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On circuit diameter bounds via circuit imbalances 通过电路失衡论电路直径边界
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-08 DOI: 10.1007/s10107-024-02107-x
Daniel Dadush, Zhuan Khye Koh, Bento Natura, László A. Végh

We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIAM J. Discrete Math. 29(1), 113–121 (2015)) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system ({xin mathbb {R}^n:, Ax=b,, mathbb {0}le xle u}) for (Ain mathbb {R}^{mtimes n}) is bounded by (O(mmin {m, n - m}log (m+kappa _A)+nlog n)), where (kappa _A) is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., all entries of A have polynomially bounded encoding length in n. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in (O(mn^2log (n+kappa _A))) augmentation steps.

我们研究由 Borgwardt、Finhold 和 Hemmecke(SIAM J. Discrete Math.29(1), 113-121 (2015))作为组合直径的放松而引入的。我们证明了一个系统的电路直径({xin mathbb {R}^n:Ax=b,, mathbb {0}le xle u}) for (Ain mathbb {R}^{mtimes n}) is bounded by (O(mmin {m, n - m}log (m+kappa _A)+nlog n)), where (kappa _A) is the circuit imbalance measure of the constraint matrix.如果 A 的所有条目在 n 中的编码长度都是多项式约束的,那么这就产生了一个强多项式电路直径约束。尽管标准的最小比值电路消除算法在一般情况下不是有限的,但我们的变体可以在(O(mn^2log (n+kappa _A))增强步骤内解决 LP。
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引用次数: 0
Monoidal strengthening and unique lifting in MIQCPs MIQCP 中的单值加强和唯一提升
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-08 DOI: 10.1007/s10107-024-02112-0
Antonia Chmiela, Gonzalo Muñoz, Felipe Serrano

Using the recently proposed maximal quadratic-free sets and the well-known monoidal strengthening procedure, we show how to improve intersection cuts for quadratically-constrained optimization problems by exploiting integrality requirements. We provide an explicit construction that allows an efficient implementation of the strengthened cuts along with computational results showing their improvements over the standard intersection cuts. We also show that, in our setting, there is unique lifting which implies that our strengthening procedure is generating the best possible cut coefficients for the integer variables.

我们利用最近提出的最大无二次型集合和著名的单环加强程序,展示了如何通过利用积分性要求来改进二次型约束优化问题的交叉切分。我们提供了一种明确的构造,可以高效地实现强化切分,同时还提供了计算结果,显示了强化切分与标准交集切分相比的改进。我们还证明,在我们的设置中,存在唯一的提升,这意味着我们的强化程序正在为整数变量生成最佳的切分系数。
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引用次数: 0
Optimal methods for convex nested stochastic composite optimization 凸嵌套随机复合优化的最优方法
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-05 DOI: 10.1007/s10107-024-02090-3
Zhe Zhang, Guanghui Lan

Recently, convex nested stochastic composite optimization (NSCO) has received considerable interest for its applications in reinforcement learning and risk-averse optimization. However, existing NSCO algorithms have worse stochastic oracle complexities, by orders of magnitude, than those for simpler stochastic optimization problems without nested structures. Additionally, these algorithms require all outer-layer functions to be smooth, a condition violated by some important applications. This raises a question regarding whether the nested composition make stochastic optimization more difficult in terms of oracle complexity. In this paper, we answer the question by developing order-optimal algorithms for convex NSCO problems constructed from an arbitrary composition of smooth, structured non-smooth, and general non-smooth layer functions. When all outer-layer functions are smooth, we propose a stochastic sequential dual (SSD) method to achieve an oracle complexity of (mathcal {O}(1/epsilon ^2)) (resp., (mathcal {O}(1/epsilon ))) when the problem is convex (resp., strongly convex). If any outer-layer function is non-smooth, we propose a non-smooth stochastic sequential dual (nSSD) method to achieve an (mathcal {O}(1/epsilon ^2)) oracle complexity. We provide a lower complexity bound to show the latter (mathcal {O}(1/epsilon ^2)) complexity to be unimprovable, even under a strongly convex setting. All these complexity results seem to be new in the literature, and they indicate that convex NSCO problems have the same order of oracle complexity as problems without nested composition, except in the strongly convex and outer non-smooth cases.

