The complexity of the partial order dimension problem: Closing the gap
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
The complexity of the partial order dimension problem: Closing the gap
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
<a href="/en/project/GA14-10799S" target="_blank" >GA14-10799S: Hybercubic, graph and hypergraph structures</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Discrete Mathematics
ISSN
0895-4801
e-ISSN
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Volume of the periodical
31
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
18
Pages from-to
172-189
UT code for WoS article
000398542500009
EID of the result in the Scopus database
2-s2.0-85018668443