A tight lower bound for steiner orientation

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Authors

Colleges, School and Institutes

External organisations

  • University of Warwick
  • Charles University

Abstract

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) ∈ 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 nO(k) time for all k ≥ 1. Pilipczuk and Wahlström [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) · no(k/logk) algorithm for any computable function f. That is, the nO(k) algorithm of Cygan et al. is almost optimal. In this paper, we give a short and easy proof that the nO(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) · no(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) · nO(√k) for Planar Steiner Orientation, or does the lower bound of f(k) · no(k) also translate to planar graphs?.

Details

Original languageEnglish
Title of host publicationComputer Science - Theory and Applications
Subtitle of host publication13th International Computer Science Symposium in Russia, CSR 2018, Moscow, Russia, June 6–10, 2018, Proceedings
EditorsVladimir V. Podolskii, Fedor V. Fomin
Publication statusPublished - 25 Apr 2018
Event13th International Computer Science Symposium in Russia, CSR 2018 - Moscow, Russian Federation
Duration: 6 Jun 201810 Jun 2018

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume10846
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference13th International Computer Science Symposium in Russia, CSR 2018
Country/TerritoryRussian Federation
CityMoscow
Period6/06/1810/06/18