最近,凸嵌套随机复合优化(NSCO)因其在强化学习和风险规避优化中的应用而受到广泛关注。然而,现有的 NSCO 算法的随机甲骨文复杂度比那些没有嵌套结构的简单随机优化问题的复杂度要差,差了几个数量级。此外,这些算法要求所有外层函数都是平滑的,而一些重要应用却违反了这一条件。这就提出了一个问题:嵌套结构是否会增加随机优化的甲骨文复杂度?在本文中,我们通过开发由光滑层函数、结构非光滑层函数和一般非光滑层函数任意组成的凸 NSCO 问题的阶优算法来回答这个问题。当所有外层函数都是平滑的时候,我们提出了一种随机顺序对偶(SSD)方法,当问题是凸的(或者说,强凸的)时,它的oracle复杂度为(mathcal {O}(1/epsilon ^2))(或者说,(mathcal {O}(1/epsilon )))。如果任何外层函数是非光滑的,我们提出了一种非光滑随机顺序对偶(nSSD)方法,以实现 (mathcal {O}(1/epsilon ^2)) oracle 复杂性。我们提供了一个较低的复杂度约束,以证明后者的 (mathcal {O}(1/epsilon ^2))复杂度是不可改进的,即使在强凸设置下也是如此。所有这些复杂度结果在文献中似乎都是新的,它们表明,除了强凸和外部非光滑情况外,凸NSCO问题与没有嵌套组合的问题具有相同的oracle复杂度阶数。
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引用次数: 0
On the directional asymptotic approach in optimization theory 论优化理论中的定向渐近方法
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-05 DOI: 10.1007/s10107-024-02089-w
Matúš Benko, Patrick Mehlitz

As a starting point of our research, we show that, for a fixed order (gamma ge 1), each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order 1), satisfies stationarity conditions in terms of a coderivative construction of order (gamma ), or is asymptotically stationary with respect to a critical direction as well as order (gamma ) in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of orders 1 and (gamma ). These abstract findings are carved out for the broad class of geometric constraints and (gamma :=2), and visualized by examples from complementarity-constrained and nonlinear semidefinite optimization. As a byproduct of the particular setting (gamma :=1), our general approach yields new so-called directional asymptotic regularity conditions which serve as constraint qualifications guaranteeing M-stationarity of local minimizers. We compare these new regularity conditions with standard constraint qualifications from nonsmooth optimization. Further, we extend directional concepts of pseudo- and quasi-normality to arbitrary set-valued mappings. It is shown that these properties provide sufficient conditions for the validity of directional asymptotic regularity. Finally, a novel coderivative-like variational tool is used to construct sufficient conditions for the presence of directional asymptotic regularity. For geometric constraints, it is illustrated that all appearing objects can be calculated in terms of initial problem data.

作为我们研究的起点,我们证明了对于一个固定的阶(gamma),欧几里得空间中一个相当一般的非光滑优化问题的每个局部最小值要么是经典意义上的M静止(对应于阶1的静止)、满足阶 (gamma ) 的编码构造的静止条件,或者相对于临界方向以及阶 (gamma ) 在一定意义上是渐近静止的。通过用不强于方向度量次规则性的约束条件排除后一种情况,我们最终得到了新的必要最优性条件,包括阶1和阶(gamma )的极限变分工具的混合物。这些抽象发现是针对广泛的几何约束和 (gamma :=2) 类而提出的,并通过来自互补约束和非线性半有限优化的例子加以形象化。作为 (gamma :=1) 这一特殊设置的副产品,我们的一般方法产生了新的所谓方向渐近正则性条件,作为保证局部最小化 M-stationarity 的约束条件。我们将这些新的正则性条件与非光滑优化的标准约束条件进行了比较。此外,我们还将伪正则和准正则的方向性概念扩展到了任意的集值映射。结果表明,这些性质为方向渐近正则性的有效性提供了充分条件。最后,利用一种新颖的类似于 coderivative 的变分工具来构建方向渐近正则性存在的充分条件。对于几何约束来说,所有出现的对象都可以通过初始问题数据计算出来。
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引用次数: 0
Matrix discrepancy and the log-rank conjecture 矩阵差异和对数秩猜想
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-05 DOI: 10.1007/s10107-024-02117-9
Benny Sudakov, István Tomon

