Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10385422" target="_blank" >RIV/00216208:11320/18:10385422 - isvavai.cz</a>

Výsledek na webu

<a href="http://drops.dagstuhl.de/opus/volltexte/2018/9051" target="_blank" >http://drops.dagstuhl.de/opus/volltexte/2018/9051</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.47" target="_blank" >10.4230/LIPIcs.ICALP.2018.47</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes

Popis výsledku v původním jazyce

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k &gt;= 1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k &gt;= 1 and 1 &lt;=beta &lt;=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta chi(G)+(4-3 beta)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes. We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present an additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

Název v anglickém jazyce

Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes

Popis výsledku anglicky

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k &gt;= 1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k &gt;= 1 and 1 &lt;=beta &lt;=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta chi(G)+(4-3 beta)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes. We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present an additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

Druh

D - Stať ve sborníku

CEP obor

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OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GA17-04611S" target="_blank" >GA17-04611S: Ramseyovské aspekty barvení grafů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018

ISBN

978-3-95977-076-7

ISSN

1868-8969

e-ISSN

neuvedeno

Počet stran výsledku

12

Strana od-do

1-12

Název nakladatele

Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Místo vydání

Dagstuhl, Germany

Místo konání akce

Prague, Czech Republi

Datum konání akce

9. 7. 2018

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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