两个独立厄尔多斯-雷尼图最大重叠的多项式时间近似方案

Random Structures and Algorithms Pub Date : 2024-03-07 DOI:10.1002/rsa.21212

Jian Ding, Hang Du, Shuyang Gong

{"title":"两个独立厄尔多斯-雷尼图最大重叠的多项式时间近似方案","authors":"Jian Ding, Hang Du, Shuyang Gong","doi":"10.1002/rsa.21212","DOIUrl":null,"url":null,"abstract":"For two independent Erdős–Rényi graphs <mjx-container aria-label=\"bold upper G left parenthesis n comma p right parenthesis\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,7\" data-semantic-content=\"8,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"bold upper G left parenthesis n comma p right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"bold\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"9\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"9\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"6\" data-semantic-content=\"1,5\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"7\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"2,3,4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"6\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"7\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/2faa26af-1d72-4b99-9369-f9fefb9bf168/rsa21212-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,7\" data-semantic-content=\"8,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"bold upper G left parenthesis n comma p right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"bold\" data-semantic-operator=\"appl\" data-semantic-parent=\"9\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\" mathvariant=\"bold\">G</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"9\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\">⁡</mo><mrow data-semantic-=\"\" data-semantic-children=\"6\" data-semantic-content=\"1,5\" data-semantic-parent=\"9\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"7\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mrow data-semantic-=\"\" data-semantic-children=\"2,3,4\" data-semantic-content=\"3\" data-semantic-parent=\"7\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"6\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">p</mi></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"7\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$$ \\mathbf{G}\\left(n,p\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, we study the maximal overlap (i.e., the number of common edges) of these two graphs over all possible vertex correspondence. We present a polynomial-time algorithm which finds a vertex correspondence whose overlap approximates the maximal overlap up to a multiplicative factor that is arbitrarily close to 1. As a by-product, we prove that the maximal overlap is asymptotically <mjx-container aria-label=\"StartFraction n Over 2 alpha minus 1 EndFraction\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mfrac data-semantic-children=\"0,7\" data-semantic- data-semantic-role=\"division\" data-semantic-speech=\"StartFraction n Over 2 alpha minus 1 EndFraction\" data-semantic-type=\"fraction\"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow data-semantic-children=\"6,4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"subtraction\" data-semantic-type=\"infixop\" size=\"s\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"5\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"6\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"7\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/1e243f5a-74ad-471b-b6ab-5e5cdad937f6/rsa21212-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mfrac data-semantic-=\"\" data-semantic-children=\"0,7\" data-semantic-role=\"division\" data-semantic-speech=\"StartFraction n Over 2 alpha minus 1 EndFraction\" data-semantic-type=\"fraction\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mrow data-semantic-=\"\" data-semantic-children=\"6,4\" data-semantic-content=\"3\" data-semantic-parent=\"8\" data-semantic-role=\"subtraction\" data-semantic-type=\"infixop\"><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"1,2\" data-semantic-content=\"5\" data-semantic-parent=\"7\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,⁢\" data-semantic-parent=\"6\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\">⁢</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">α</mi></mrow><mo data-semantic-=\"\" data-semantic-operator=\"infixop,−\" data-semantic-parent=\"7\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" form=\"prefix\">−</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></mrow></mfrac></mrow>$$ \\frac{n}{2\\alpha -1} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> for <mjx-container aria-label=\"p equals n Superscript negative alpha\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,6\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"p equals n Superscript negative alpha\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"7\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\"2,5\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\" size=\"s\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/fd33064b-fec1-487b-a93d-bb698834109e/rsa21212-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,6\" data-semantic-content=\"1\" data-semantic-role=\"equality\" data-semantic-speech=\"p equals n Superscript negative alpha\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">p</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"7\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><msup data-semantic-=\"\" data-semantic-children=\"2,5\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"superscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4\" data-semantic-content=\"3\" data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" form=\"prefix\">−</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">α</mi></mrow></msup></mrow>$$ p={n}^{-\\alpha } $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> with some constant <mjx-container aria-label=\"alpha element of left parenthesis 1 divided by 2 comma 1 right parenthesis\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,11\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"element\" data-semantic-speech=\"alpha element of left parenthesis 1 divided by 2 comma 1 right parenthesis\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"infixop,∈\" data-semantic-parent=\"12\" data-semantic-role=\"element\" data-semantic-type=\"operator\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"10\" data-semantic-content=\"2,8\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"11\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"9,6,7\" data-semantic-content=\"6\" data-semantic- data-semantic-parent=\"11\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mjx-mrow