17 Search Results for "Majumdar, Diptapriyo"

Document
DOI: 10.4230/LIPIcs.WG.2026.18
Polynomial Kernels for Spanning Tree with Diversity Requirements

Authors: Petr A. Golovach, Diptapriyo Majumdar, and Saket Saurabh

Published in: LIPIcs, Volume 376, 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)

Abstract
Given a connected undirected graph G, a spanning tree is a subgraph T of G such that V(T) = V(G) and T is a tree. A collection of 𝓁 spanning trees T₁,…,T_{𝓁} is {{pairwise k-diverse}} if for every i ≠ j, |E(T_i) △ E(T_j)| ≥ k. Given a connected undirected graph G and integers p, q, k, 𝓁, {Leaf&Internal-Constrained Diverse Spanning Trees} asks whether there are 𝓁 distinct spanning trees T₁,…,T_{𝓁} of G that are {{pairwise k-diverse}} such that each tree has at least p leaves and at least q internal vertices. Similarly, {Leaf&Non-terminal-Constrained Diverse Spanning Trees} takes a connected undirected graph G, V_NT ⊆ V(G), and three integers p, k, 𝓁, and asks if G has 𝓁 spanning trees that are {{pairwise k-diverse}}, and each has at least p leaves and contains the vertices of V_NT as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for {Leaf&Internal-Constrained Diverse Spanning Trees} and {Leaf&Non-terminal-Constrained Diverse Spanning Trees}, when parameterized by p + q + k + 𝓁 and p + |V_NT| + k + 𝓁, respectively.
Cite as

Petr A. Golovach, Diptapriyo Majumdar, and Saket Saurabh. Polynomial Kernels for Spanning Tree with Diversity Requirements. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{golovach_et_al:LIPIcs.WG.2026.18,
  author =	{Golovach, Petr A. and Majumdar, Diptapriyo and Saurabh, Saket},
  title =	{{Polynomial Kernels for Spanning Tree with Diversity Requirements}},
  booktitle =	{52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-430-7},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{376},
  editor =	{Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.18},
  URN =		{urn:nbn:de:0030-drops-261840},
  doi =		{10.4230/LIPIcs.WG.2026.18},
  annote =	{Keywords: Parameterized Complexity, Kernelization, Diverse Solutions, Diverse Spanning Trees}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.78
Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors

Authors: Roohani Sharma and Michał Włodarczyk

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Let ℱ be a finite family of graphs. In the ℱ-Deletion problem, one is given a graph G and an integer k, and the goal is to find k vertices whose deletion results in a graph with no minor from the family ℱ. This may be regarded as a far-reaching generalization of Vertex Cover and Feedback vertex Set. In their seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family ℱ contains a planar graph. As the size of their kernel is g(ℱ) ⋅ k^{f(ℱ)}, a natural follow-up question was whether the dependence on ℱ in the exponent of k can be avoided. The answer turned out to be negative: Giannopoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] proved that this is already inevitable for the special case of the Treewidth-η-Deletion problem. In this work, we show that this non-uniformity can be avoided at the expense of a small loss. First, we present a simple 2-approximate kernelization algorithm for Treewidth-η-Deletion with a kernel size g(η) ⋅ k⁶. Next, we show that the approximation factor can be made arbitrarily close to 1, if we settle for a kernelization protocol with 𝒪(1) calls to an oracle that solves instances of size bounded by a uniform polynomial in k. We extend the above results to general ℱ-Deletion, whenever ℱ contains a planar graph, as long as an oracle for Treewidth-η-Deletion is available for small instances. Notably, all our constants are computable functions of ℱ and our techniques work also when some graphs in ℱ may be disconnected. Our results rely on two novel techniques. First, we transform so-called "near-protrusion decompositions" into true protrusion decompositions by sacrificing a small accuracy loss. Secondly, we show how to optimally compress such a decomposition with respect to general ℱ-Deletion. Using our second technique, we also obtain linear kernels on sparse graph classes when ℱ contains a planar graph, whereas the previously known theorems required all graphs in ℱ to be connected. Specifically, we generalize the kernelization algorithm by Kim, Langer, Paul, Reidl, Rossmanith, Sau & Sikdar [TALG 2015] on graph classes that exclude a topological minor.
Cite as

