加权顶点覆盖的参数化近似算法

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science Pub Date : 2024-09-12 DOI:10.1016/j.tcs.2024.114870

Soumen Mandal , Pranabendu Misra , Ashutosh Rai , Saket Saurabh

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An <math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math>-bi-criteria approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover either produces a vertex cover S such that <math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≤</mo><mi>a</mi><mi>k</mi></math> and <math><mi>w</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>≤</mo><mi>b</mi><mi>W</mi></math>, or decides that there is no vertex cover of size at most k of weight at most W. We obtain the following results.<ul><li>•A simple <math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>-bi-criteria approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover in polynomial time by modifying the standard LP-rounding algorithm.</li><li>•A simple exact parameterized algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.4656</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> time1.</li><li>•A <math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>,</mo><mn>2</mn><mo>)</mo></math>-approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.4656</mn></mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo><mi>k</mi></mrow></msup><mo>)</mo></math> time.</li><li>•A <math><mo>(</mo><mn>1.5</mn><mo>,</mo><mn>1.5</mn><mo>)</mo></math>-approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.414</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> time.</li><li>•A <math><mo>(</mo><mn>2</mn><mo>−</mo><mi>δ</mi><mo>,</mo><mn>2</mn><mo>−</mo><mi>δ</mi><mo>)</mo></math>-approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mrow><mo>(</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mfrac><mrow><mi>δ</mi><mi>k</mi><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>δ</mi></mrow></mfrac></mrow><mrow><mfrac><mrow><mi>δ</mi><mi>k</mi><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi><mo>)</mo></mrow><mrow><mn>2</mn><mi>δ</mi></mrow></mfrac></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>δ</mi><mi>k</mi><mo>+</mo><mi>i</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>δ</mi><mi>k</mi><mo>−</mo><mfrac><mrow><mn>2</mn><mi>i</mi><mi>δ</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi></mrow></mfrac></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>)</mo></mrow></math> time for any <math><mi>δ</mi><mo><</mo><mn>0.5</mn></math>. For example, for <math><mo>(</mo><mn>1.75</mn><mo>,</mo><mn>1.75</mn><mo>)</mo></math> and <math><mo>(</mo><mn>1.9</mn><mo>,</mo><mn>1.9</mn><mo>)</mo></math>-approximation algorithms, we get running times of <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.272</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> and <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.151</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> respectively.</li></ul>Our algorithms (expectedly) do not improve upon the running times of the existing algorithms for the unweighted version of Vertex Cover. When compared to algorithms for the weighted version, our algorithms are the first ones to the best of our knowledge which work with arbitrary weights, and they perform well when the solution size is much smaller than the total weight of the desired solution.</div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114870"},"PeriodicalIF":0.9000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Parameterized approximation algorithms for weighted vertex cover\",\"authors\":\"Soumen Mandal , Pranabendu Misra , Ashutosh Rai , Saket Saurabh\",\"doi\":\"10.1016/j.tcs.2024.114870\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>A vertex cover of a graph is a set of vertices of the graph such that every edge has at least one endpoint in it. 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We obtain the following results.<ul><li>•A simple <math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>-bi-criteria approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover in polynomial time by modifying the standard LP-rounding algorithm.</li><li>•A simple exact parameterized algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.4656</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> time1.</li><li>•A <math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>,</mo><mn>2</mn><mo>)</mo></math>-approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.4656</mn></mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo><mi>k</mi></mrow></msup><mo>)</mo></math> time.</li><li>•A <math><mo>(</mo><mn>1.5</mn><mo>,</mo><mn>1.5</mn><mo>)</mo></math>-approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.414</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> time.</li><li>•A <math><mo>(</mo><mn>2</mn><mo>−</mo><mi>δ</mi><mo>,</mo><mn>2</mn><mo>−</mo><mi>δ</mi><mo>)</mo></math>-approximation algorithm for <math><mo>(</mo><mi>k</mi><mo>,</mo><mi>W</mi><mo>)</mo></math>-Vertex Cover running in <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mrow><mo>(</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mfrac><mrow><mi>δ</mi><mi>k</mi><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>δ</mi></mrow></mfrac></mrow><mrow><mfrac><mrow><mi>δ</mi><mi>k</mi><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi><mo>)</mo></mrow><mrow><mn>2</mn><mi>δ</mi></mrow></mfrac></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>δ</mi><mi>k</mi><mo>+</mo><mi>i</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>δ</mi><mi>k</mi><mo>−</mo><mfrac><mrow><mn>2</mn><mi>i</mi><mi>δ</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi></mrow></mfrac></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>)</mo></mrow></math> time for any <math><mi>δ</mi><mo><</mo><mn>0.5</mn></math>. For example, for <math><mo>(</mo><mn>1.75</mn><mo>,</mo><mn>1.75</mn><mo>)</mo></math> and <math><mo>(</mo><mn>1.9</mn><mo>,</mo><mn>1.9</mn><mo>)</mo></math>-approximation algorithms, we get running times of <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.272</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> and <math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.151</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math> respectively.</li></ul>Our algorithms (expectedly) do not improve upon the running times of the existing algorithms for the unweighted version of Vertex Cover. 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引用次数: 0