data-semantic-children=\"3,5\" data-semantic-content=\"4\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"9\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"10\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"11\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/493558b9-2d91-4205-b21f-f1385aaafe7e/rsa21212-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,11\" data-semantic-content=\"1\" data-semantic-role=\"element\" data-semantic-speech=\"alpha element of left parenthesis 1 divided by 2 comma 1 right parenthesis\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"12\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">α</mi><mo data-semantic-=\"\" data-semantic-operator=\"infixop,∈\" data-semantic-parent=\"12\" data-semantic-role=\"element\" data-semantic-type=\"operator\">∈</mo><mrow data-semantic-=\"\" data-semantic-children=\"10\" data-semantic-content=\"2,8\" data-semantic-parent=\"12\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"11\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mrow data-semantic-=\"\" data-semantic-children=\"9,6,7\" data-semantic-content=\"6\" data-semantic-parent=\"11\" data-semantic-role=\"sequence\" data-semantic-type=\"punctuated\"><mrow data-semantic-=\"\" data-semantic-children=\"3,5\" data-semantic-content=\"4\" data-semantic-parent=\"10\" data-semantic-role=\"division\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"9\" data-semantic-role=\"division\" data-semantic-type=\"operator\" stretchy=\"false\">/</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"9\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"10\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"10\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"11\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$$ \\alpha \\in \\left(1/2,1\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"25 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A polynomial-time approximation scheme for the maximal overlap of two independent Erdős–Rényi graphs\",\"authors\":\"Jian Ding, Hang Du, Shuyang Gong\",\"doi\":\"10.1002/rsa.21212\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For two independent Erdős–Rényi graphs <mjx-container aria-label=\\\"bold upper G left parenthesis n comma p right parenthesis\\\" ctxtmenu_counter=\\\"0\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,7\\\" data-semantic-content=\\\"8,0\\\" data-semantic- data-semantic-role=\\\"simple function\\\" data-semantic-speech=\\\"bold upper G left parenthesis n comma p right parenthesis\\\" data-semantic-type=\\\"appl\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"bold\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"6\\\" data-semantic-content=\\\"1,5\\\" data-semantic- data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"2,3,4\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\" rspace=\\\"3\\\" style=\\\"margin-left: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/2faa26af-1d72-4b99-9369-f9fefb9bf168/rsa21212-math-0001.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,7\\\" data-semantic-content=\\\"8,0\\\" data-semantic-role=\\\"simple function\\\" data-semantic-speech=\\\"bold upper G left parenthesis n comma p right parenthesis\\\" data-semantic-type=\\\"appl\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"bold\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"simple function\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"bold\\\">G</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"appl\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"application\\\" data-semantic-type=\\\"punctuation\\\">⁡</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"6\\\" data-semantic-content=\\\"1,5\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">(</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"2,3,4\\\" data-semantic-content=\\\"3\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\">,</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">p</mi></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">)</mo></mrow></mrow>$$ \\\\mathbf{G}\\\\left(n,p\\\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, we study the maximal overlap (i.e., the number of common edges) of these two graphs over all possible vertex correspondence. We present a polynomial-time algorithm which finds a vertex correspondence whose overlap approximates the maximal overlap up to a multiplicative factor that is arbitrarily close to 1. As a by-product, we prove that the maximal overlap is asymptotically <mjx-container aria-label=\\\"StartFraction n Over 2 alpha minus 1 EndFraction\\\" ctxtmenu_counter=\\\"1\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow><mjx-mfrac data-semantic-children=\\\"0,7\\\" data-semantic- data-semantic-role=\\\"division\\\" data-semantic-speech=\\\"StartFraction n Over 2 alpha minus 1 EndFraction\\\" data-semantic-type=\\\"fraction\\\"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size=\\\"s\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow data-semantic-children=\\\"6,4\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"infixop\\\" size=\\\"s\\\"><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple;clearspeak:unit\\\" data-semantic-children=\\\"1,2\\\" data-semantic-content=\\\"5\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,−\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/1e243f5a-74ad-471b-b6ab-5e5cdad937f6/rsa21212-math-0002.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mfrac data-semantic-=\\\"\\\" data-semantic-children=\\\"0,7\\\" data-semantic-role=\\\"division\\\" data-semantic-speech=\\\"StartFraction n Over 2 alpha minus 1 EndFraction\\\" data-semantic-type=\\\"fraction\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"6,4\\\" data-semantic-content=\\\"3\\\" data-semantic-parent=\\\"8\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"infixop\\\"><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple;clearspeak:unit\\\" data-semantic-children=\\\"1,2\\\" data-semantic-content=\\\"5\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,⁢\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\">⁢</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\">α</mi></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,−\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" form=\\\"prefix\\\">−</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">1</mn></mrow></mfrac></mrow>$$ \\\\frac{n}{2\\\\alpha -1} $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> for <mjx-container aria-label=\\\"p equals n Superscript negative alpha\\\" ctxtmenu_counter=\\\"2\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,6\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"equality\\\" data-semantic-speech=\\\"p equals n Superscript negative alpha\\\" data-semantic-type=\\\"relseq\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"relseq,=\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"equality\\\" data-semantic-type=\\\"relation\\\" rspace=\\\"5\\\" space=\\\"5\\\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\\\"2,5\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"superscript\\\"><mjx-mrow><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\\\"vertical-align: 0.363em;\\\"><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"negative\\\" data-semantic-type=\\\"prefixop\\\" size=\\\"s\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"prefixop,−\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/fd33064b-fec1-487b-a93d-bb698834109e/rsa21212-math-0003.