The complexity of the partial order dimension problem: Closing the gap
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10366551" target="_blank" >RIV/00216208:11320/17:10366551 - isvavai.cz</a>
Výsledek na webu
<a href="http://dx.doi.org/10.1137/15M1007720" target="_blank" >http://dx.doi.org/10.1137/15M1007720</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/15M1007720" target="_blank" >10.1137/15M1007720</a>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
The complexity of the partial order dimension problem: Closing the gap
Popis výsledku v původním jazyce
The dimension of a partial order P is the minimum number of linear orders whose intersection is P. There are efficient algorithms to test if a partial order has dimension at most 2. In 1982 Yannakakis [SIAM J.Algebraic Discrete Methods, 3 (1982), pp. 351-358] showed that for k >= 3 to test if a partial order has dimension <= k is NP-complete. The height of a partial order P is the maximum size of a chain in P. Yannakakis also showed that for k >= 4 to test if a partial order of height 2 has dimension <= k is NP-complete. The complexity of deciding whether an order of height 2 has dimension 3 was left open. This question became one of the best known open problems in dimension theory for partial orders. We show that the problem is NP-complete. Technically, we show that the decision problem (3DH2) for dimension is equivalent to deciding for the existence of bipartite triangle containment representations (BTCon). This problem then allows a reduction from a class of planar satis fi ability problems (P-3-CON-3-SAT(4)) which is known to be NP-hard.
Název v anglickém jazyce
The complexity of the partial order dimension problem: Closing the gap
Popis výsledku anglicky
The dimension of a partial order P is the minimum number of linear orders whose intersection is P. There are efficient algorithms to test if a partial order has dimension at most 2. In 1982 Yannakakis [SIAM J.Algebraic Discrete Methods, 3 (1982), pp. 351-358] showed that for k >= 3 to test if a partial order has dimension <= k is NP-complete. The height of a partial order P is the maximum size of a chain in P. Yannakakis also showed that for k >= 4 to test if a partial order of height 2 has dimension <= k is NP-complete. The complexity of deciding whether an order of height 2 has dimension 3 was left open. This question became one of the best known open problems in dimension theory for partial orders. We show that the problem is NP-complete. Technically, we show that the decision problem (3DH2) for dimension is equivalent to deciding for the existence of bipartite triangle containment representations (BTCon). This problem then allows a reduction from a class of planar satis fi ability problems (P-3-CON-3-SAT(4)) which is known to be NP-hard.
Klasifikace
Druh
J<sub>imp</sub> - Článek v periodiku v databázi Web of Science
CEP obor
—
OECD FORD obor
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Návaznosti výsledku
Projekt
<a href="/cs/project/GA14-10799S" target="_blank" >GA14-10799S: Hyperkrychlové, grafové a hypergrafové struktury</a><br>
Návaznosti
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Ostatní
Rok uplatnění
2017
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název periodika
SIAM Journal on Discrete Mathematics
ISSN
0895-4801
e-ISSN
—
Svazek periodika
31
Číslo periodika v rámci svazku
1
Stát vydavatele periodika
US - Spojené státy americké
Počet stran výsledku
18
Strana od-do
172-189
Kód UT WoS článku
000398542500009
EID výsledku v databázi Scopus
2-s2.0-85018668443