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DOI: 10.4230/LIPIcs.FSTTCS.2025.39

Improved Approximation for Pathwidth One Vertex Deletion and Parameterized Complexity of Its Variants

Authors: Satyabrata Jana, Soumen Mandal, Ashutosh Rai, and Saket Saurabh

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

The pathwidth of a graph is a measure of how path-like the graph is. The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph G and an integer k, one can delete at most k vertices from G so that the remaining graph has pathwidth at most one. This is a natural variation of the classical Feedback vertex Set (FVS) problem, where the deletion of at most k vertices results in a graph of treewidth at most one. In this work, we investigate POVD in the realm of approximation algorithms. We first design a 3-approximation algorithm for POVD running in polynomial time. Then, using this constant factor approximation algorithm, we obtain a randomized parameterized approximation algorithm for POVD running in time 𝒪^*((h_β)^k), that improves the fastest existing running times for approximation ratios in the range (1.76147,3). Here the constant h_β depends on the approximation factor β alone and has value 2^{(3-β)}, which lies in the range (1,2.3596), when β ∈ (1.76147,3). Taking inspiration from two extensively studied problems, namely Connected FVS and Independent FVS, we investigate two variations of the POVD problem from the perspective of parameterized algorithms. These variations are the connected variant, called Connected pathwidth One Vertex Deletion (CPOVD) and the independent variant, called Independent Pathwidth One Vertex Deletion (IPOVD). While in CPOVD the subgraph G[S] induced by the vertices to be deleted needs to be connected, in IPOVD it needs to be independent. Specifically, we show the following results. - CPOVD can be solved in {𝒪}^*(14^k) time and admits no polynomial kernel unless NP ⊆ {co-NP/poly}. - IPOVD can be solved in {𝒪}^*(7^k) time, and admits a kernel of size 𝒪(k³).

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Satyabrata Jana, Soumen Mandal, Ashutosh Rai, and Saket Saurabh. Improved Approximation for Pathwidth One Vertex Deletion and Parameterized Complexity of Its Variants. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jana_et_al:LIPIcs.FSTTCS.2025.39,
  author =	{Jana, Satyabrata and Mandal, Soumen and Rai, Ashutosh and Saurabh, Saket},
  title =	{{Improved Approximation for Pathwidth One Vertex Deletion and Parameterized Complexity of Its Variants}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{39:1--39:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.39},
  URN =		{urn:nbn:de:0030-drops-251192},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.39},
  annote =	{Keywords: Pathwidth, Parameterized complexity, Approximation, Kernelization}
}

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DOI: 10.4230/LIPIcs.MFCS.2023.56

Parameterized Approximation Scheme for Feedback Vertex Set

Authors: Satyabrata Jana, Daniel Lokshtanov, Soumen Mandal, Ashutosh Rai, and Saket Saurabh

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract

Feedback Vertex Set (FVS) is one of the most studied vertex deletion problems in the field of graph algorithms. In the decision version of the problem, given a graph G and an integer k, the question is whether there exists a set S of at most k vertices in G such that G-S is acyclic. It is one of the first few problems which were shown to be NP-complete, and has been extensively studied from the viewpoint of approximation and parameterized algorithms. The best-known polynomial time approximation algorithm for FVS is a 2-factor approximation, while the best known deterministic and randomized FPT algorithms run in time 𝒪^*(3.460^k) and 𝒪^*(2.7^k) respectively. In this paper, we contribute to the newly established area of parameterized approximation, by studying FVS in this paradigm. In particular, we combine the approaches of parameterized and approximation algorithms for the study of FVS, and achieve an approximation guarantee with a factor better than 2 in randomized FPT running time, that improves over the best known parameterized algorithm for FVS. We give three simple randomized (1+ε) approximation algorithms for FVS, running in times 𝒪^*(2^{εk}⋅ 2.7^{(1-ε)k}), 𝒪^*(({(4/(1+ε))^{(1+ε)}}⋅{(ε/3)^ε})^k), and 𝒪^*(4^{(1-ε)k}) respectively for every ε ∈ (0,1). Combining these three algorithms, we obtain a factor (1+ε) approximation algorithm for FVS, which has better running time than the best-known (randomized) FPT algorithm for every ε ∈ (0, 1). This is the first attempt to look at a parameterized approximation of FVS to the best of our knowledge. Our algorithms are very simple, and they rely on some well-known reduction rules used for arriving at FPT algorithms for FVS.

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Satyabrata Jana, Daniel Lokshtanov, Soumen Mandal, Ashutosh Rai, and Saket Saurabh. Parameterized Approximation Scheme for Feedback Vertex Set. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jana_et_al:LIPIcs.MFCS.2023.56,
  author =	{Jana, Satyabrata and Lokshtanov, Daniel and Mandal, Soumen and Rai, Ashutosh and Saurabh, Saket},
  title =	{{Parameterized Approximation Scheme for Feedback Vertex Set}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{56:1--56:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.56},
  URN =		{urn:nbn:de:0030-drops-185902},
  doi =		{10.4230/LIPIcs.MFCS.2023.56},
  annote =	{Keywords: Feedback Vertex Set, Parameterized Approximation}
}

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Improved Approximation for Pathwidth One Vertex Deletion and Parameterized Complexity of Its Variants

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Parameterized Approximation Scheme for Feedback Vertex Set

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