{"id":"https://openalex.org/W7162503524","doi":"https://doi.org/10.48550/arxiv.2605.26852","title":"Recognizing Level-k-Based Phylogenetic Networks is NP-Complete","display_name":"Recognizing Level-k-Based Phylogenetic Networks is NP-Complete","publication_year":2026,"publication_date":"2026-05-26","ids":{"openalex":"https://openalex.org/W7162503524","doi":"https://doi.org/10.48550/arxiv.2605.26852"},"language":null,"primary_location":{"id":"pmh:oai:arXiv.org:2605.26852","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2605.26852","pdf_url":"https://arxiv.org/pdf/2605.26852","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2605.26852","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029279861","display_name":"Takatora Suzuki","orcid":"https://orcid.org/0009-0007-3425-7006","observed_orcids":["https://orcid.org/0009-0007-3425-7006"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Suzuki, Takatora","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5029279861"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10015","display_name":"Genomics and Phylogenetic Studies","score":0.9320999979972839,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T10015","display_name":"Genomics and Phylogenetic Studies","score":0.9320999979972839,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.015399999916553497,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10385","display_name":"Plant Diversity and Evolution","score":0.006099999882280827,"subfield":{"id":"https://openalex.org/subfields/1105","display_name":"Ecology, Evolution, Behavior and Systematics"},"field":{"id":"https://openalex.org/fields/11","display_name":"Agricultural and Biological Sciences"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/phylogenetic-networks","display_name":"phylogenetic networks","score":0.9879999756813049},{"id":"https://openalex.org/keywords/tree-networks","display_name":"tree networks","score":0.777999997138977},{"id":"https://openalex.org/keywords/social-support-networks","display_name":"social support networks","score":0.5440000295639038},{"id":"https://openalex.org/keywords/np-completeness","display_name":"NP-completeness","score":0.9049999713897705},{"id":"https://openalex.org/keywords/computational-complexity","display_name":"computational complexity","score":0.9399999976158142},{"id":"https://openalex.org/keywords/reticulate-evolution","display_name":"reticulate evolution","score":0.9449999928474426},{"id":"https://openalex.org/keywords/computational-phylogenetics","display_name":"computational phylogenetics","score":0.5490000247955322}],"concepts":[{"id":"https://openalex.org/C193252679","wikidata":"https://www.wikidata.org/wiki/Q242125","display_name":"Phylogenetic tree","level":3,"score":0.8567000031471252},{"id":"https://openalex.org/C26619641","wikidata":"https://www.wikidata.org/wiki/Q3142246","display_name":"Phylogenetic network","level":4,"score":0.7677000164985657},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.59170001745224},{"id":"https://openalex.org/C41168302","wikidata":"https://www.wikidata.org/wiki/Q3772859","display_name":"Computational phylogenetics","level":5,"score":0.5600000023841858},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5447999835014343},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5084999799728394},{"id":"https://openalex.org/C55919133","wikidata":"https://www.wikidata.org/wiki/Q7644297","display_name":"Supertree","level":4,"score":0.48669999837875366},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.462799996137619},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.4481000006198883},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4320000112056732},{"id":"https://openalex.org/C2780432395","wikidata":"https://www.wikidata.org/wiki/Q21149008","display_name":"Reticulate","level":2,"score":0.42309999465942383},{"id":"https://openalex.org/C56086750","wikidata":"https://www.wikidata.org/wiki/Q6042592","display_name":"Integer programming","level":2,"score":0.39430001378059387},{"id":"https://openalex.org/C90132467","wikidata":"https://www.wikidata.org/wiki/Q171184","display_name":"Phylogenetics","level":3,"score":0.365200012922287},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3465999960899353},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.32440000772476196},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3012000024318695},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2994000017642975},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.26179999113082886},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.2574999928474426},{"id":"https://openalex.org/C2992072516","wikidata":"https://www.wikidata.org/wiki/Q171184","display_name":"Phylogenetic relationship","level":4,"score":0.251800000667572},{"id":"https://openalex.org/C36647736","wikidata":"https://www.wikidata.org/wiki/Q5418752","display_name":"Evolving networks","level":3,"score":0.25110000371932983}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2605.26852","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2605.26852","pdf_url":"https://arxiv.org/pdf/2605.26852","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2605.26852","doi":"https://doi.org/10.48550/arxiv.2605.26852","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.26852","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2605.26852","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2605.26852","pdf_url":"https://arxiv.org/pdf/2605.26852","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7162503524.pdf","grobid_xml":"https://content.openalex.org/works/W7162503524.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Phylogenetic":[0],"networks":[1,10,22,30],"generalize":[2],"phylogenetic":[3,55],"trees":[4,14],"by":[5,75],"representing":[6],"reticulate":[7],"evolution.":[8],"Tree-based":[9],"and":[11,36,64],"their":[12],"support":[13,45],"have":[15],"been":[16],"extensively":[17],"studied,":[18],"but":[19],"not":[20],"all":[21],"are":[23,31],"tree-based.":[24],"To":[25],"measure":[26],"how":[27],"far":[28],"such":[29],"from":[32],"being":[33],"tree-based,":[34],"Suzuki":[35],"Hayamizu":[37],"(2025)":[38],"formulated":[39],"the":[40,44,91],"problem":[41,61],"of":[42,50],"finding":[43],"network":[46],"with":[47],"minimum":[48,92],"level":[49,93],"a":[51],"given":[52],"rooted":[53],"almost-binary":[54],"network.":[56],"They":[57],"conjectured":[58],"that":[59],"this":[60,69,73],"is":[62,86,94],"NP-hard":[63],"provided":[65],"exponential-time":[66],"algorithms.":[67],"In":[68],"paper,":[70],"we":[71],"prove":[72],"conjecture":[74],"showing":[76],"that,":[77],"for":[78],"every":[79],"fixed":[80],"integer":[81],"$k":[82],"\\geq":[83],"1$,":[84],"it":[85],"NP-complete":[87],"to":[88],"decide":[89],"whether":[90],"at":[95],"most":[96],"$k$.":[97]},"counts_by_year":[],"updated_date":"2026-09-26T08:19:03.415552","created_date":"2026-05-28T00:00:00"}