On the concentration of the number of solutions of random satisfiability formulas. Issue 3 (12th April 2013)

• Record Type: Journal Article
• Authors: Abbe, Emmanuel; Montanari, Andrea
• Is Part Of: Random structures & algorithms. Volume 45:Issue 3(2014)
• Page Start: 362
• Page End: 382
• Publication Date: 2013-04-12
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20501
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>Let <italic>Z</italic>(<italic>F</italic>) be the number of solutions of a random <italic>k</italic>‐satisfiability formula <italic>F</italic> with <italic>n</italic> variables and clause density <italic>α</italic>. Assume that the probability that <italic>F</italic> is unsatisfiable is <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9b5np" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20501:rsa20501-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>log</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>δ</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> for some <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9b5qs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20501:rsa20501-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>δ</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. We show that (possibly excluding a countable set of "exceptional" <italic>α</italic>'s) the number of solutions concentrates, i.e., there exists a non‐random function <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9b5tf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20501:rsa20501-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>α</mml:mo><mml:mo>↦</mml:mo><mml:msub><mml:mo>ϕ</mml:mo><mml:mi>s</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mo>α</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> such that, for any <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9b5v0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20501:rsa20501-math-0004" 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>, we have <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9b5zn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20501:rsa20501-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mo>ϕ</mml:mo><mml:mi>s</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mo>ε</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo>, </mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mo>ϕ</mml:mo><mml:mi>s</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mo>ε</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> with high probability. In particular, the assumption holds for all <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9b618" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20501:rsa20501-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>α</mml:mo><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>, which proves the above concentration claim in the whole satisfiability regime of random 2‐SAT. We also extend these results to a broad class of constraint satisfaction problems. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 362–382, 2014</p> </abstract>