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DKUM
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Title:
KVAZIPRIREJANJA V DVODELNIH GRAFIH
Authors:
ID
Kren, Matej
(Author)
ID
Brešar, Boštjan
(Mentor)
More about this mentor...
Files:
MAG_Kren_Matej_2014.pdf
(1,03 MB)
MD5: AB2C8976A287231A023CBD39F603C22E
Language:
Slovenian
Work type:
Master's thesis/paper
Typology:
2.09 - Master's Thesis
Organization:
FNM - Faculty of Natural Sciences and Mathematics
Abstract:
V magistrskem delu obravnavamo posplošitve problema iskanja največjega prirejanja v dvodelnem grafu. Dan je dvodelen graf G=(A+B,E) in funkcija potreb, ki vsakemu vozlišču v množici B priredi t.i. potrebo vozlišča. V problemu kvaziprirejanja v dvodelnem grafu G iščemo takšno podmnožico F množice povezav E, da ima vsako vozlišče iz B vsaj toliko F-incidenčnih povezav kot ima potrebo, vozlišča iz množice A pa imajo kar se da uravnoteženo število pripadajočih F-incidenčnih povezav. Problem lahko variiramo tako, da vozliščem iz množice A omejimo število F-incidenčnih povezav s kapacitetno funkcijo in tedaj govorimo o f,g-kvaziprirejanju. V tem primeru nas zanima ali obstaja množica F, ki zadošča kapacitetni funkciji in funkciji potreb v danem dvodelnem grafu. V prvem poglavju so opisani osnovni pojmi in definicije, ki jih potrebujemo v nadaljevanju. V drugem poglavju posplošimo definicijo prirejanja in nekaterih pripadajočih pojmov, ki nam pomagajo dokazati lastnosti kvaziprirejanj. V tretjem poglavju poiščemo učinkovit algoritem za iskanje g-kvaziprirejanja, ki mu dokažemo pravilnost delovanja ter linearno časovno in prostorsko zahtevnost. Algoritem v nadaljevanju dopolnimo tako, da učinkovito poišče optimalno g-kvaziprirejanje ob dodajanju ali odvzemanju vozlišča. V zadnjem poglavju predstavimo odločitveni problem obstoja f,g-kvaziprirejanja. Kot rezultat navedemo široko posplošitev Hallovega poročnega izreka.
Keywords:
prirejanje
,
dvodelen graf
,
kvaziprirejanje
,
madžarska metoda
,
Hallov poročni izrek.
Place of publishing:
Maribor
Publisher:
[M. Kren]
Year of publishing:
2014
PID:
20.500.12556/DKUM-44100
UDC:
519.172.5(043.2)
COBISS.SI-ID:
20528136
NUK URN:
URN:SI:UM:DK:GL7XTFWN
Publication date in DKUM:
14.05.2014
Views:
1820
Downloads:
161
Metadata:
Categories:
FNM
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Vancouver
:
KREN, Matej, 2014,
KVAZIPRIREJANJA V DVODELNIH GRAFIH
[online]. Master’s thesis. Maribor : M. Kren. [Accessed 12 April 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=44100
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Secondary language
Language:
English
Title:
QUASI-MATCHINGS IN BIPARTITE GRAPHS
Abstract:
In master thesis we discuss generalizations of the problem of determining maximum matching in bipartite graphs. A bipartite graph G=(A+B,E), and a need function that assigns the so-called need of every vertex in B, are given. In the quasi-matching problem in a bipartite graph G we seek a subset of edges F, where every vertex from B has at least as many F-incident edges as its need, and vertices in A have the number of F-incident edges as balanced as possible. We can vary the problem by limiting the number of F-incident edges with a capacity function to vertices from A, and we talk about f,g-quasi-matching. In that case we are interested in, whether a set F that fulfils the capacity and need function in the bipartite graph, actually exists. The first chapter introduces basic concepts and definitions that are needed for further understanding of the thesis. In the second chapter we generalize the definition of matching and some concepts that help us to prove some quasi-matching properties. In third chapter we present an efficient algorithm for finding g-quasi-matching in a bipartite graph, for which we prove correctness and linear time and space complexity. In the last part of the chapter we improve the algorithm so that it efficiently finds g-quasi-matching if a vertex is added or taken. In the last chapter we present decision problem of existence of f,g-quasi-matching. As a result generalization of Hall's marriage theorem is given.
Keywords:
matching
,
bipartite graph
,
quasi-matching
,
Hungarian method
,
Hall's marriage theorem
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