul等人扩展了Garg的原始对偶子程序,以改进预算奖励旅行推销员和最小生成树问题的近似比率。我们遵循他们的算法和分析,提供一个更清晰的Garg结果。此外,我们引入了核的新概念,它可以更容易地可视化算法的各个阶段,并更清楚地理解修剪阶段。其他值得注意的更新包括提出k -MST问题的线性规划公式,包括伪代码,用更简单的中性集概念取代Garg使用的着色方案,并提供显式的势函数。
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引用次数: 1
Min-Max Optimization Made Simple: Approximating the Proximal Point Method via Contraction Maps 简化最小-最大优化:通过收缩映射逼近近点方法
Pub Date : 2023-01-10 DOI: 10.48550/arXiv.2301.03931
V. Cevher, G. Piliouras, Ryann Sim, Stratis Skoulakis
In this paper we present a first-order method that admits near-optimal convergence rates for convex/concave min-max problems while requiring a simple and intuitive analysis. Similarly to the seminal work of Nemirovski and the recent approach of Piliouras et al. in normal form games, our work is based on the fact that the update rule of the Proximal Point method (PP) can be approximated up to accuracy $epsilon$ with only $O(log 1/epsilon)$ additional gradient-calls through the iterations of a contraction map. Then combining the analysis of (PP) method with an error-propagation analysis we establish that the resulting first order method, called Clairvoyant Extra Gradient, admits near-optimal time-average convergence for general domains and last-iterate convergence in the unconstrained case.
本文给出了一种一阶方法,该方法允许凸/凹最小-最大问题的近似最优收敛速率,同时需要简单直观的分析。与Nemirovski的开创性工作和Piliouras等人最近在正常形式游戏中的方法类似,我们的工作是基于这样一个事实,即Proximal Point method (PP)的更新规则可以通过收缩地图的迭代仅$O(log 1/epsilon)$额外的梯度调用来近似达到$epsilon$的精度。然后将(PP)方法的分析与误差传播分析相结合,证明了所得到的一阶方法Clairvoyant Extra Gradient在一般情况下具有近似最优的时间平均收敛性,在无约束情况下具有最后迭代收敛性。
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引用次数: 4
Simpler and faster algorithms for detours in planar digraphs 平面有向图中绕行的更简单、更快的算法
Pub Date : 2023-01-06 DOI: 10.48550/arXiv.2301.02421
Meike Hatzel, Konrad Majewski, Michal Pilipczuk, Marek Sokolowski
In the directed detour problem one is given a digraph $G$ and a pair of vertices $s$ and~$t$, and the task is to decide whether there is a directed simple path from $s$ to $t$ in $G$ whose length is larger than $mathsf{dist}_{G}(s,t)$. The more general parameterized variant, directed long detour, asks for a simple $s$-to-$t$ path of length at least $mathsf{dist}_{G}(s,t)+k$, for a given parameter $k$. Surprisingly, it is still unknown whether directed detour is polynomial-time solvable on general digraphs. However, for planar digraphs, Wu and Wang~[Networks, '15] proposed an $mathcal{O}(n^3)$-time algorithm for directed detour, while Fomin et al.~[STACS 2022] gave a $2^{mathcal{O}(k)}cdot n^{mathcal{O}(1)}$-time fpt algorithm for directed long detour. The algorithm of Wu and Wang relies on a nontrivial analysis of how short detours may look like in a plane embedding, while the algorithm of Fomin et al.~is based on a reduction to the ${S}$-disjoint paths problem on planar digraphs. This latter problem is solvable in polynomial time using the algebraic machinery of Schrijver~[SIAM~J.~Comp.,~'94], but the degree of the obtained polynomial factor is huge. In this paper we propose two simple algorithms: we show how to solve, in planar digraphs, directed detour in time $mathcal{O}(n^2)$ and directed long detour in time $2^{mathcal{O}(k)}cdot n^4 log n$. In both cases, the idea is to reduce to the $2$-disjoint paths problem in a planar digraph, and to observe that the obtained instances of this problem have a certain topological structure that makes them amenable to a direct greedy strategy.