Roohani Sharma and Michał Włodarczyk. Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 78:1-78:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{sharma_et_al:LIPIcs.STACS.2026.78,
  author =	{Sharma, Roohani and W{\l}odarczyk, Micha{\l}},
  title =	{{Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{78:1--78:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.78},
  URN =		{urn:nbn:de:0030-drops-255674},
  doi =		{10.4230/LIPIcs.STACS.2026.78},
  annote =	{Keywords: kernelization, graph minors, treewidth, uniform kernels, minor hitting}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.23
Enumeration Kernels for Vertex Cover and Feedback Vertex Set

Authors: Marin Bougeret, Guilherme C. M. Gomes, Vinicius F. dos Santos, and Ignasi Sau

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Enumerative kernelization is a recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms. Its study began with the paper of Creignou et al. [Theory Comput. Syst., 2017], and development in the area has started to accelerate with the work of Golovach et al. [J. Comput. Syst. Sci., 2022]. The latter introduced polynomial-delay enumeration kernels and applied them in the study of structural parameterizations of the Matching Cut problem and some variants. Few other results, mostly on Longest Path and some generalizations of Matching Cut, have also been developed. However, little success has been seen in enumeration versions of Vertex Cover and Feedback Vertex Set, some of the most studied problems in kernelization. In this paper, we address this shortcoming. Our first result is a polynomial-delay enumeration kernel with 2k vertices for Enum Vertex Cover, where we wish to list all solutions with at most k vertices. This is obtained by developing a non-trivial lifting algorithm for the classical crown decomposition reduction rule, and directly improves upon the kernel with 𝒪(k²) vertices derived from the work of Creignou et al. Our other result is a polynomial-delay enumeration kernel with 𝒪(k³) vertices and edges for Enum Feedback Vertex Set; the proof is inspired by some ideas of Thomassé [TALG, 2010], but with a weaker bound on the kernel size due to difficulties in applying the q-expansion technique.
Cite as

Marin Bougeret, Guilherme C. M. Gomes, Vinicius F. dos Santos, and Ignasi Sau. Enumeration Kernels for Vertex Cover and Feedback Vertex Set. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bougeret_et_al:LIPIcs.IPEC.2025.23,
  author =	{Bougeret, Marin and C. M. Gomes, Guilherme and dos Santos, Vinicius F. and Sau, Ignasi},
  title =	{{Enumeration Kernels for Vertex Cover and Feedback Vertex Set}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.23},
  URN =		{urn:nbn:de:0030-drops-251552},
  doi =		{10.4230/LIPIcs.IPEC.2025.23},
  annote =	{Keywords: Kernelization, Enumeration, Vertex cover, Crown decomposition, Feedback vertex set}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.43
A Finer View of the Parameterized Landscape of Labeled Graph Contractions

Authors: Yashaswini Mathur and Prafullkumar Tale

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

Abstract
We study the Labeled Contractibility problem, where the input consists of two vertex-labeled graphs G and H, and the goal is to determine whether H can be obtained from G via a sequence of edge contractions. Lafond and Marchand [WADS 2025] initiated the parameterized complexity study of this problem, showing it to be W[1]-hard when parameterized by the number k of allowed contractions. They also proved that the problem is fixed-parameter tractable when parameterized by the tree-width tw of G, via an application of Courcelle’s theorem resulting in a non-constructive algorithm. In this work, we present a constructive fixed-parameter algorithm for Labeled Contractibility with running time 2^{𝒪(tw²)} ⋅ |V(G)|^{𝒪(1)}. We also prove that unless the Exponential Time Hypothesis ({ETH}) fails, it does not admit an algorithm running in time 2^{o(tw²)} ⋅ |V(G)|^{𝒪(1)}. This result adds Labeled Contractibility to a small list of problems that admit such a lower bound and matching algorithm. We further strengthen existing hardness results by showing that the problem remains NP-complete even when both input graphs have bounded maximum degree. We also investigate parameterizations by (k + δ(G)) where δ(G) denotes the degeneracy of G, and rule out the existence of subexponential-time algorithms. This answers question raised in Lafond and Marchand [WADS 2025]. We additionally provide an improved FPT algorithm with better dependence on (k + δ(G)) than previously known. Finally, we analyze a brute-force algorithm for Labeled Contractibility with running time |V(H)|^{𝒪(|V(G)|)}, and show that this running time is optimal under {ETH}.
Cite as

