{"id":"https://openalex.org/W7171948847","doi":"https://doi.org/10.48550/arxiv.2607.27264","title":"Improved Lower Bounds on Tree Covers","display_name":"Improved Lower Bounds on Tree Covers","publication_year":2026,"publication_date":"2026-07-29","ids":{"openalex":"https://openalex.org/W7171948847","doi":"https://doi.org/10.48550/arxiv.2607.27264"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2607.27264","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.27264","pdf_url":"https://arxiv.org/pdf/2607.27264","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2607.27264","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160057513","display_name":"Yu Chen","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chen, Yu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5160027376","display_name":"Shengtang Huang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Huang, Shengtang","raw_affiliation_strings":["Center on Frontiers of Computing Studies , Peking University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center on Frontiers of Computing Studies , Peking University","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072885035","display_name":"Zihan Tan","orcid":"https://orcid.org/0000-0003-4844-8480","observed_orcids":["https://orcid.org/0000-0003-4844-8480","https://orcid.org/0009-0008-1754-3841"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tan, Zihan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5126837429","display_name":"Hangyu Xu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xu, Hangyu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.5985999703407288,"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"}},"topics":[{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.5985999703407288,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.15160000324249268,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.1200999990105629,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/tree-cover","display_name":"tree cover","score":0.9679999947547913},{"id":"https://openalex.org/keywords/metric-distortion","display_name":"metric distortion","score":0.5709999799728394},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.8140000104904175},{"id":"https://openalex.org/keywords/index-of-agreement","display_name":"index of agreement","score":0.9589999914169312},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"simplicial complexes","score":0.9990000128746033},{"id":"https://openalex.org/keywords/metric-embeddings","display_name":"metric embeddings","score":0.460999995470047},{"id":"https://openalex.org/keywords/algebraic-topology","display_name":"algebraic topology","score":0.6610000133514404}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8321999907493591},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7466999888420105},{"id":"https://openalex.org/C70943382","wikidata":"https://www.wikidata.org/wiki/Q505356","display_name":"Antipodal point","level":2,"score":0.7430999875068665},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6675000190734863},{"id":"https://openalex.org/C171036898","wikidata":"https://www.wikidata.org/wiki/Q256355","display_name":"Equivariant map","level":2,"score":0.5662999749183655},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.5055999755859375},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4431999921798706},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.43959999084472656},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.3885999917984009},{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.3813000023365021},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.3774000108242035},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.35670000314712524},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.3346000015735626},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.32499998807907104},{"id":"https://openalex.org/C126780896","wikidata":"https://www.wikidata.org/wiki/Q899871","display_name":"Distortion (music)","level":4,"score":0.3228999972343445},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.31310001015663147},{"id":"https://openalex.org/C2780897414","wikidata":"https://www.wikidata.org/wiki/Q7600592","display_name":"Star (game theory)","level":2,"score":0.3043000102043152},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.3025999963283539},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.29580000042915344},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.2921000123023987},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.26840001344680786},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.2615000009536743},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.25859999656677246},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.25760000944137573}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2607.27264","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.27264","pdf_url":"https://arxiv.org/pdf/2607.27264","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2607.27264","doi":"https://doi.org/10.48550/arxiv.2607.27264","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.27264","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2607.27264","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.27264","pdf_url":"https://arxiv.org/pdf/2607.27264","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7171948847.pdf","grobid_xml":"https://content.openalex.org/works/W7171948847.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W6910664452","https://openalex.org/W2946166375","https://openalex.org/W7118693498","https://openalex.org/W7075994079","https://openalex.org/W7203931208","https://openalex.org/W4243780282","https://openalex.org/W4411319933","https://openalex.org/W4416862315","https://openalex.org/W2726117250","https://openalex.org/W7143334208"],"abstract_inverted_index":{"A":[0],"tree":[1,96,113],"cover":[2],"of":[3,11,54,72,88,106],"an":[4],"$n$-point":[5],"metric":[6],"space":[7],"is":[8,19,36,44],"a":[9,51,73,89],"collection":[10],"$k$":[12],"trees":[13],"such":[14],"that":[15,78],"every":[16],"pairwise":[17,93],"distance":[18],"approximately":[20],"preserved":[21],"in":[22],"at":[23],"least":[24],"one":[25],"tree.":[26],"For":[27],"fixed":[28],"$k$,":[29],"the":[30,39,62,65,81,84,102],"best":[31,40],"known":[32,41],"distortion":[33],"upper":[34,66],"bound":[35,43,53],"$\\tilde{O}(n^{1/k})$":[37],"and":[38,104,108,115],"lower":[42,52],"$\u03a9_k(n^{1/2^{k-1}})$.":[45],"In":[46],"this":[47],"paper,":[48],"we":[49,79,116],"prove":[50],"$\u03a9_k(n^{1/(k-1)^2})$":[55],"for":[56,111],"all":[57],"$k\\ge":[58],"3$,":[59],"significantly":[60],"narrowing":[61],"gap":[63],"towards":[64],"bound.":[67],"Our":[68],"approach":[69],"makes":[70],"use":[71],"novel":[74],"algebraic":[75,107],"topological":[76,86,109],"notion":[77,99],"call":[80],"\\emph{agreement":[82],"index}:":[83],"equivariant":[85],"index":[87],"simplicial":[90],"complex":[91],"encoding":[92],"agreement":[94],"among":[95],"labels.":[97],"This":[98],"helps":[100],"characterize":[101],"power":[103],"limit":[105],"methods":[110],"analyzing":[112],"covers,":[114],"believe":[117],"it":[118],"will":[119],"also":[120],"find":[121],"applications":[122],"to":[123],"other":[124],"problems.":[125]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-08-01T00:00:00"}