Generalisations of matrix partitions: Complexity and obstructions
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10488015" target="_blank" >RIV/00216208:11320/24:10488015 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=eJ8X5ouR1f" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=eJ8X5ouR1f</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.tcs.2024.114652" target="_blank" >10.1016/j.tcs.2024.114652</a>
Alternative languages
Result language
angličtina
Original language name
Generalisations of matrix partitions: Complexity and obstructions
Original language description
A trigraph is a graph where each pair of vertices is labelled either 0 (a non -arc), 1 (an arc) or * (both an arc and a non -arc). In a series of papers, Hell and co-authors (see for instance [P. Hell, 2014 [21]]) proposed to study the complexity of homomorphisms from graphs to trigraphs, called Matrix Partition Problems , where arcs and non -arcs can be both mapped to *-arcs, while a non -arc cannot be mapped to an arc, and vice -versa. Even though Matrix Partition Problems are generalisations of Constraint Satisfaction Problems (CSPs) , they share with them the property of being "intrinsically" combinatorial. So, the question of a possible P -time vs NP -complete dichotomy is a very natural one and was raised in Hell et al.'s papers. We propose a generalisation of Matrix Partition Problems to relational structures and study them with respect to the question of a dichotomy. We first show that trigraph homomorphisms and Matrix Partition Problems are P -time equivalent, and then prove that one can also restrict (with respect to having a dichotomy) to relational structures with a single relation. Failing in proving that Matrix Partition Problems on directed graphs are not P -time equivalent to Matrix Partitions on relational structures, we give some evidence that it might be unlikely by formalising the reductions used in the case of CSPs and by showing that such reductions cannot work for the case of Matrix Partition Problems. We then turn our attention to Matrix Partition problems that can be described by finite sets of (inducedsubgraph) obstructions. We show, in particular, that any such problem has finitely many minimal obstructions if and only if it has finite duality. We conclude by showing that on trees (seen as trigraphs) it is NP -complete to decide whether a given tree has a homomorphism to another input trigraph. The latter shows a notable difference on tractability between CSP and Matrix Partition Problems as it is well-known that CSP is tractable on the class of trees.
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
10101 - Pure mathematics
Result continuities
Project
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Continuities
R - Projekt Ramcoveho programu EK
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Theoretical Computer Science
ISSN
0304-3975
e-ISSN
1879-2294
Volume of the periodical
1007
Issue of the periodical within the volume
29. července 2024
Country of publishing house
NL - THE KINGDOM OF THE NETHERLANDS
Number of pages
19
Pages from-to
114652
UT code for WoS article
001251156300001
EID of the result in the Scopus database
2-s2.0-85195093129