摘要

图的顶点覆盖是指图中每个边至少有一个端点的顶点集合。在这项工作中，我们研究的是以解决方案大小为参数的加权顶点覆盖问题。形式上，在(k,W)-顶点覆盖问题中，给定一个图 G、一个整数 k、一个正有理数 W 和一个权重函数 w:V(G)→Q+，问题是 G 是否有一个大小至多为 k、权重至多为 W（k 为参数）的顶点覆盖。(k,W)-Vertex Cover 的 (a,b)-bi-criteria 近似算法要么产生一个顶点覆盖 S，使得 |S|≤ak 且 w(S)≤bW，要么判定不存在大小至多为 k 权重至多为 W 的顶点覆盖。我们得到了以下结果。-运行于 O⁎(1.4656k) time1 的 (k,W)-Vertex Cover 的简单精确参数化算法。对于任意 δ<0.5，(k,W)-顶点覆盖的(2-δ,2-δ)-近似计算法在 O⁎(∑i=δk(1-2δ)1+2δδk(1-2δ)2δ(δk+iδk-2iδ1-2δ))时间内运行。例如，对于 (1.75,1.75) 和 (1.9,1.9) 近似算法，我们得到的运行时间分别为 O⁎(1.272k) 和 O⁎(1.151k)。与有权重版本的算法相比，我们的算法是我们所知的第一种可以使用任意权重的算法，而且在解的大小远小于所需解的总权重时，我们的算法表现良好。

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Parameterized approximation algorithms for weighted vertex cover

A vertex cover of a graph is a set of vertices of the graph such that every edge has at least one endpoint in it. In this work, we study Weighted Vertex Cover with solution size as a parameter. Formally, in the $(k, W)$ -Vertex Cover problem, given a graph G, an integer k, a positive rational W, and a weight function $w : V (G) \to Q^{+}$ , the question is whether G has a vertex cover of size at most k of weight at most W, with k being the parameter. An $(a, b)$ -bi-criteria approximation algorithm for $(k, W)$ -Vertex Cover either produces a vertex cover S such that $| S | \leq a k$ and $w (S) \leq b W$ , or decides that there is no vertex cover of size at most k of weight at most W. We obtain the following results.

•
A simple $(2, 2)$ -bi-criteria approximation algorithm for $(k, W)$ -Vertex Cover in polynomial time by modifying the standard LP-rounding algorithm.
•
A simple exact parameterized algorithm for $(k, W)$ -Vertex Cover running in $O^{⁎} ({1.4656}^{k})$ time¹.
•
A $(1 + ϵ, 2)$ -approximation algorithm for $(k, W)$ -Vertex Cover running in $O^{⁎} ({1.4656}^{(1 - ϵ) k})$ time.
•
A $(1.5, 1.5)$ -approximation algorithm for $(k, W)$ -Vertex Cover running in $O^{⁎} ({1.414}^{k})$ time.
•
A $(2 - δ, 2 - δ)$ -approximation algorithm for $(k, W)$ -Vertex Cover running in $O^{⁎} (\sum_{i = \frac{δ k (1 - 2 δ)}{1 + 2 δ}}^{\frac{δ k (1 - 2 δ)}{2 δ}} (\begin{matrix} δ k + i \\ δ k - \frac{2 i δ}{1 - 2 δ} \end{matrix}))$ time for any $δ < 0.5$ . For example, for $(1.75, 1.75)$ and $(1.9, 1.9)$ -approximation algorithms, we get running times of $O^{⁎} ({1.272}^{k})$ and $O^{⁎} ({1.151}^{k})$ respectively.

Our algorithms (expectedly) do not improve upon the running times of the existing algorithms for the unweighted version of Vertex Cover. When compared to algorithms for the weighted version, our algorithms are the first ones to the best of our knowledge which work with arbitrary weights, and they perform well when the solution size is much smaller than the total weight of the desired solution.

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来源期刊

Theoretical Computer Science 工程技术-计算机：理论方法

CiteScore

2.60

自引率

18.20%

发文量

471

审稿时长

12.6 months

期刊介绍： Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.