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

  • Rajesh Hemant ChitnisEmail author
  • Hossein Esfandiari
  • MohammadTaghi Hajiaghayi
  • Rohit Khandekar
  • Guy Kortsarz
  • Saeed Seddighin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)

Abstract

Given an edge-weighted directed graph \(G=(V,E)\) on \(n\) vertices and a set \(T=\{t_1, t_2, \ldots t_p\}\) of \(p\) terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find an edge set \(H\subseteq E\) of minimum weight such that \(G[H]\) contains a \(t_{i}\rightarrow t_j\) path for each \(1\le i\ne j\le p\). The problem is NP-hard, but Feldman and Ruhl [FOCS ’99; SICOMP ’06] gave a novel \(n^{O(p)}\) 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-\((k_1, k_2)\) problem is defined as follows: given an edge-weighted directed graph \(G=(V,E)\) with weight function \(\omega : E\rightarrow \mathbb {R}_{\ge 0}\), two terminal vertices \(s, t\), and integers \(k_1, k_2\) ; the objective is to find a set of \(k_1\) paths \(F_1, F_2, \ldots , F_{k_1}\) from \(s\leadsto t\) and \(k_2\) paths \(B_1, B_2, \ldots , B_{k_2}\) from \(t\leadsto s\) such that \(\sum _{e\in E} \omega (e)\cdot \phi (e)\) is minimized, where \(\phi (e)= \max \Big \{|\{i : i\in [k_1], e\in F_i\}|\ ;\ |\{j : j\in [k_2], e\in B_j\}|\Big \}\). For each \(k\ge 1\), we show the following:
  • The \(2\)-SCSS-\((k,1)\) problem can be solved in \(n^{O(k)}\) time.

  • A matching lower bound for our algorithm: the \(2\)-SCSS-\((k,1)\) problem does not have an \(f(k)\cdot n^{o(k)}\) 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-\((k_1, k_2)\) problem if \(\min \{k_1, k_2\}\ge 2\). 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].

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Rajesh Hemant Chitnis
    • 1
    Email author
  • Hossein Esfandiari
    • 1
  • MohammadTaghi Hajiaghayi
    • 1
  • Rohit Khandekar
    • 2
  • Guy Kortsarz
    • 3
  • Saeed Seddighin
    • 1
  1. 1.Department of Computer ScienceUniversity of Maryland at College ParkCollege ParkUSA
  2. 2.KCG Holdings Inc.New YorkUSA
  3. 3.Department of Computer ScienceRutgers University-CamdenCamdenUSA

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