NP-hard sets have many hard instances
Identifieur interne : 001807 ( Istex/Corpus ); précédent : 001806; suivant : 001808NP-hard sets have many hard instances
Auteurs : Martin MundhenkSource :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 1997.
Abstract
Abstract: The notion of instance complexity was introduced by Ko, Orponen, Schöning, and Watanabe [9] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they stated the “instance complexity conjecture,” that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Buhrman and Orponen [4] showed that E-complete sets have exponentially dense hard instances, and Fortnow and Kummer [5] proved that NP-hard sets have p-hard instances unless P = NP. They left open whether the p-hard instances of NP-hard sets must be dense. In this work, we introduce a slightly weaker notion of hard instances and obtain a superpolynomial lower bound on the density of hard instances in the case of NP-hard sets. We additionally show that NP-hard sets cannot consist of hard instances only, unless P = NP. Kummer [10] proved that the class of recursive sets cannot be characterized by a respective version of the instance complexity conjecture, i.e. there exist nonrecursive sets without hard instances. We give a complete characterization of the class of recursive sets comparing the instance complexity to a relativized Kolmogorov complexity of strings. A set A is shown to be recursive iff ic $$\left( {x:A} \right) \leqslant C^{K_0 \oplus A} \left( x \right)$$ for almost all x. This translates to a characterization of P.
Url:
DOI: 10.1007/BFb0029986
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ISTEX:93811378C4C92F8FAF325B7FF522DD646CFD34F4Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">NP-hard sets have many hard instances</title><author><name sortKey="Mundhenk, Martin" sort="Mundhenk, Martin" uniqKey="Mundhenk M" first="Martin" last="Mundhenk">Martin Mundhenk</name><affiliation><mods:affiliation>FB IV — Informatik, Universität Trier, D-54286, Trier, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:93811378C4C92F8FAF325B7FF522DD646CFD34F4</idno><date when="1997" year="1997">1997</date><idno type="doi">10.1007/BFb0029986</idno><idno type="url">https://api.istex.fr/document/93811378C4C92F8FAF325B7FF522DD646CFD34F4/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001807</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001807</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">NP-hard sets have many hard instances</title><author><name sortKey="Mundhenk, Martin" sort="Mundhenk, Martin" uniqKey="Mundhenk M" first="Martin" last="Mundhenk">Martin Mundhenk</name><affiliation><mods:affiliation>FB IV — Informatik, Universität Trier, D-54286, Trier, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="s">Lecture Notes in Computer Science</title><imprint><date>1997</date></imprint><idno type="ISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="ISSN">0302-9743</idno></series><idno type="istex">93811378C4C92F8FAF325B7FF522DD646CFD34F4</idno><idno type="DOI">10.1007/BFb0029986</idno><idno type="ChapterID">43</idno><idno type="ChapterID">Chap43</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0302-9743</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: The notion of instance complexity was introduced by Ko, Orponen, Schöning, and Watanabe [9] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they stated the “instance complexity conjecture,” that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Buhrman and Orponen [4] showed that E-complete sets have exponentially dense hard instances, and Fortnow and Kummer [5] proved that NP-hard sets have p-hard instances unless P = NP. They left open whether the p-hard instances of NP-hard sets must be dense. In this work, we introduce a slightly weaker notion of hard instances and obtain a superpolynomial lower bound on the density of hard instances in the case of NP-hard sets. We additionally show that NP-hard sets cannot consist of hard instances only, unless P = NP. Kummer [10] proved that the class of recursive sets cannot be characterized by a respective version of the instance complexity conjecture, i.e. there exist nonrecursive sets without hard instances. We give a complete characterization of the class of recursive sets comparing the instance complexity to a relativized Kolmogorov complexity of strings. A set A is shown to be recursive iff ic $$\left( {x:A} \right) \leqslant C^{K_0 \oplus A} \left( x \right)$$ for almost all x. This translates to a characterization of P.