Complexity of the Steiner Network Problem with Respect to the Number of Terminals

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F19%3A00334098" target="_blank" >RIV/68407700:21240/19:00334098 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.4230/LIPIcs.STACS.2019.25" target="_blank" >http://dx.doi.org/10.4230/LIPIcs.STACS.2019.25</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.STACS.2019.25" target="_blank" >10.4230/LIPIcs.STACS.2019.25</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Complexity of the Steiner Network Problem with Respect to the Number of Terminals

Popis výsledku v původním jazyce

In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals $Tsubseteq V(G)$ with |T|=q, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph $H subseteq G$ of the minimum cost such that there is a directed path from s to t in H for all st in A(R). It is known that the problem can be solved in time $|V(G)|^{O(|A(R)|)}$ [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time $|V(G)|^{o(|A(R)|)}$ even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time $|V(G)|^{o(q)}$, unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent. We show that textsc{Directed Steiner Network} is solvable in time $f(q)cdot |V(G)|^{O(c_g cdot q)}$, where $c_g$ is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no $f(q)cdot |V(G)|^{o(q^2/ log q)}$ algorithm for any function f for the problem on general graphs, unless ETH fails.

Název v anglickém jazyce

Complexity of the Steiner Network Problem with Respect to the Number of Terminals

Popis výsledku anglicky

In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals $Tsubseteq V(G)$ with |T|=q, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph $H subseteq G$ of the minimum cost such that there is a directed path from s to t in H for all st in A(R). It is known that the problem can be solved in time $|V(G)|^{O(|A(R)|)}$ [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time $|V(G)|^{o(|A(R)|)}$ even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time $|V(G)|^{o(q)}$, unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent. We show that textsc{Directed Steiner Network} is solvable in time $f(q)cdot |V(G)|^{O(c_g cdot q)}$, where $c_g$ is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no $f(q)cdot |V(G)|^{o(q^2/ log q)}$ algorithm for any function f for the problem on general graphs, unless ETH fails.

Druh

D - Stať ve sborníku

CEP obor

—

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/GA17-20065S" target="_blank" >GA17-20065S: Těsné parametrizované výsledky pro problémy orientované souvislosti</a><br>

Návaznosti

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

Rok uplatnění

2019

Kód důvěrnosti údajů

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

Název statě ve sborníku

36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany

ISBN

978-3-95977-100-9

ISSN

—

e-ISSN

1868-8969

Počet stran výsledku

17

Strana od-do

"25:1"-"25:17"

Název nakladatele

Schloss Dagstuhl - Leibniz Center for Informatics

Místo vydání

Wadern

Místo konání akce

Berlín

Datum konání akce

13. 3. 2019

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

000472795800024