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An ETH-Tight Algorithm for Bidirected Steiner Connectivity

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Algorithms and Data Structures (WADS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14079))

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Abstract

In the Strongly Connected Steiner Subgraph problem, we are given an n-vertex digraph D, a weight function \(w:A(D)\mapsto {\mathbb {R}}^+\) on the arc set of D, and a set of k terminals \(Q\subseteq V(D)\), and our objective is to find a strongly connected subgraph of D containing Q with minimum total weight. The problem is known to be W[1]-hard on general digraphs. However on bi-directed graphs (digraphs where, if uv is an arc then so is vu) with symmetric weight function \(w:A(D)\mapsto {\mathbb {R}}^+\) (i.e., \(w(uv)=w(vu)\) for any \(uv\in A(D)\)), Chitnis, Feldmann and Manurangsi [TALG 2021] showed that the problem is fixed parameter tractable (FPT) with running time \(2^{\mathcal {O}({k^2})}n^{\mathcal {O}(1)}\), where n is the input length. They also show that, unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm for the problem on bi-directed graphs with running time \(2^{o(k)}n^{\mathcal {O}(1)}\). They left the existence of a single-exponential in k time algorithm as an open problem. We resolve this question, by designing an algorithm for the problem running in time \(2^{\mathcal {O}(k)}n^{\mathcal {O}(1)}\) that is asymptotically tight under ETH, thereby closing the gap between the upper and lower-bounds for this problem.

Chitnis, Feldmann and Manurangsi [TALG 2021] showed that an optimum solution to this problem can always be described by a collection of trees, that are mapped to the input graph via homomorphisms, and glued together at the terminal vertices. This structural result allows us to design an algorithm via the combination of a Dreyfus-Wagner style dynamic programming algorithm and the notion of representative sets over linear matroids.

Supported by NSF award CCF-2008838, Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18, and ERC grants LOPRE (grant agreement No. 819416) and PARAPATH.

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Notes

  1. 1.

    The proofs of the results marked with \(\star \) are deferred to the full version of the paper.

  2. 2.

    Here, we slightly abuse the notation and consider z to be a labelled vertex in D with label \(k+1\) if \(\ell _R(h)=k+1\).

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Author information

Authors and Affiliations

  1. University of California, Santa Barbara, USA

    Daniel Lokshtanov

  2. Chennai Mathematical Institute, Chennai, India

    Pranabendu Misra

  3. Indian Institute of Technology, Hyderbad, India

    Fahad Panolan

  4. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Saket Saurabh

  5. University of Bergen, Bergen, Norway

    Saket Saurabh

  6. Ben-Gurion University, Beersheba, Israel

    Meirav Zehavi

Authors
  1. Daniel Lokshtanov
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  2. Pranabendu Misra
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  3. Fahad Panolan
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  4. Saket Saurabh
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  5. Meirav Zehavi
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Corresponding author

Correspondence to Fahad Panolan .

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Editors and Affiliations

  1. Carleton University, Ottawa, ON, Canada

    Pat Morin

  2. University of California, Santa Barbara, CA, USA

    Subhash Suri

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Lokshtanov, D., Misra, P., Panolan, F., Saurabh, S., Zehavi, M. (2023). An ETH-Tight Algorithm for Bidirected Steiner Connectivity. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_39

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  • DOI: https://doi.org/10.1007/978-3-031-38906-1_39

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