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,6\\\" data-semantic-content=\\\"1\\\" data-semantic-role=\\\"equality\\\" data-semantic-speech=\\\"p equals n Superscript negative alpha\\\" data-semantic-type=\\\"relseq\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">p</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"relseq,=\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"equality\\\" data-semantic-type=\\\"relation\\\">=</mo><msup data-semantic-=\\\"\\\" data-semantic-children=\\\"2,5\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"superscript\\\"><mrow><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">n</mi></mrow><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"negative\\\" data-semantic-type=\\\"prefixop\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"prefixop,−\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" form=\\\"prefix\\\">−</mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\">α</mi></mrow></msup></mrow>$$ p={n}^{-\\\\alpha } $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> with some constant <mjx-container aria-label=\\\"alpha element of left parenthesis 1 divided by 2 comma 1 right parenthesis\\\" ctxtmenu_counter=\\\"3\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"0,11\\\" data-semantic-content=\\\"1\\\" data-semantic- data-semantic-role=\\\"element\\\" data-semantic-speech=\\\"alpha element of left parenthesis 1 divided by 2 comma 1 right parenthesis\\\" data-semantic-type=\\\"infixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,∈\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"element\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"5\\\" space=\\\"5\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"10\\\" data-semantic-content=\\\"2,8\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"11\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\\\"9,6,7\\\" data-semantic-content=\\\"6\\\" data-semantic- data-semantic-parent=\\\"11\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mjx-mrow data-semantic-children=\\\"3,5\\\" data-semantic-content=\\\"4\\\" data-semantic- data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\" space=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\" rspace=\\\"3\\\" style=\\\"margin-left: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"11\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\\\"true\\\" display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"/cms/asset/493558b9-2d91-4205-b21f-f1385aaafe7e/rsa21212-math-0004.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"0,11\\\" data-semantic-content=\\\"1\\\" data-semantic-role=\\\"element\\\" data-semantic-speech=\\\"alpha element of left parenthesis 1 divided by 2 comma 1 right parenthesis\\\" data-semantic-type=\\\"infixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\">α</mi><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,∈\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"element\\\" data-semantic-type=\\\"operator\\\">∈</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"10\\\" data-semantic-content=\\\"2,8\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"leftright\\\" data-semantic-type=\\\"fenced\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"11\\\" data-semantic-role=\\\"open\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">(</mo><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"9,6,7\\\" data-semantic-content=\\\"6\\\" data-semantic-parent=\\\"11\\\" data-semantic-role=\\\"sequence\\\" data-semantic-type=\\\"punctuated\\\"><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"3,5\\\" data-semantic-content=\\\"4\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"infixop\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">1</mn><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" stretchy=\\\"false\\\">/</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"9\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"punctuated\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"comma\\\" data-semantic-type=\\\"punctuation\\\">,</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"10\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">1</mn></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"fenced\\\" data-semantic-parent=\\\"11\\\" data-semantic-role=\\\"close\\\" data-semantic-type=\\\"fence\\\" stretchy=\\\"false\\\">)</mo></mrow></mrow>$$ \\\\alpha \\\\in \\\\left(1/2,1\\\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>.\",\"PeriodicalId\":20948,\"journal\":{\"name\":\"Random Structures and Algorithms\",\"volume\":\"25 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Structures and 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引用次数: 0

摘要

对于两个独立的厄尔多斯-雷尼图 G(n,p)$$ \mathbf{G}\left(n,p\right) $$，我们研究了这两个图在所有可能的顶点对应关系中的最大重合度（即公共边的数量）。我们提出了一种多项式时间算法，它能找到一种顶点对应关系，其重叠度近似于最大重叠度，其乘法因子可任意接近于 1。作为副产品，我们证明在 p=n-α$$ p={n}^{-\alpha } $$ 时，最大重叠度近似为 n2α-1$$ \frac{n}{2\alpha -1} $$ 。$$ 有一些常数 α∈(1/2,1)$$ \alpha \in \left(1/2,1\right) $$.

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A polynomial-time approximation scheme for the maximal overlap of two independent Erdős–Rényi graphs

For two independent Erdős–Rényi graphs

G (n, p) $$ \mathbf{G}\left(n,p\right) $$

, we study the maximal overlap (i.e., the number of common edges) of these two graphs over all possible vertex correspondence. We present a polynomial-time algorithm which finds a vertex correspondence whose overlap approximates the maximal overlap up to a multiplicative factor that is arbitrarily close to 1. As a by-product, we prove that the maximal overlap is asymptotically