Given an (mtimes n) binary matrix M with (|M|=pcdot mn) (where |M| denotes the number of 1 entries), define the discrepancy of M as ({{,textrm{disc},}}(M)=displaystyle max nolimits _{Xsubset [m], Ysubset [n]}big ||M[Xtimes Y]|-p|X|cdot |Y|big |). Using semidefinite programming and spectral techniques, we prove that if ({{,textrm{rank},}}(M)le r) and (ple 1/2), then

$$begin{aligned}{{,textrm{disc},}}(M)ge Omega (mn)cdot min left{ p,frac{p^{1/2}}{sqrt{r}}right} .end{aligned}$$

We use this result to obtain a modest improvement of Lovett’s best known upper bound on the log-rank conjecture. We prove that any (mtimes n) binary matrix M of rank at most r contains an ((mcdot 2^{-O(sqrt{r})})times (ncdot 2^{-O(sqrt{r})})) sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank r is at most (O(sqrt{r})).

给定一个 (mtimes n) 二进制矩阵 M,其 (|M|=pcdot mn) (其中 |M| 表示 1 条目的数量),定义 M 的差异为 ({{、(M)=displaystyle max nolimits _{Xsubset [m], Ysubset [n]}big |||M[Xtimes Y]|-p|X|cdot |Y|big |)。利用半定量编程和谱技术,我们证明如果({{,textrm{rank},}}(M)le r) and(ple 1/2)、then $$begin{aligned}{{,textrm{disc},}}(M)ge Omega (mn)cdot min left{ p,frac{p^{1/2}}{sqrt{r}}right} .end{aligned}$$我们利用这个结果对洛维特最著名的对数秩猜想的上界进行了适度的改进。我们证明了任何秩为 r 的二进制矩阵 M 都包含一个 ((mcdot 2^{-O(sqrt{r})})times (ncdot 2^{-O(sqrt{r})})) 大小的 all-1 或 all-0 子矩阵,这意味着任何秩为 r 的布尔函数的确定性通信复杂度最多为 (O(sqrt{r}))。
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引用次数: 0
On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models 通过等效因子化列稀疏正则化模型求解秩正则化最小化问题
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-03 DOI: 10.1007/s10107-024-02103-1
Wenjing Li, Wei Bian, Kim-Chuan Toh

Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column-sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column-sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its nonconvex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.

秩正则化最小化问题是低秩矩阵补全/恢复问题的理想模型。矩阵因子化方法可以将高维秩正则化问题转化为低维因子化列稀疏正则化问题。后者可以极大地促进适用算法的快速计算,但需要同时克服损失函数和正则化函数的非凸性。本文考虑因子化列稀疏正则化模型。首先,我们对该模型进行了带约束条件的优化,并在优化后的因子化问题和秩正则化问题之间建立了一定的等价关系。此外,我们还强化了因式分解问题静止点的最优性条件,并定义了强静止点的概念。此外,我们还在全局最小值和强驻点的意义上建立了因式分解问题与其非凸松弛之间的等价性。为了解决因式分解问题,我们设计了两类算法,并给出了减少其计算量的自适应方法。第一种算法是从松弛的角度出发,它的迭代拥有有限迭代后因式分解问题全局最小值的一些特性。我们对其迭代数收敛到强静止点进行了一些分析。第二种算法是为直接求解因式分解问题而设计的。我们针对因式分解问题改进了 Bolte 等人(Math Program Ser A 146:459-494, 2014)介绍的 PALM 算法,并给出了其改进后的收敛结果。最后,我们进行了数值实验,展示了所提出的低秩矩阵补全模型和算法的良好性能。
{"title":"On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models","authors":"Wenjing Li, Wei Bian, Kim-Chuan Toh","doi":"10.1007/s10107-024-02103-1","DOIUrl":"https://doi.org/10.1007/s10107-024-02103-1","url":null,"abstract":"<p>Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column-sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column-sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its nonconvex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"46 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints 论具有三个实数或四个复数同质约束条件的同质 QCQP 的 SDP 松弛的紧密性
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-21 DOI: 10.1007/s10107-024-02105-z
Wenbao Ai, Wei Liang, Jianhua Yuan

