{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/108615"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/108615","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Improving the smoothed complexity of flip for max cut problems","abstract":"Finding locally optimal solutions for max-cut and max-k-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Roglin [1] showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres and Wei [2] showed that the smoothed complexity of FLIP for maxcut in complete graphs is O(φ^5 n^15.1), where φ is an upper bound on the random edge-weight density and n is the number of vertices in the input graph. While Angel, Bubeck, Peres and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress towards improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O(φ n^7.83). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for max-3-cut in complete graphs is polynomial and for max-k-cut in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest towards showing smoothed polynomial complexity of FLIP for max-k-cut in complete graphs for larger constants k.","abstract_html":"Finding locally optimal solutions for max-cut and max-k-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Roglin [1] showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres and Wei [2] showed that the smoothed complexity of FLIP for maxcut in complete graphs is O(φ^5 n^15.1), where φ is an upper bound on the random edge-weight density and n is the number of vertices in the input graph. While Angel, Bubeck, Peres and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress towards improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O(φ n^7.83). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for max-3-cut in complete graphs is polynomial and for max-k-cut in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest towards showing smoothed polynomial complexity of FLIP for max-k-cut in complete graphs for larger constants k.","abstract_has_math":false,"creators":["Bibaksereshkeh, Seyedali"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Industrial Engineering","degree_department":null,"school":null,"contributors":["Chandrasekaran , Karthekeyan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-10-07T22:44:37Z","date_published":"2020-10-07T22:44:37Z","updated_at":"2026-07-22T22:24:48Z","subjects":["max cut","flip","graph theory","smoothed complexity"],"languages":["en"],"rights":["Copyright 2020 SeyedAli BibakSereshkeh"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/108615","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Chandrasekaran , Karthekeyan"]},{"key":"dc:creator","label":"Author","values":["Bibaksereshkeh, Seyedali"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-10-07T22:44:37Z","2022-10-07T22:44:53Z","2020-07-21","2020-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Industrial Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["max cut","flip","graph theory","smoothed complexity"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 SeyedAli BibakSereshkeh"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/108615"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Finding locally optimal solutions for max-cut and max-k-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Roglin [1] showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres and Wei [2] showed that the smoothed complexity of FLIP for maxcut in complete graphs is O(φ^5 n^15.1), where φ is an upper bound on the random edge-weight density and n is the number of vertices in the input graph. While Angel, Bubeck, Peres and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress towards improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O(φ n^7.83). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for max-3-cut in complete graphs is polynomial and for max-k-cut in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest towards showing smoothed polynomial complexity of FLIP for max-k-cut in complete graphs for larger constants k.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2022-08-01","The student, Seyedali Bibaksereshkeh, accepted the attached license on 2020-07-15 at 10:37.","The student, Seyedali Bibaksereshkeh, submitted this Thesis for approval on 2020-07-15 at 10:51.","This Thesis was approved for publication on 2020-07-21 at 15:50.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15631 on 2020-10-02 at 15:33:22","Made available in DSpace on 2020-10-07T22:44:37Z (GMT). No. of bitstreams: 2 BIBAKSERESHKEH-THESIS-2020.pdf: 449339 bytes, checksum: ec9a83a3d8a9974fef53978c233ddaef (MD5) LICENSE.txt: 4215 bytes, checksum: 486a8b036fe71caef044610afb2912bd (MD5) Previous issue date: 2020-07-21","Embargo set by: Seth Robbins for item 116242 Lift date: 2022-10-07T22:44:53Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Improving the smoothed complexity of flip for max cut problems"]}]}],"canonical_facts":{"dc:contributor":["Chandrasekaran , Karthekeyan"],"dc:creator":["Bibaksereshkeh, Seyedali"],"dc:date":["2020-10-07T22:44:37Z","2022-10-07T22:44:53Z","2020-07-21","2020-08"],"dc:description":["Finding locally optimal solutions for max-cut and max-k-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Roglin [1] showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres and Wei [2] showed that the smoothed complexity of FLIP for maxcut in complete graphs is O(φ^5 n^15.1), where φ is an upper bound on the random edge-weight density and n is the number of vertices in the input graph. While Angel, Bubeck, Peres and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress towards improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O(φ n^7.83). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for max-3-cut in complete graphs is polynomial and for max-k-cut in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest towards showing smoothed polynomial complexity of FLIP for max-k-cut in complete graphs for larger constants k.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2022-08-01","The student, Seyedali Bibaksereshkeh, accepted the attached license on 2020-07-15 at 10:37.","The student, Seyedali Bibaksereshkeh, submitted this Thesis for approval on 2020-07-15 at 10:51.","This Thesis was approved for publication on 2020-07-21 at 15:50.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15631 on 2020-10-02 at 15:33:22","Made available in DSpace on 2020-10-07T22:44:37Z (GMT). No. of bitstreams: 2 BIBAKSERESHKEH-THESIS-2020.pdf: 449339 bytes, checksum: ec9a83a3d8a9974fef53978c233ddaef (MD5) LICENSE.txt: 4215 bytes, checksum: 486a8b036fe71caef044610afb2912bd (MD5) Previous issue date: 2020-07-21","Embargo set by: Seth Robbins for item 116242 Lift date: 2022-10-07T22:44:53Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/108615"],"dc:language":["en"],"dc:rights":["Copyright 2020 SeyedAli BibakSereshkeh"],"dc:subject":["max cut","flip","graph theory","smoothed complexity"],"dc:title":["Improving the smoothed complexity of flip for max cut problems"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Industrial Engineering"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:48Z"}