Yashaswini Mathur and Prafullkumar Tale. A Finer View of the Parameterized Landscape of Labeled Graph Contractions. 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. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mathur_et_al:LIPIcs.FSTTCS.2025.43,
  author =	{Mathur, Yashaswini and Tale, Prafullkumar},
  title =	{{A Finer View of the Parameterized Landscape of Labeled Graph Contractions}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{43:1--43:19},
  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.2025.43},
  URN =		{urn:nbn:de:0030-drops-251237},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.43},
  annote =	{Keywords: Labeled Contraction, ETH Lower-bound, Treewidth, NP-hard}
}
Document
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).
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.5
Graph Coloring Below Guarantees via Co-Triangle Packing

Authors: Shyan Akmal and Tomohiro Koana

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
In the 𝓁-Coloring problem, we are given a graph on n nodes, and tasked with determining if its vertices can be properly colored using 𝓁 colors. In this paper we study below-guarantee graph coloring, which tests whether an n-vertex graph can be properly colored using g-k colors, where g is a trivial upper bound such as n. We introduce an algorithmic framework that builds on a packing of co-triangles K₃ (independent sets of three vertices): the algorithm greedily finds co-triangles and employs a win-win analysis. If many are found, we immediately return yes; otherwise these co-triangles form a small co-triangle modulator, whose deletion makes the graph co-triangle-free. Extending the work of [Gutin et al., SIDMA 2021], who solved 𝓁-Coloring (for any 𝓁) in randomized O^∗(2^k) time when given a K₂-free modulator of size k, we show that this problem can likewise be solved in randomized O^*(2^{k}) time when given a K₃-free modulator of size k. This result in turn yields a randomized O^*(2^{3k/2}) algorithm for (n-k)-Coloring (also known as Dual Coloring), improving the previous O^*(4^k) bound. We then introduce a smaller parameterization, (ω+μ-k)-Coloring, where ω is the clique number and μ is the size of a maximum matching in the complement graph; since ω+μ ≤ n for any graph, this problem is strictly harder. Using the same co-triangle-packing argument, we obtain a randomized O^*(2^{6k}) algorithm, establishing its fixed-parameter tractability for a smaller parameter. Complementing this finding, we show that no fixed-parameter tractable algorithm exists for (ω-k)-Coloring or (μ-k)-Coloring under standard complexity assumptions.
Cite as

Shyan Akmal and Tomohiro Koana. Graph Coloring Below Guarantees via Co-Triangle Packing. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{akmal_et_al:LIPIcs.ISAAC.2025.5,
  author =	{Akmal, Shyan and Koana, Tomohiro},
  title =	{{Graph Coloring Below Guarantees via Co-Triangle Packing}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{5:1--5:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.5},
  URN =		{urn:nbn:de:0030-drops-249130},
  doi =		{10.4230/LIPIcs.ISAAC.2025.5},
  annote =	{Keywords: coloring, parameterized algorithms, algebraic algorithms, above-guarantee, below-guarantee, subset convolution, determinants}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.48
Quadratic Kernel for Cliques or Trees Vertex Deletion

Authors: Soh Kumabe

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
We consider Cliques or Trees Vertex Deletion, which is a hybrid of two fundamental parameterized problems: Cluster Vertex Deletion and Feedback Vertex Set. In this problem, we are given an undirected graph G and an integer k, and asked to find a vertex subset X of size at most k such that each connected component of G-X is either a clique or a tree. Jacob et al. (ISAAC, 2024) provided a kernel of O(k⁵) vertices for this problem, which was recently improved to O(k⁴) by Tsur (IPL, 2025). Our main result is a kernel of O(k²) vertices. This result closes the gap between the kernelization result for Feedback Vertex Set, which corresponds to the case where each connected component of G-X must be a tree. Although both cluster vertex deletion number and feedback vertex set number are well-studied structural parameters, little attention has been given to parameters that generalize both of them. In fact, the lowest common well-known generalization of them is clique-width, which is a highly general parameter. To fill the gap here, we initiate the study of the cliques or trees vertex deletion number as a structural parameter. We prove that Longest Cycle, which is a fundamental problem that does not admit o(n^k)-time algorithm unless ETH fails when k is the clique-width, becomes fixed-parameter tractable when parameterized by the cliques or trees vertex deletion number.
Cite as