In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and necessary test condition to detect whether its standard semi-definite programming (SDP) relaxation is tight or not. This test condition is based on only an optimal solution pair of the SDP relaxation and its dual. When the tightness is confirmed, a global optimal solution of the original problem is found simultaneously in polynomial-time. While the tightness does not hold, the SDP relaxation and its dual are proved to have the unique optimal solutions. Moreover, the Lagrangian version of such the test condition is specified for non-homogeneous cases. Based on the Lagrangian version, it is proved that several latest sufficient conditions to test the SDP tightness are contained by our test condition under the situation of two constraints. Thirdly, as an application of the test condition, S-lemma and Yuan’s lemma are generalized to three real and four complex quadratic forms first under certain exact conditions, which improves some classical results in literature. Finally, a counterexample is presented to show that the test condition cannot be simply extended to four real or five complex homogeneous quadratic constraints.

在本文中,我们考虑的问题是在三个实数或四个复数同质二次函数不等式或相等约束条件下,最小化一个一般同质二次函数。对于这个问题,我们提出了一个充分且必要的检验条件,以检测其标准半有限编程(SDP)松弛是否紧密。该检验条件仅基于 SDP 松弛及其对偶的最优解对。当紧密性得到确认时,就能同时在多项式时间内找到原始问题的全局最优解。当严密性不成立时,则证明 SDP 松弛及其对偶具有唯一最优解。此外,还为非均质情况指定了这种测试条件的拉格朗日版本。基于拉格朗日版本,证明了在两个约束的情况下,我们的检验条件包含了检验 SDP 紧缩性的几个最新充分条件。第三,作为检验条件的应用,在一定的精确条件下,S-lemma 和 Yuan's lemma 首先被推广到三实四复二次型,从而改进了文献中的一些经典结果。最后,提出了一个反例,说明检验条件不能简单地扩展到四实数或五复数同质二次约束。
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引用次数: 0
A slope generalization of Attouch theorem 阿图什定理的斜率一般化
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-20 DOI: 10.1007/s10107-024-02108-w
Aris Daniilidis, David Salas, Sebastián Tapia-García

A classical result of variational analysis, known as Attouch theorem, establishes an equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we show that in finite dimensions and under a mild boundedness assumption, we can replace subdifferentials (sets of vectors) by slopes (scalars, corresponding to the distance of the subdifferentials to zero) and still obtain the same characterization: namely, the epigraphical convergence of functions is equivalent to the epigraphical convergence of their slopes. This surprising result goes in line with recent developments on slope determination (Boulmezaoud et al. in SIAM J Optim 28(3):2049–2066, 2018; Pérez-Aros et al. in Math Program 190(1–2):561-583, 2021) and slope sensitivity (Daniilidis and Drusvyatskiy in Proc Am Math Soc 151(11):4751-4756, 2023) for convex functions.

