On independent sets in random graphs1. Issue 3 (30th May 2014)

• Record Type: Journal Article
• Authors: Coja‐Oghlan, Amin; Efthymiou, Charilaos
• Is Part Of: Random structures & algorithms. Volume 47:Issue 3(2015)
• Page Start: 436
• Page End: 486
• Publication Date: 2014-05-30
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20550
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The independence number of a sparse random graph <italic>G</italic>(<italic>n, m</italic>) of average degree <italic>d</italic> = 2<italic>m</italic>/<italic>n</italic> is well‐known to be <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9ts4cz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20550:rsa20550-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mo>ε</mml:mo><mml:mi>d</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>n</mml:mi><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>d</mml:mi><mml:mo>≤</mml:mo><mml:mo>α</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mo>ε</mml:mo><mml:mi>d</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi>n</mml:mi><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> with high probability, with <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9ts4f2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20550:rsa20550-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mo>ε</mml:mo><mml:mi>d</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> in the limit of large <italic>d</italic>. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9ts4gm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20550:rsa20550-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>, i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9ts4jq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20550:rsa20550-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>ε</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mi>n</mml:mi><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> for any <italic>fixed</italic><inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9ts4k8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20550:rsa20550-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ε</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> (independent of both <italic>d</italic> and <italic>n</italic>). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size <italic>k</italic> of the independent sets passes the point <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9ts4qg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20550:rsa20550-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∼</mml:mo><mml:mi>n</mml:mi><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>. Roughly speaking, we prove that independent sets of size <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9ts4nc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20550:rsa20550-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>ε</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mi>n</mml:mi><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 436–486, 2015</p> </abstract>