{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/122771"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/122771","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Imperfect gaps in Gap-ETH and PCPs","abstract":"In this thesis we study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a new way to transform a PCP with imperfect completeness to a PCP with perfect completeness, when the initial gap is a constant. In particular, we show that PCP[subscript c,s][r, q] [mathematical symbol] PCP[subscript 1,s'][r + 0(1), q+ 0 (r)] for c - s = [omega](1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NTIME[0(n)] with a 0(log n) additive loss in the query complexity q. We show our result by constructing a \"robust circuit\" using threshold gates. These results are a gap amplification procedure for PCPs (when completeness is imperfect), analogous to questions studied in parallel repetition [21] and pseudorandomness [141. We also investigate the time complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness; that is, MAX 3SAT(1 - [epsilon], 1 - [delta]) for [delta] > [epsilon] has 2⁰([superscript n])-time algorithms if and only if MAX 3SAT(1, 1 - [delta]) has 2⁰([superscript n])-time algorithms. We also relate the time complexities of these two problems in a more fine-grained way, to show that T₂ (n) </= T₁ (n(log log n)⁰(¹)), where T₁(n), T₂(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness, respectively. This is joint work with Mitali Bafna.","abstract_html":"In this thesis we study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a new way to transform a PCP with imperfect completeness to a PCP with perfect completeness, when the initial gap is a constant. In particular, we show that PCP[subscript c,s][r, q] [mathematical symbol] PCP[subscript 1,s&#x27;][r + 0(1), q+ 0 (r)] for c - s = [omega](1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NTIME[0(n)] with a 0(log n) additive loss in the query complexity q. We show our result by constructing a &quot;robust circuit&quot; using threshold gates. These results are a gap amplification procedure for PCPs (when completeness is imperfect), analogous to questions studied in parallel repetition [21] and pseudorandomness [141. We also investigate the time complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness; that is, MAX 3SAT(1 - [epsilon], 1 - [delta]) for [delta] &gt; [epsilon] has 2⁰([superscript n])-time algorithms if and only if MAX 3SAT(1, 1 - [delta]) has 2⁰([superscript n])-time algorithms. We also relate the time complexities of these two problems in a more fine-grained way, to show that T₂ (n) &lt;/= T₁ (n(log log n)⁰(¹)), where T₁(n), T₂(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness, respectively. This is joint work with Mitali Bafna.","abstract_has_math":false,"creators":["Vyas, Nikhil(Electrical engineer and computer scientist)Massachusetts Institute of Technology."],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","school":null,"contributors":[],"advisors":["Richard Ryan Williams."],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019","date_published":"2019","updated_at":"2026-07-22T22:21:37Z","subjects":["Electrical Engineering and Computer Science."],"languages":["eng"],"rights":["MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/122771","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Richard Ryan Williams."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","EECS"]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. 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They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1721.1/122771"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 45-47)."]},{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis we study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a new way to transform a PCP with imperfect completeness to a PCP with perfect completeness, when the initial gap is a constant. In particular, we show that PCP[subscript c,s][r, q] [mathematical symbol] PCP[subscript 1,s'][r + 0(1), q+ 0 (r)] for c - s = [omega](1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NTIME[0(n)] with a 0(log n) additive loss in the query complexity q. We show our result by constructing a \"robust circuit\" using threshold gates. These results are a gap amplification procedure for PCPs (when completeness is imperfect), analogous to questions studied in parallel repetition [21] and pseudorandomness [141. We also investigate the time complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness; that is, MAX 3SAT(1 - [epsilon], 1 - [delta]) for [delta] > [epsilon] has 2⁰([superscript n])-time algorithms if and only if MAX 3SAT(1, 1 - [delta]) has 2⁰([superscript n])-time algorithms. We also relate the time complexities of these two problems in a more fine-grained way, to show that T₂ (n) </= T₁ (n(log log n)⁰(¹)), where T₁(n), T₂(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness, respectively. This is joint work with Mitali Bafna."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["Imperfect gaps in Gap-ETH and PCPs"]}]}],"canonical_facts":{"dc:contributor.advisor":["Richard Ryan Williams."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","EECS"],"dc:contributor.other":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science."],"dc:creator":["Vyas, Nikhil(Electrical engineer and computer scientist)Massachusetts Institute of Technology."],"dc:date.accessioned":["2019-11-04T20:23:27Z"],"dc:date.available":["2019-11-04T20:23:27Z"],"dc:date.issued":["2019"],"dc:description":["Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 45-47)."],"dc:description.abstract":["In this thesis we study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a new way to transform a PCP with imperfect completeness to a PCP with perfect completeness, when the initial gap is a constant. In particular, we show that PCP[subscript c,s][r, q] [mathematical symbol] PCP[subscript 1,s'][r + 0(1), q+ 0 (r)] for c - s = [omega](1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NTIME[0(n)] with a 0(log n) additive loss in the query complexity q. We show our result by constructing a \"robust circuit\" using threshold gates. These results are a gap amplification procedure for PCPs (when completeness is imperfect), analogous to questions studied in parallel repetition [21] and pseudorandomness [141. We also investigate the time complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness; that is, MAX 3SAT(1 - [epsilon], 1 - [delta]) for [delta] > [epsilon] has 2⁰([superscript n])-time algorithms if and only if MAX 3SAT(1, 1 - [delta]) has 2⁰([superscript n])-time algorithms. We also relate the time complexities of these two problems in a more fine-grained way, to show that T₂ (n) </= T₁ (n(log log n)⁰(¹)), where T₁(n), T₂(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness, respectively. This is joint work with Mitali Bafna."],"dc:description.degree":["S.M."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/122771"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Electrical Engineering and Computer Science."],"dc:title":["Imperfect gaps in Gap-ETH and PCPs"],"dc:type":["Thesis"],"thesis:degree_name":["Master"]},"updated_at":"2026-07-22T22:21:37Z"}