变分分析的一个经典结果,即阿图什(Attouch)定理,确定了适当凸下半连续函数序列的表观收敛性与相应次微分映射的图形收敛性之间的等价性,但须满足一个固定积分常数的归一化条件。在这项工作中,我们证明了在有限维度和温和的有界性假设下,我们可以用斜率(标量,对应于子微分到零的距离)替换子微分(向量集),并仍然得到相同的特征:即函数的图解收敛等同于其斜率的图解收敛。这一令人惊讶的结果与凸函数的斜率确定(Boulmezaoud 等人,发表于 SIAM J Optim 28(3):2049-2066, 2018;Pérez-Aros 等人,发表于 Math Program 190(1-2):561-583, 2021)和斜率敏感性(Daniilidis 和 Drusvyatskiy,发表于 Proc Am Math Soc 151(11):4751-4756, 2023)的最新进展一致。
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引用次数: 0
Generalized scaling for the constrained maximum-entropy sampling problem 受限最大熵抽样问题的广义缩放
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-20 DOI: 10.1007/s10107-024-02101-3
Zhongzhu Chen, Marcia Fampa, Jon Lee

The best practical techniques for exact solution of instances of the constrained maximum-entropy sampling problem, a discrete-optimization problem arising in the design of experiments, are via a branch-and-bound framework, working with a variety of concave continuous relaxations of the objective function. A standard and computationally-important bound-enhancement technique in this context is (ordinary) scaling, via a single positive parameter. Scaling adjusts the shape of continuous relaxations to reduce the gaps between the upper bounds and the optimal value. We extend this technique to generalized scaling, employing a positive vector of parameters, which allows much more flexibility and thus potentially reduces the gaps further. We give mathematical results aimed at supporting algorithmic methods for computing optimal generalized scalings, and we give computational results demonstrating the performance of generalized scaling on benchmark problem instances.

受限最大熵采样问题是实验设计中出现的离散优化问题,精确求解该问题实例的最佳实用技术是通过分支与边界框架,利用目标函数的各种凹连续松弛来实现的。在这种情况下,一种标准的、在计算上非常重要的边界增强技术是通过单个正参数进行(普通)缩放。缩放可以调整连续松弛的形状,从而缩小上限与最优值之间的差距。我们将这一技术扩展到广义缩放,即采用一个正向参数向量,这样就有了更大的灵活性,从而有可能进一步缩小差距。我们给出的数学结果旨在支持计算最优广义缩放的算法方法,我们给出的计算结果证明了广义缩放在基准问题实例上的性能。
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引用次数: 0
A PTAS for the horizontal rectangle stabbing problem 水平矩形刺入问题的 PTAS
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-13 DOI: 10.1007/s10107-024-02106-y
Arindam Khan, Aditya Subramanian, Andreas Wiese

We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a ((2+varepsilon ))-approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are (delta )-large, that is, have at least one edge whose length is at least (delta ), while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.

我们研究的是矩形切割问题,在这个问题中,我们要切割的是平面上 n 个轴线对齐的矩形,也就是说,我们要选择线段,使得每个矩形都有一条线段与它的两条相对边相交。在水平矩形刺入问题(刺入)中,我们的目标是找到一组总长度最小的水平线段,从而刺入所有矩形。在水平-垂直刺入问题(HV-Stabbing)中,目标是找到一组总长度最小的直线(即垂直或水平)线段,使所有矩形都被刺入。这两个变体都是 NP 难。Chan 等人(ISAAC,2018)通过提供恒定近似算法,开始了对这些问题的研究。最近,Eisenbrand 等人(A QPTAS for stabbing rectangles, 2021)提出了针对 Stabbing 问题的 QPTAS 和多项式时间 8 近似算法,但该问题是否存在 PTAS 尚无定论。在本文中,我们得到了 Stabbing 的 PTAS,从而解决了这个问题。对于HV-Stabbing,我们得到了一个((2+varepsilon ))近似值。我们还得到了 HV-Stabbing 特殊情况下的 PTAS:(i) 所有矩形都是正方形,(ii) 每个矩形的宽度最多等于它的高度,(iii) 所有矩形都是(Δ )大的,也就是说,至少有一条边的长度至少是(Δ ),而所有边的长度最多是 1。 我们的结果还意味着对其他问题的近似值的改进,比如广义最小曼哈顿网络。
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引用次数: 0
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Mathematical Programming
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