Soh Kumabe. Quadratic Kernel for Cliques or Trees Vertex Deletion. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kumabe:LIPIcs.ISAAC.2025.48,
  author =	{Kumabe, Soh},
  title =	{{Quadratic Kernel for Cliques or Trees Vertex Deletion}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{48:1--48:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.48},
  URN =		{urn:nbn:de:0030-drops-249568},
  doi =		{10.4230/LIPIcs.ISAAC.2025.48},
  annote =	{Keywords: Fixed-Parameter Tractability, Kernelization, Deletion to Scattered Graph Classes, Cluster Vertex Deletion, Feedback Vertex Set}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.23
Precoloring Extension with Demands on Paths

Authors: Arun Kumar Das, Michal Opler, and Tomáš Valla

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
Let G be a graph with a set of precolored vertices, and let us be given an integer distance parameter d and a set of integer demands d₁,… ,d_c. The Distance Precoloring Extension with Demands (DPED) problem is to compute a vertex c-coloring of G such that the following three conditions hold: (i) the resulting coloring respects the colors of the precolored vertices, (ii) the distance of two vertices of the same color is at least d, and (iii) the number of vertices colored by color i is exactly d_i. This problem is motivated by a program scheduling in commercial broadcast channels with constraints on content repetition and placement, which leads precisely to the DPED problem for paths. In this paper, we study DPED on paths and present a polynomial time exact algorithm when precolored vertices are restricted to the two ends of the path and devise an approximation algorithm for DPED with an additive approximation factor polynomially bounded by d and the number of precolored vertices. Then, we prove that the Distance Precoloring Extension problem on paths, a less restrictive version of DPED without the demand constraints, and then DPED itself, is NP-complete. Motivated by this result, we further study the parameterized complexity of DPED on paths. We establish that the DPED problem on paths is W[1]-hard when parameterized by the number of colors and the distance. On the positive side, we devise a fixed parameter tractable (FPT) algorithm for DPED on paths when the number of colors, the distance, and the number of precolored vertices are considered as the parameters. Moreover, we prove that Distance Precoloring Extension is FPT parameterized by the distance. As a byproduct, we also obtain several results for the Distance List Coloring problem on paths.
Cite as

Arun Kumar Das, Michal Opler, and Tomáš Valla. Precoloring Extension with Demands on Paths. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{das_et_al:LIPIcs.ISAAC.2025.23,
  author =	{Das, Arun Kumar and Opler, Michal and Valla, Tom\'{a}\v{s}},
  title =	{{Precoloring Extension with Demands on Paths}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.23},
  URN =		{urn:nbn:de:0030-drops-249319},
  doi =		{10.4230/LIPIcs.ISAAC.2025.23},
  annote =	{Keywords: precoloring extension, distance coloring, FPT, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2024.19
Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

Authors: Dipayan Chakraborty, Florent Foucaud, Diptapriyo Majumdar, and Prafullkumar Tale

