Limits of Local Search: Quality and Efficiency
Limits of Local Search: Quality and Efficiency
复制标题
本地搜索的局限性：质量和效率
DOI：
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发表时间：
2016
影响因子：
0.8
通讯作者：
Saurabh Ray
中科院分区：
文献类型：
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作者：
N. Bus;S. Garg;Nabil H. Mustafa;Saurabh Ray
在过去的几十年中12pt] {minimal} usepackage {amsmath} usepackage {wasysym} usepackage {amsfonts} $$ MATHCAL {d } $$ end {document}几何对象，计算pdocumentClass [12pt] {minimal} usepackage {amsmath} usepackage {wasySym {wasySym} usepackage} usepackage {amsfonts} amsfontpacky {amssmmb} {amsfontpacky {amsfontpagage {amssmb} {amssymb} } usepackage {Mathrsfs} usepackage {upgreek} setLength {oddSidemargin} { - 69pt} egin {document} $$ MATHCAL {d} $$ {d} $ end end eend {document {document {case} {Wasysym} usepackage {amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage} usepackage {upgreek} upgreek} setLength从Hochbaum的第二幅作品开始（Siam J Comput 11：555-556，1982），飞机上的磁盘在飞机上的磁盘终于在Mustafa和Ray中实现895，2010）。 12pt] {minimal} usepackage {amsmath} usepackage {wasysym} usepackage {amsfonts} $ $ k-1 $$ end {document}点;调用这样的算法a（k，k-1）documentclass [12pt] {minimal} usepackage {amsmath} usepackage {wasysym} useymym} usepackage} } use-package {upgreek} setLength {oddsidemargin} { - 69pt} egin {document} $$（k，k-k-1）$$ end eend {document} -local搜索算法。对于几何问题，用于几何独立集问题，主导集，地形保护问题以及其他几个算法的算法方法（对于几何独立集问题），不幸的是，所有这些算法都具有相同的限制：本地搜索能够给出PTAS，可以给出PTAS，但是尤其是在很大的运行时间。 {upgreek} setLength {oddSideMargin} { - 69pt} egin {document} $$ k ge 30 $$ end {document {document}，那么本地搜索能够给出一个恒定因素（作为k）近似值（fraser in in Fraser in不幸的是，几何覆盖和刺穿问题的算法都意味着，本地搜索完全正确工作的运行时间是ω（n30）documentClass [12pt] amsfonts} usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs}作为当前搜索是唯一可以在实践中提供可能有用的近似因素的方法，在效率和质量中探索局部搜索很重要。 2，1）本地搜索不能给出恒定的因子近似值，我们表明（3，2）本地搜索能够给出恒定的因素近似值； （3，2）的限制 - 局部搜索：该因子8近似值。对于几何独立设定的问题，用于统治的集合，对于地形守卫问题，其他几个问题（3，2） - 局部搜索算法就成为有效且优质算法的关键瓶颈。改进的算法。
Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set Ddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {D}$$end{document} of geometric objects, compute the minimum-sized subset of P that hits all objects in Ddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {D}$$end{document}. For the case where Ddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathcal {D}$$end{document} is a set of disks in the plane, the 30-year quest for a PTAS, starting from the seminal work of Hochbaum (SIAM J Comput 11:555–556, 1982), was finally achieved in Mustafa and Ray (Discret Comput Geom 44:883–895, 2010). Surprisingly, the algorithm to achieve the PTAS is simple: local-search. In particular, the algorithm starts with any hitting set, and iteratively tries to decrease its size by trying to replace some k points by k-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$k-1$$end{document} points; call such an algorithm a (k,k-1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(k, k-1)$$end{document}-local search algorithm. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. In particular, the current best work shows that if k≥30documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$k ge 30$$end{document}, then local-search is able to give a constant factor (as a function of k) approximation ratio (Fraser in Algorithms for Geometric Covering and Piercing Problems, 2012). Unfortunately this then implies that the running time for local-search to provably work at all is Ω(n30)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Omega (n^{30})$$end{document} using the current framework. As currently local search is the only known method that gives approximation factors that could be useful in practice, it becomes important to explore the limits—in both efficiency and quality—of local search. Simple examples show that (1, 0) and (2, 1) local search cannot give constant factor approximations. In this paper, we show that, surprisingly, just (3, 2) local search is able to give a constant-factor approximation; in fact we are able to get the precise quality limit of (3, 2)-local search: factor 8 approximation. This simplest working instance of local search already gives an approximation factor that is better than all known other methods! In fact, our improvement applies to all algorithms that use local-search for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others. Finding efficient (3, 2)-local search algorithms then becomes the key bottleneck in efficient and good-quality algorithms. In this paper we present such improved algorithms.