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>Martin Mundhenk</name><affiliations><json:string>FB IV — Informatik, Universität Trier, D-54286, Trier, Germany</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>ReviewPaper</json:string></originalGenre><abstract>Abstract: The notion of instance complexity was introduced by Ko, Orponen, Schöning, and Watanabe [9] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they stated the “instance complexity conjecture,” that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Buhrman and Orponen [4] showed that E-complete sets have exponentially dense hard instances, and Fortnow and Kummer [5] proved that NP-hard sets have p-hard instances unless P = NP. They left open whether the p-hard instances of NP-hard sets must be dense. In this work, we introduce a slightly weaker notion of hard instances and obtain a superpolynomial lower bound on the density of hard instances in the case of NP-hard sets. We additionally show that NP-hard sets cannot consist of hard instances only, unless P = NP. Kummer [10] proved that the class of recursive sets cannot be characterized by a respective version of the instance complexity conjecture, i.e. there exist nonrecursive sets without hard instances. We give a complete characterization of the class of recursive sets comparing the instance complexity to a relativized Kolmogorov complexity of strings. A set A is shown to be recursive iff ic $$\left( {x:A} \right) \leqslant C^{K_0 \oplus A} \left( x \right)$$ for almost all x. This translates to a characterization of P.</abstract><qualityIndicators><score>7.479</score><pdfVersion>1.3</pdfVersion><pdfPageSize>440 x 666 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>1419</abstractCharCount><pdfWordCount>4779</pdfWordCount><pdfCharCount>20818</pdfCharCount><pdfPageCount>10</pdfPageCount><abstractWordCount>225</abstractWordCount></qualityIndicators><title>NP-hard sets have many hard instances</title><chapterId><json:string>43</json:string><json:string>Chap43</json:string></chapterId><refBibs><json:item><author><json:item><name>V Arvind</name></json:item><json:item><name>J Ksbler</name></json:item><json:item><name>M Mundhenk</name></json:item></author><host><pages><last>151</last><first>140</first></pages><author></author><title>Proceedings 12th Conference on the Foundations of Software Technology 8~ Theoretical Computer Science</title><publicationDate>1992</publicationDate></host><title>On bounded 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Buhrman</name></json:item><json:item><name>P Orponen</name></json:item></author><host><pages><last>222</last><first>217</first></pages><author></author><title>Proc. 9th Structure in Complexity Theory Conference</title><publicationDate>1994</publicationDate></host><title>Random strings make hard instances</title><publicationDate>1994</publicationDate></json:item><json:item><author><json:item><name>L Fortnow</name></json:item><json:item><name>M Kummer</name></json:item></author><host><author></author><title>Proceedings of 12th Symposium on Theoretical Aspects of Computer Science</title><publicationDate>1995</publicationDate></host><title>Resource-bounded instance complexity</title><publicationDate>1995</publicationDate></json:item><json:item><author><json:item><name>J Hartmanis</name></json:item></author><host><pages><last>445</last><first>439</first></pages><author></author><title>Proc. 24th IEEE Syrup. on Foundations of Computer Science</title><publicationDate>1983</publicationDate></host><title>Generalized Kotmogorov complexity and the structure of feasible computations</title><publicationDate>1983</publicationDate></json:item><json:item><author><json:item><name>R Karp</name></json:item><json:item><name>R Lipton</name></json:item></author><host><pages><last>309</last><first>302</first></pages><author></author><title>tn Prooceedings of the 12th A CM Symposium on Theory of Computing</title><publicationDate>1980-04</publicationDate></host><title>Some relations between nonuniform and uniform complexity classes</title><publicationDate>1980-04</publicationDate></json:item><json:item><author><json:item><name>K Ko</name></json:item></author><host><pages><last>337</last><first>327</first></pages><author></author><title>Proc. 7th Structure in Complexity Theory Conference</title><publicationDate>1992</publicationDate></host><title>A note on the instance complexity of pseudorandom sets</title><publicationDate>1992</publicationDate></json:item><json:item><author><json:item><name>K Ko</name></json:item><json:item><name>P Orponen</name></json:item><json:item><name>U Schsning</name></json:item><json:item><name>O Watanabe</name></json:item></author><host><volume>41</volume><pages><last>121</last><first>96</first></pages><author></author><title>Journal of the ACM</title><publicationDate>1994</publicationDate></host><title>Instance complexity</title><publicationDate>1994</publicationDate></json:item><json:item><author><json:item><name>M Kummer</name></json:item></author><host><pages><last>124</last><first>111</first></pages><author></author><title>Proc. lOth Structure in Complexity Theory Conference</title><publicationDate>1995</publicationDate></host><title>The instance complexity conjecture</title><publicationDate>1995</publicationDate></json:item><json:item><host><author><json:item><name>M Li</name></json:item><json:item><name>P Vit£nyi</name></json:item></author><title>An introduction to Kolmogorov complexity and its applications</title><publicationDate>1993</publicationDate></host></json:item><json:item><author><json:item><name>N Lynch</name></json:item></author><host><volume>22</volume><pages><last>345</last><first>341</first></pages><author></author><title>Journal of the ACM</title><publicationDate>1975</publicationDate></host><title>On reducibility to complex or sparse