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
Foucaud et al. [ICALP 2024] demonstrated that some problems in NP can admit (tight) double-exponential lower bounds when parameterized by treewidth or vertex cover number. They showed these first-of-their-kind results by proving conditional lower bounds for certain graph problems, in particular, the metric-based identification problems (Strong) Metric Dimension. We continue this line of research and highlight the usefulness of this type of problems, to prove relatively rare types of (tight) lower bounds. We investigate fine-grained algorithmic aspects of classical (non-metric based) identification problems in graphs, namely Locating-Dominating Set, and in set systems, namely Test Cover. In the first problem, an input is a graph G on n vertices and an integer k, and the objective is to decide whether there is a subset S of k vertices such that any two distinct vertices not in S are dominated by distinct subsets of S. In the second problem, an input is a set of items U, a collection of subsets ℱ of U called tests, and an integer k, and the objective is to select a set S of at most k tests such that any two distinct items are contained in a distinct subset of tests of S. For our first result, we adapt the techniques introduced by Foucaud et al. [ICALP 2024] to prove similar (tight) lower bounds for these two problems. - Locating-Dominating Set (respectively, Test Cover) parameterized by the treewidth of the input graph (respectively, the natural auxiliary graph) does not admit an algorithm running in time 2^{2^o(tw)} ⋅ poly(n) (respectively, 2^{2^o(tw)} ⋅ poly(|U| + |ℱ|))), unless the ETH fails. This augments the short list of NP-Complete problems that admit tight double-exponential lower bounds when parameterized by treewidth, and shows that "local" (non-metric-based) problems can also admit such bounds. We show that these lower bounds are tight by designing treewidth-based dynamic programming schemes with matching running times. Next, we prove that these two problems also admit "exotic" (and tight) lower bounds, when parameterized by the solution size k. We prove that unless the ETH fails, - Locating-Dominating Set does not admit an algorithm running in time 2^o(k²) ⋅ poly(n), nor a polynomial-time kernelization algorithm that reduces the solution size and outputs a kernel with 2^o(k) vertices, and - Test Cover does not admit an algorithm running in time 2^{2^o(k)} ⋅ poly(|U| + |ℱ|) nor a kernel with 2^{2^o(k)} vertices. Again, we show that these lower bounds are tight by designing (kernelization) algorithms with matching running times. To the best of our knowledge, Locating-Dominating Set is the first known problem which is FPT when parameterized by solution size k, where the optimal running time has a quadratic function in the exponent. These results also extend the (very) small list of problems that admit an ETH-based lower bound on the number of vertices in a kernel, and (for Test Cover) a double-exponential lower bound when parameterized by the solution size. Whereas it is the first example, to the best of our knowledge, that admit a double exponential lower bound for the number of vertices.
Cite as

Dipayan Chakraborty, Florent Foucaud, Diptapriyo Majumdar, and Prafullkumar Tale. Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chakraborty_et_al:LIPIcs.ISAAC.2024.19,
  author =	{Chakraborty, Dipayan and Foucaud, Florent and Majumdar, Diptapriyo and Tale, Prafullkumar},
  title =	{{Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.19},
  URN =		{urn:nbn:de:0030-drops-221469},
  doi =		{10.4230/LIPIcs.ISAAC.2024.19},
  annote =	{Keywords: Identification Problems, Locating-Dominating Set, Test Cover, Double Exponential Lower Bound, ETH, Kernelization Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2024.41
A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees

Authors: Ashwin Jacob, Diptapriyo Majumdar, and Meirav Zehavi

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
The class of graph deletion problems has been extensively studied in theoretical computer science, particularly in the field of parameterized complexity. Recently, a new notion of graph deletion problems was introduced, called deletion to scattered graph classes, where after deletion, each connected component of the graph should belong to at least one of the given graph classes. While fixed-parameter algorithms were given for a wide variety of problems, little progress has been made on the kernelization complexity of any of them. Here, we present the first non-trivial polynomial kernel for one such deletion problem, where, after deletion, each connected component should be a clique or a tree - that is, as dense as possible or as sparse as possible (while being connected). We develop a kernel of O(k⁵) vertices for the same.
Cite as

Ashwin Jacob, Diptapriyo Majumdar, and Meirav Zehavi. A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jacob_et_al:LIPIcs.ISAAC.2024.41,
  author =	{Jacob, Ashwin and Majumdar, Diptapriyo and Zehavi, Meirav},
  title =	{{A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{41:1--41:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.41},
  URN =		{urn:nbn:de:0030-drops-221687},
  doi =		{10.4230/LIPIcs.ISAAC.2024.41},
  annote =	{Keywords: Parameterized Complexity, Kernelization, Scattered Graph Classes, New Expansion Lemma, Cliques or Trees Vertex Deletion}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.16
Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints

