Recoverable values for independent sets. Issue 1 (5th March 2013)

• Record Type: Journal Article
• Authors: Feige, Uriel; Reichman, Daniel
• Is Part Of: Random structures & algorithms. Volume 46:Issue 1(2015)
• Page Start: 142
• Page End: 159
• Publication Date: 2013-03-05
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20492
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The notion of <italic>recoverable value</italic> was advocated in the work of Feige, Immorlica, Mirrokni and Nazerzadeh (APPROX 2009) as a measure of quality for approximation algorithms. There, this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has <italic>recoverable factor ρ</italic>, if for every graph it recovers an independent set of size at least <disp-formula content-type="mathematics" id="rsa20492-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcq6q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10429832:media:rsa20492:rsa20492-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mi>max</mml:mi></mml:mrow><mml:mi>I</mml:mi></mml:msub><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mi>min</mml:mi></mml:mrow></mml:mstyle><mml:mrow><mml:mo stretchy="true">[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>, </mml:mo><mml:mfrac><mml:mo>ρ</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow><mml:mo stretchy="true">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula> where <italic>d</italic>(<italic>v</italic>) is the degree of vertex <italic>v</italic>, and <italic>I</italic> ranges over all independent sets in <italic>G</italic>. Hence, in a sense, from every vertex <italic>v</italic> in the maximum independent set the algorithm recovers a value of at least <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcq4m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20492:rsa20492-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ρ</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> toward the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. A simple greedy algorithm achieves <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcqfk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20492:rsa20492-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ρ</mml:mo><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. We design a new randomized algorithm for MIS that ensures an expected recoverable factor of at least <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcq9c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20492:rsa20492-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ρ</mml:mo><mml:mo>≥</mml:mo><mml:mn>7</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. In passing, we prove that approximating MIS in graphs with a given <italic>k</italic>‐coloring within a ratio larger than 2/ <italic>k</italic> is unique‐games hard. This rules out an alternative approach for obtaining <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rcqks" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20492:rsa20492-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ρ</mml:mo><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 142–159, 2015</p> </abstract>