sets</title><publicationDate>1975</publicationDate></json:item><json:item><author><json:item><name>S Mahaney</name></json:item></author><host><volume>25</volume><pages><last>143</last><first>130</first></pages><issue>2</issue><author></author><title>Journal of Computer and System Sciences</title><publicationDate>1982</publicationDate></host><title>Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis</title><publicationDate>1982</publicationDate></json:item><json:item><author><json:item><name>P Orponen</name></json:item><json:item><name>U Schsning</name></json:item></author><host><author></author><title>11th Syrup</title><publicationDate>1984</publicationDate></host><title>The structure of polynomial complexity cores</title><publicationDate>1984</publicationDate></json:item><json:item><author><json:item><name>O Watanabe</name></json:item></author><host><pages><last>146</last><first>138</first></pages><author></author><title>Proc. 1987 Structure in Complexity Theory Conference</title><publicationDate>1987</publicationDate></host><title>Polynomial time reducibility to a set of small density</title><publicationDate>1987</publicationDate></json:item></refBibs><genre><json:string>conference</json:string></genre><serie><editor><json:item><name>Gerhard Goos</name></json:item><json:item><name>Juris Hartmanis</name></json:item><json:item><name>Jan van 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ident="en">en</language></langUsage><abstract xml:lang="en"><p>Abstract: The notion of instance complexity was introduced by Ko, Orponen, Schöning, and Watanabe [9] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they stated the “instance complexity conjecture,” that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Buhrman and Orponen [4] showed that E-complete sets have exponentially dense hard instances, and Fortnow and Kummer [5] proved that NP-hard sets have p-hard instances unless P = NP. They left open whether the p-hard instances of NP-hard sets must be dense. In this work, we introduce a slightly weaker notion of hard instances and obtain a superpolynomial lower bound on the density of hard instances in the case of NP-hard sets. We additionally show that NP-hard sets cannot consist of hard instances only, unless P = NP. Kummer [10] proved that the class of recursive sets cannot be characterized by a respective version of the instance complexity conjecture, i.e. there exist nonrecursive sets without hard instances. We give a complete characterization of the class of recursive sets comparing the instance complexity to a relativized Kolmogorov complexity of strings. A set A is shown to be recursive iff ic $$\left( {x:A} \right) \leqslant C^{K_0 \oplus A} \left( x \right)$$ for almost all x. 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Papers</ChapterCategory><ChapterFirstPage>428</ChapterFirstPage><ChapterLastPage>437</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag</CopyrightHolderName><CopyrightYear>1997</CopyrightYear></ChapterCopyright><ChapterHistory><OnlineDate><Year>2005</Year><Month>6</Month><Day>17</Day></OnlineDate></ChapterHistory><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><BookID>3540634371</BookID><BookTitle>Mathematical Foundations of Computer Science 1997</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Martin</GivenName><FamilyName>Mundhenk</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgDivision>FB IV — Informatik</OrgDivision><OrgName>Universität Trier</OrgName><OrgAddress><Postcode>D-54286</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>The notion of <Emphasis Type="Italic">instance complexity</Emphasis> was introduced by Ko, Orponen, Schöning, and Watanabe [9] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they stated the “instance complexity conjecture,” that every set not in P has <Emphasis Type="Italic">p</Emphasis>-hard instances. Whereas this conjecture is still unsettled, Buhrman and Orponen [4] showed that E-complete sets have exponentially dense hard instances, and Fortnow and Kummer [5] proved that NP-hard sets have <Emphasis Type="Italic">p</Emphasis>-hard instances unless P = NP. They left open whether the <Emphasis Type="Italic">p</Emphasis>-hard instances of NP-hard sets must be dense. In this work, we introduce a slightly weaker notion of hard instances and obtain a superpolynomial lower bound on the density of hard instances in the case of NP-hard sets. We additionally show that NP-hard sets cannot consist of hard instances only, unless P = NP. Kummer [10] proved that the class of recursive sets cannot be characterized by a respective version of the instance complexity conjecture, i.e. there exist nonrecursive sets without hard instances. We give a complete characterization of the class of recursive sets comparing the instance complexity to a relativized Kolmogorov complexity of strings. A set A is shown to be recursive iff ic<InlineEquation ID="IE1"><InlineMediaObject><ImageObject FileRef="558_3540634371_Chapter_43_TeX2GIFIE1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX"> $$\left( {x:A} \right) \leqslant C^{K_0 \oplus A} \left( x \right)$$ </EquationSource></InlineEquation> for almost all <Emphasis Type="Italic">x</Emphasis>. This translates to a characterization of P.