A Tight Algorithm for Strongly Connected Steiner Subgraph on Two Terminals with Demands (Extended Abstract)

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

Authors

  • Hossein Esfandiari
  • Mohammad Taghi Hajiaghayi
  • Rohit Khandekar
  • Guy Kortsarz
  • Saeed Seddighin

Colleges, School and Institutes

External organisations

  • University of Maryland
  • KCG Holdings Inc., New York, USA
  • Rutgers, The State University of New Jersey

Abstract

Given an edge-weighted directed graph G = (V, E) on n vertices and a set T = {t1, t2,… tp} of p terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find an edge set H ⊆ E of minimum weight such that G[H] contains a ti → tj path for each 1 ≤ i _= j ≤ p. The problem is NP-hard, but Feldman and Ruhl [FOCS’99; SICOMP’06] gave a novel nO(p) algorithm for the p-SCSS problem. In this paper, we investigate the computational complexity of a variant of 2-SCSS where we have demands for the number of paths between each terminal pair. Formally, the 2-SCSS-(k1, k2) problem is defined as follows: given an edge-weighted directed graph G = (V, E) with weight function ω: E → R≥0, two terminal vertices s, t, and integers k1, k2; the objective is to find a set of k1 paths F1, F2,…, Fk1 from s → t and k2 paths B1, B2,…, Bk2 from t → s such that ∑e∈E ω(e)·φ(e) is minimized, where φ(e) = max{|{i: i ∈ [k1], e ∈ Fi}|; |{j: j ∈ [k2], e ∈ Bj}|}. For each k ≥ 1, we show the following:-The 2-SCSS-(k, 1) problem can be solved in nO(k) time.-A matching lower bound for our algorithm: the 2-SCSS-(k, 1) problem does not have an f(k) · no(k) algorithm for any computable function f, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for 2-SCSS-(k, 1) relies on a structural result regarding the optimal solution followed by using the idea of a “token game” similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the 2-SCSS-(k1, k2) problem if min{k1, k2} ≥ 2. Therefore 2-SCSS-(k, 1) is the most general problem one can attempt to solve with our techniques. To obtain the lower bound matching the algorithm, we reduce from a special variant of the Grid Tiling problem introduced by Marx [FOCS’07; ICALP’12].

Details

Original languageEnglish
Title of host publicationParameterized and Exact Computation
Subtitle of host publication9th International Symposium, IPEC 2014, Wroclaw, Poland, September 10-12, 2014. Revised Selected Papers
EditorsMarek Cygan, Pinar Heggernes
Publication statusPublished - 3 Dec 2014
Event9th International Symposium on Parameterized and Exact Computation, IPEC 2014 - Wroclaw, Poland
Duration: 10 Sep 201412 Sep 2014

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume8894
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference9th International Symposium on Parameterized and Exact Computation, IPEC 2014
Country/TerritoryPoland
CityWroclaw
Period10/09/1412/09/14