在有向绕路问题中,给定一个有向图$G$和一对顶点$s$和~$t$,任务是确定$G$中是否存在一条从$s$到$t$的有向简单路径,其长度大于$mathsf{dist}_{G}(s,t)$。对于给定的参数$k$,更一般的参数化变体,定向长迂回,要求一个简单的$s$到$t$的路径,其长度至少为$mathsf{dist}_{G}(s,t)+k$。令人惊讶的是,在一般有向图上,定向绕行是否多项式时间可解仍然是未知的。然而,对于平面有向图,Wu和Wang~[Networks, '15]提出了$mathcal{O}(n^3)$ time算法用于有向绕路,而Fomin等人~[STACS 2022]给出了$2^{mathcal{O}(k)}cdot n^{mathcal{O}(1)}$ time fpt算法用于有向长绕路。Wu和Wang的算法依赖于对平面嵌入中的短弯路的非平凡分析,而Fomin等人的算法是基于对平面有向图上的${S}$-不相交路径问题的简化。用Schrijver~[SIAM~J.~Comp]的代数机制在多项式时间内求解后一个问题。,~'94],但得到的多项式因子的程度是巨大的。本文提出了两种简单的算法:我们展示了如何求解平面有向图中时间$mathcal{O}(n^2)$的有向绕路和时间$2^{mathcal{O}(k)}cdot n^4 log n$的有向长绕路。在这两种情况下,思想都是将其简化为平面有向图中的$2$-不相交路径问题,并观察该问题的实例具有一定的拓扑结构,使其适用于直接贪婪策略。
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引用次数: 4
A Simple Optimal Algorithm for the 2-Arm Bandit Problem 双臂强盗问题的一种简单优化算法
Pub Date : 2023-01-01 DOI: 10.1137/1.9781611977585.ch33
Maxime Larcher, Robert Meier, A. Steger
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引用次数: 0
A Simple Algorithm for Submodular Minimum Linear Ordering 一种简单的次模最小线性排序算法
Pub Date : 2023-01-01 DOI: 10.1137/1.9781611977585.ch3
Dor Katzelnick, Roy Schwartz
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引用次数: 0
Proximity Search in the Greedy Tree 贪婪树中的邻近搜索
Pub Date : 2023-01-01 DOI: 10.1137/1.9781611977585.ch29
Oliver A. Chubet, Parth Parikh, Don Sheehy, S. Sheth
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引用次数: 0
On the Fine-Grained Complexity of Approximating k-Center in Sparse Graphs 稀疏图中k-中心逼近的细粒度复杂度
Pub Date : 2023-01-01 DOI: 10.1137/1.9781611977585.ch14
Amir Abboud, Vincent Cohen-Addad, Euiwoong Lee, Pasin Manurangsi
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引用次数: 2
An Improved Online Reduction from PAC Learning to Mistake-Bounded Learning 从PAC学习到错误边界学习的改进在线还原
Pub Date : 2023-01-01 DOI: 10.1137/1.9781611977585.ch34
Lucas Gretta, Eric Price
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引用次数: 0
Simple Random Order Contention Resolution for Graphic Matroids with Almost no Prior Information 几乎没有先验信息的图形拟阵的简单随机顺序争用解决
Pub Date : 2022-11-28 DOI: 10.48550/arXiv.2211.15146
Richard Santiago, I. Sergeev, R. Zenklusen
Random order online contention resolution schemes (ROCRS) are structured online rounding algorithms with numerous applications and links to other well-known online selection problems, like the matroid secretary conjecture. We are interested in ROCRS subject to a matroid constraint, which is among the most studied constraint families. Previous ROCRS required to know upfront the full fractional point to be rounded as well as the matroid. It is unclear to what extent this is necessary. Fu, Lu, Tang, Turkieltaub, Wu, Wu, and Zhang (SOSA 2022) shed some light on this question by proving that no strong (constant-selectable) online or even offline contention resolution scheme exists if the fractional point is unknown, not even for graphic matroids. In contrast, we show, in a setting with slightly more knowledge and where the fractional point reveals one by one, that there is hope to obtain strong ROCRS by providing a simple constant-selectable ROCRS for graphic matroids that only requires to know the size of the ground set in advance. Moreover, our procedure holds in the more general adversarial order with a sample setting, where, after sampling a random constant fraction of the elements, all remaining (non-sampled) elements may come in adversarial order.