Authors: Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, and Abhishek Sahu

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Given an undirected graph G and a set A ⊆ V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G)⧵A. An A-path is called an (A,𝓁)-path if the length of the path is exactly 𝓁. In the (A, 𝓁)-Path Packing problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, 𝓁)-paths in G or not. The problem is already known to be fixed-parmeter tractable when parameterized by k+𝓁 via color coding while it remains Para-NP-hard when parameterized by k (Hamiltonian Path) or 𝓁 (P₃-Partition) alone. Therefore, a logical direction to pursue this problem is to examine it in relation to structural parameters. Belmonte et al. initiated a study along these lines and proved that ALPP parameterized by pw+|A| is W[1]-hard where pw is the pathwidth of G. In this paper, we strengthen their result and prove that it is unlikely that ALPP is fixed-parameter tractable even with respect to a bigger parameter (|A|+dtp) where dtp denotes the distance between G and a path graph (distance to path). We use a randomized reduction to achieve the mentioned result. Toward this, we prove a lemma similar to the influential "isolation lemma": Given a set system (X,ℱ) if the elements of X are assigned a weight uniformly at random from a set of values fairly large, then each subset in ℱ will have a unique weight with high probability. We believe that this result will be useful beyond the scope of this paper. ALPP being hard even for structural parameters like distance to path+|A| rules out the possibility of any FPT algorithms for many well-known other structural parameters, including FVS+|A| and treewidth+|A|. There is a straightforward FPT algorithm for ALPP parameterized by vc, the vertex cover number of the input graph. Following this, we consider the parameters CVD(cluster vertex deletion)+|A| and CVD+|𝓁| and show the problem to be FPT with respect to these parameters. Note that CVD is incomparable to the treewidth of a graph and has been in vogue recently.
Cite as

Susobhan Bandopadhyay, Aritra Banik, Diptapriyo Majumdar, and Abhishek Sahu. Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bandopadhyay_et_al:LIPIcs.MFCS.2024.16,
  author =	{Bandopadhyay, Susobhan and Banik, Aritra and Majumdar, Diptapriyo and Sahu, Abhishek},
  title =	{{Tractability of Packing Vertex-Disjoint A-Paths Under Length Constraints}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.16},
  URN =		{urn:nbn:de:0030-drops-205725},
  doi =		{10.4230/LIPIcs.MFCS.2024.16},
  annote =	{Keywords: Parameterized complexity, (A,𝓁)-Path Packing, Kernelization, Randomized-Exponential Time Hypothesis, Graph Classes}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.45
Finding a Highly Connected Steiner Subgraph and its Applications

Authors: Eduard Eiben, Diptapriyo Majumdar, and M. S. Ramanujan

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

Abstract
Given a (connected) undirected graph G, a set X ⊆ V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ⊇ X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness). Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.
Cite as

Eduard Eiben, Diptapriyo Majumdar, and M. S. Ramanujan. Finding a Highly Connected Steiner Subgraph and its Applications. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.MFCS.2023.45,
  author =	{Eiben, Eduard and Majumdar, Diptapriyo and Ramanujan, M. S.},
  title =	{{Finding a Highly Connected Steiner Subgraph and its Applications}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{45:1--45:15},
  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.2023.45},
  URN =		{urn:nbn:de:0030-drops-185793},
  doi =		{10.4230/LIPIcs.MFCS.2023.45},
  annote =	{Keywords: Parameterized Complexity, Steiner Subgraph Extension, p-edge-connected graphs, Matroids, Representative Families}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.18
Parameterized Complexity of Deletion to Scattered Graph Classes

