Separable discrete functions: Recognition and sufficient conditions
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10407349" target="_blank" >RIV/00216208:11320/19:10407349 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=9rdw_zRayV" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=9rdw_zRayV</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.disc.2018.12.026" target="_blank" >10.1016/j.disc.2018.12.026</a>
Alternative languages
Result language
angličtina
Original language name
Separable discrete functions: Recognition and sufficient conditions
Original language description
A discrete function of n variables is a mapping g : X-1 x . . . x X-n -> A, where X-1, . . . , X-n, and A are arbitrary finite sets. Function g is called separable if there exist n functions g(i) : X-i -> A for i = 1, . . . , n, such that for every input x(1), . . . , x(n) the function g(x(1), . . . , x(n)) takes one of the values g(1)(x(1)), . . . , g(n)(x(n)). Given a discrete function g, it is an interesting problem to ask whether g is separable or not. Although this seems to be a very basic problem concerning discrete functions, the complexity of recognition of separable discrete functions of n variables is known only for n = 2. In this paper we will show that a slightly more general recognition problem, when g is not fully but only partially defined, is NP-complete for n >= 3. We will then use this result to show that the recognition of fully defined separable discrete functions is NP-complete for n >= 4. The general recognition problem contains the above mentioned special case for n = 2. This case is well-studied in the context of game theory, where (separable) discrete functions of n variables are referred to as (assignable) n-person game forms. There is a known sufficient condition for assignability (separability) of two-person game forms (discrete functions of two variables) called (weak) total tightness of a game form. This property can be tested in polynomial time, and can be easily generalized both to higher dimension and to partially defined functions. We will prove in this paper that weak total tightness implies separability for (partially defined) discrete functions of n variables for any n, thus generalizing the above result known for n = 2. Our proof is constructive. Using a graph-based discrete algorithm we show how for a given weakly totally tight (partially defined) discrete function g of n variables one can construct separating functions g(1), . . . , g(n) in polynomial time with respect to the size of the input function. (C) 2019 Elsevier B.V. All rights reserved.
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/GA15-15511S" target="_blank" >GA15-15511S: Boolean techniques in knowledge representation</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Discrete Mathematics
ISSN
0012-365X
e-ISSN
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Volume of the periodical
342
Issue of the periodical within the volume
5
Country of publishing house
NL - THE KINGDOM OF THE NETHERLANDS
Number of pages
18
Pages from-to
1275-1292
UT code for WoS article
000462691000006
EID of the result in the Scopus database
2-s2.0-85060874537