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>Supported in part by the Office of the Vice Chancellor for Research and Graduate Studies at the University of Kentucky, and by the Deutsche Forschungsgemeinschaft (DFG), grant Mu 1226/2-1. Parts of the work done at University of Kentucky.</SimplePara></ArticleNote></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>NP-hard sets have many hard instances</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>NP-hard sets have many hard instances</title></titleInfo><name type="personal"><namePart type="given">Martin</namePart><namePart type="family">Mundhenk</namePart><affiliation>FB IV — Informatik, Universität Trier, D-54286, Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="conference" displayLabel="ReviewPaper"></genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2005-06-17</dateIssued><copyrightDate encoding="w3cdtf">1997</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: The notion of instance complexity was introduced by Ko, Orponen, Schöning, and Watanabe [9] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they stated the “instance complexity conjecture,” that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Buhrman and Orponen [4] showed that E-complete sets have exponentially dense hard instances, and Fortnow and Kummer [5] proved that NP-hard sets have p-hard instances unless P = NP. They left open whether the p-hard instances of NP-hard sets must be dense. In this work, we introduce a slightly weaker notion of hard instances and obtain a superpolynomial lower bound on the density of hard instances in the case of NP-hard sets. We additionally show that NP-hard sets cannot consist of hard instances only, unless P = NP. Kummer [10] proved that the class of recursive sets cannot be characterized by a respective version of the instance complexity conjecture, i.e. there exist nonrecursive sets without hard instances. We give a complete characterization of the class of recursive sets comparing the instance complexity to a relativized Kolmogorov complexity of strings. A set A is shown to be recursive iff ic $$\left( {x:A} \right) \leqslant C^{K_0 \oplus A} \left( x \right)$$ for almost all x. This translates to a characterization of P.</abstract><relatedItem type="host"><titleInfo><title>Mathematical Foundations of Computer Science 1997</title><subTitle>22nd International Symposium, MFCS '97 Bratislava, Slovakia, August 25–29, 1997 Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Igor</namePart><namePart type="family">Prívara</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Peter</namePart><namePart type="family">Ružička</namePart><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Proceedings"></genre><originInfo><copyrightDate encoding="w3cdtf">1997</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11645">Computer Science</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="I">Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I16005">Theory of Computation</topic><topic authority="SpringerSubjectCodes" authorityURI="I14029">Software Engineering</topic><topic authority="SpringerSubjectCodes" authorityURI="I14037">Programming Languages, Compilers, Interpreters</topic><topic authority="SpringerSubjectCodes" authorityURI="I17028">Discrete Mathematics in Computer Science</topic></subject><identifier type="DOI">10.1007/BFb0029943</identifier><identifier type="ISBN">978-3-540-63437-9</identifier><identifier type="eISBN">978-3-540-69547-9</identifier><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="BookTitleID">46525</identifier><identifier type="BookID">3540634371</identifier><identifier type="BookChapterCount">51</identifier><identifier type="BookVolumeNumber">1295</identifier><part><date>1997</date><detail type="volume"><number>1295</number><caption>vol.</caption></detail><extent unit="pages"><start>428</start><end>437</end></extent></part><recordInfo><recordOrigin>Springer-Verlag, 1997</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Lecture Notes in Computer Science</title></titleInfo><name type="personal"><namePart type="given">Gerhard</namePart><namePart type="family">Goos</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Juris</namePart><namePart type="family">Hartmanis</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jan</namePart><namePart type="family">van Leeuwen</namePart><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><copyrightDate encoding="w3cdtf">1997</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="SeriesID">558</identifier><recordInfo><recordOrigin>Springer-Verlag, 1997</recordOrigin></recordInfo></relatedItem><identifier type="istex">93811378C4C92F8FAF325B7FF522DD646CFD34F4</identifier><identifier type="DOI">10.1007/BFb0029986</identifier><identifier type="ChapterID">43</identifier><identifier type="ChapterID">Chap43</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag, 1997</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Springer-Verlag, 1997</recordOrigin></recordInfo></mods></metadata></istex></record>Pour manipuler ce document sous Unix (Dilib)
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