A Tight Lower Bound for Steiner Orientation
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10386943" target="_blank" >RIV/00216208:11320/18:10386943 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/978-3-319-90530-3_7" target="_blank" >https://doi.org/10.1007/978-3-319-90530-3_7</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/978-3-319-90530-3_7" target="_blank" >10.1007/978-3-319-90530-3_7</a>
Alternative languages
Result language
angličtina
Original language name
A Tight Lower Bound for Steiner Orientation
Original language description
In the Steiner Orientation problem, the input is a mixed graph G (it has both directed and undirected edges) and a set of k terminal pairs T. The question is whether we can orient the undirected edges in a way such that there is a directed s -> t path for each terminal pair (s, t) is an element of T. Arkin and Hassin [DAM' 02] showed that the STEINER ORIENTATION problem is NP-complete. They also gave a polynomial time algorithm for the special case when k = 2. From the viewpoint of exact algorithms, Cygan, Kortsarz and Nutov [ESA'12, SIDMA'13] designed an XP algorithm running in n(O(k)) time for all k >= 1. Pilipczuk andWahlstrom [SODA'16] showed that the STEINER ORIENTATION problem is W[1]-hard parameterized by k. As a byproduct of their reduction, they were able to show that under the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [JCSS'01] the Steiner Orientation problem does not admit an f(k).n(o(k/log k)) algorithm for any computable function f. That is, the n(O(k)) algorithm of Cygan et al. is almost optimal. In this paper, we give a short and easy proof that the n(O(k)) algorithm of Cygan et al. is asymptotically optimal, even if the input graph has genus 1. Formally, we show that the Steiner Orientation problem is W[1]-hard parameterized by the number k of terminal pairs, and, under ETH, cannot be solved in f(k).n(O(k)) time for any function f even if the underlying undirected graph has genus 1. We give a reduction from the Grid Tiling problem which has turned out to be very useful in proving W[1]-hardness of several problems on planar graphs. As a result of our work, the main remaining open question is whether Steiner Orientation admits the "square-root phenomenon" on planar graphs (graphs with genus 0): can one obtain an algorithm running in time f(k).n(O(root k)) for Planar Steiner Orientation, or does the lower bound of f(k).n(O(k))) also translate to planar graphs?
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
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/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
COMPUTER SCIENCE - THEORY AND APPLICATIONS, CSR 2018
ISBN
978-3-319-90529-7
ISSN
0302-9743
e-ISSN
1611-3349
Number of pages
13
Pages from-to
65-77
Publisher name
SPRINGER INTERNATIONAL PUBLISHING AG
Place of publication
CHAM
Event location
Higher Sch Economics
Event date
Jun 6, 2018
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
000445826800007