随机顺序在线争用解决方案(ROCRS)是一种结构化的在线舍入算法,有许多应用程序,并链接到其他著名的在线选择问题,如矩阵秘书猜想。我们感兴趣的是受矩阵约束的ROCRS,这是研究最多的约束族之一。以前的ROCRS要求预先知道要舍入的完整分数点以及矩阵。目前尚不清楚这在多大程度上是必要的。Fu, Lu, Tang, Turkieltaub, Wu, Wu和Zhang (SOSA 2022)通过证明如果分数点未知,甚至对于图形拟阵,不存在强大的(恒定可选的)在线甚至离线争用解决方案,从而阐明了这个问题。相比之下,我们表明,在知识略多的设置中,分数点逐一揭示,有希望通过为图形拟阵提供简单的常数可选ROCRS来获得强ROCRS,而图形拟阵只需要事先知道地面集的大小。此外,我们的程序适用于更一般的对抗顺序和样本设置,其中,在采样元素的随机常数部分之后,所有剩余的(未采样的)元素可能以对抗顺序出现。
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引用次数: 0
Optimal resizable arrays 最佳可调整大小数组
Pub Date : 2022-11-20 DOI: 10.48550/arXiv.2211.11009
R. Tarjan, Uri Zwick
A emph{resizable array} is an array that can emph{grow} and emph{shrink} by the addition or removal of items from its end, or both its ends, while still supporting constant-time emph{access} to each item stored in the array given its emph{index}. Since the size of an array, i.e., the number of items in it, varies over time, space-efficient maintenance of a resizable array requires dynamic memory management. A standard doubling technique allows the maintenance of an array of size~$N$ using only $O(N)$ space, with $O(1)$ amortized time, or even $O(1)$ worst-case time, per operation. Sitarski and Brodnik et al. describe much better solutions that maintain a resizable array of size~$N$ using only $N+O(sqrt{N})$ space, still with $O(1)$ time per operation. Brodnik et al. give a simple proof that this is best possible. We distinguish between the space needed for emph{storing} a resizable array, and accessing its items, and the emph{temporary} space that may be needed while growing or shrinking the array. For every integer $rge 2$, we show that $N+O(N^{1/r})$ space is sufficient for storing and accessing an array of size~$N$, if $N+O(N^{1-1/r})$ space can be used briefly during grow and shrink operations. Accessing an item by index takes $O(1)$ worst-case time while grow and shrink operations take $O(r)$ amortized time. Using an exact analysis of a emph{growth game}, we show that for any data structure from a wide class of data structures that uses only $N+O(N^{1/r})$ space to store the array, the amortized cost of grow is $Omega(r)$, even if only grow and access operations are allowed. The time for grow and shrink operations cannot be made worst-case, unless $r=2$.
emph{可调整大小的数组}是这样一种数组,它可以通过从其末端或两端添加或删除项来emph{增长}和emph{缩小},同时仍然支持对给定emph{索引}的数组中存储的每个项进行恒定时间emph{访问}。由于数组的大小(即其中的项数)随时间而变化,因此对可调整大小的数组进行有效的空间维护需要动态内存管理。标准的加倍技术允许只使用$O(N)$空间来维护大小为$N$的数组,每个操作的平摊时间为$O(1)$,甚至是最坏情况时间为$O(1)$。Sitarski和Brodnik等人描述了更好的解决方案,即仅使用$N+O(sqrt{N})$空间维护可调整大小的数组$N$,每次操作仍然需要$O(1)$时间。Brodnik等人给出了一个简单的证明,证明这是最好的可能。我们区分了emph{存储}可调整大小的数组和访问其项所需的空间,以及在增加或缩小数组时可能需要的emph{临时}空间。对于每个整数$rge 2$,如果$N+O(N^{1-1/r})$空间可以在增长和收缩操作期间短暂使用,那么$N+O(N^{1/r})$空间足以存储和访问大小为$N$的数组。通过索引访问项需要$O(1)$最坏情况时间,而增长和收缩操作需要$O(r)$平摊时间。通过对emph{增长博弈}的精确分析,我们表明,对于仅使用$N+O(N^{1/r})$空间来存储数组的大量数据结构中的任何数据结构,增长的平摊成本为$Omega(r)$,即使只允许增长和访问操作。增长和收缩操作的时间不能做最坏情况,除非$r=2$。
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引用次数: 1
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Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)
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