Hardness of embedding simplicial complexes in R^d

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F11%3A10099166" target="_blank" >RIV/00216208:11320/11:10099166 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.4171/JEMS/252" target="_blank" >http://dx.doi.org/10.4171/JEMS/252</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4171/JEMS/252" target="_blank" >10.4171/JEMS/252</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Hardness of embedding simplicial complexes in R^d

Popis výsledku v původním jazyce

Let EMBEDkRIGHTWARDS ARROWd be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R? Known results easily imply polynomiality of EMBEDkRIGHTWARDS ARROW2(k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDkRIGHTWARDS ARROW2k for all k GREATER-THAN OR EQUAL TO 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBEDdRIGHTWARDS ARROWd and EMBED(dMINUS SIGN 1)RIGHTWARDS ARROWd are undecidable for each d GREATER-THAN OR EQUAL TO 5. Our main result is NP-hardness of EMBED2RIGHTWARDS ARROW4 and, more generally, of EMBEDkRIGHTWARDS ARROWd for all k, d with d GREATER-THAN OR EQUAL TO 4 and d GREATER-THAN OR EQUAL TO k GREATER-THAN OR EQUAL TO (2d MINUS SIGN 2)/3. These dimensions fall outside the metastable range of a theorem of Hae?iger and Weber, which characterizes embeddability using the deleted product ob

Název v anglickém jazyce

Hardness of embedding simplicial complexes in R^d

Popis výsledku anglicky

Let EMBEDkRIGHTWARDS ARROWd be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R? Known results easily imply polynomiality of EMBEDkRIGHTWARDS ARROW2(k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDkRIGHTWARDS ARROW2k for all k GREATER-THAN OR EQUAL TO 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBEDdRIGHTWARDS ARROWd and EMBED(dMINUS SIGN 1)RIGHTWARDS ARROWd are undecidable for each d GREATER-THAN OR EQUAL TO 5. Our main result is NP-hardness of EMBED2RIGHTWARDS ARROW4 and, more generally, of EMBEDkRIGHTWARDS ARROWd for all k, d with d GREATER-THAN OR EQUAL TO 4 and d GREATER-THAN OR EQUAL TO k GREATER-THAN OR EQUAL TO (2d MINUS SIGN 2)/3. These dimensions fall outside the metastable range of a theorem of Hae?iger and Weber, which characterizes embeddability using the deleted product ob

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

<a href="/cs/project/1M0545" target="_blank" >1M0545: Institut Teoretické Informatiky</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>Z - Vyzkumny zamer (s odkazem do CEZ)

Rok uplatnění

2011

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of the European Mathematical Society

ISSN

1435-9855

e-ISSN

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Svazek periodika

13

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

37

Strana od-do

259-295

Kód UT WoS článku

000286014400001

EID výsledku v databázi Scopus

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