Authors: Ashwin Jacob, Diptapriyo Majumdar, and Venkatesh Raman

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
Graph-modification problems, where we add/delete a small number of vertices/edges to make the given graph to belong to a simpler graph class, is a well-studied optimization problem in all algorithmic paradigms including classical, approximation and parameterized complexity. Specifically, graph-deletion problems, where one needs to delete at most k vertices to place it in a given non-trivial hereditary (closed under induced subgraphs) graph class, captures several well-studied problems including Vertex Cover, Feedback Vertex Set, Odd Cycle Transveral, Cluster Vertex Deletion, and Perfect Deletion. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. While a precise characterization of the graph classes for which the problem is fixed-parameter tractable (FPT) is elusive, it has long been known that if the graph class is characterized by a finite set of forbidden graphs, then the problem is FPT. In this paper, we initiate a study of a natural variation of the problem of deletion to scattered graph classes where we need to delete at most k vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes. A simple hitting set based approach is no longer feasible even if each of the graph classes is characterized by finite forbidden sets. As our main result, we show that this problem (in the case where each graph class has a finite forbidden set) is fixed-parameter tractable by a O^*(2^(k^O(1))) algorithm, using a combination of the well-known techniques in parameterized complexity - iterative compression and important separators. Our approach follows closely that of a related problem in the context of satisfiability [Ganian, Ramanujan, Szeider, TAlg 2017], where one wants to find a small backdoor set so that the resulting CSP (constraint satisfaction problem) instance belongs to one of several easy instances of satisfiability. While we follow the main idea from this work, there are some challenges for our problem which we needed to overcome. When there are two graph classes with finite forbidden sets to get to, and if one of the forbidden sets has a path, then we show that the problem has a (better) singly exponential algorithm and a polynomial sized kernel. We also design an efficient FPT algorithm for a special case when one of the graph classes has an infinite forbidden set. Specifically, we give a O^*(4^k) algorithm to determine whether k vertices can be deleted from a given graph so that in the resulting graph, each connected component is a tree (the sparsest connected graph) or a clique (the densest connected graph).
Cite as

Ashwin Jacob, Diptapriyo Majumdar, and Venkatesh Raman. Parameterized Complexity of Deletion to Scattered Graph Classes. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jacob_et_al:LIPIcs.IPEC.2020.18,
  author =	{Jacob, Ashwin and Majumdar, Diptapriyo and Raman, Venkatesh},
  title =	{{Parameterized Complexity of Deletion to Scattered Graph Classes}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.18},
  URN =		{urn:nbn:de:0030-drops-133210},
  doi =		{10.4230/LIPIcs.IPEC.2020.18},
  annote =	{Keywords: Parameterized Complexity, Scattered Graph Classes, Important Separators}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.19
Parameterized Pre-Coloring Extension and List Coloring Problems

Authors: Gregory Gutin, Diptapriyo Majumdar, Sebastian Ordyniak, and Magnus Wahlström

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V(G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λ_P: X → Q for X ⊆ V(G), decide whether λ_P can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O*(2^k)-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n-k colors for every v ∈ V(G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k²) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k²) colors.
Cite as

Gregory Gutin, Diptapriyo Majumdar, Sebastian Ordyniak, and Magnus Wahlström. Parameterized Pre-Coloring Extension and List Coloring Problems. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gutin_et_al:LIPIcs.STACS.2020.19,
  author =	{Gutin, Gregory and Majumdar, Diptapriyo and Ordyniak, Sebastian and Wahlstr\"{o}m, Magnus},
  title =	{{Parameterized Pre-Coloring Extension and List Coloring Problems}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.19},
  URN =		{urn:nbn:de:0030-drops-118801},
  doi =		{10.4230/LIPIcs.STACS.2020.19},
  annote =	{Keywords: Parameterized Algorithms, W-hardness, Kernelization, Graph Coloring, List Coloring}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2016.21
Structural Parameterizations of Feedback Vertex Set

Authors: Diptapriyo Majumdar

Published in: LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Abstract
A feedback vertex set in an undirected graph is a subset of vertices whose removal results in an acyclic graph. It is well-known that the problem of finding a minimum sized (or k sized in case of decision version of) feedback vertex set (FVS) is polynomial time solvable in (sub)-cubic graphs, in pseudo-forests (graphs where each component has at most one cycle) and mock-forests (graph where each vertex is part of at most one cycle). In general graphs, it is known that the problem is NP-Complete, and has an O*((3.619)^k) fixed-parameter algorithm and an O(k^2) kernel where k, the solution size is the parameter. We consider the parameterized and kernelization complexity of feedback vertex set where the parameter is the size of some structure of the input. In particular, we show that * FVS is fixed-parameter tractable, but is unlikely to have polynomial sized kernel when parameterized by the number of vertices of the graph whose degree is at least 4. This answers a question asked in an earlier paper. * When parameterized by k, the number of vertices, whose deletion results in a pseudo-forest, we give an O(k^6) vertices kernel improving from the previously known O(k^{10}) bound. * When parameterized by the number k of vertices, whose deletion results in a mock-d-forest, we give a kernel consisting of O(k^{3d+3}) vertices and prove a lower bound of Omega(k^{d+2}) vertices (under complexity theoretic assumptions). Mock-d-forest for a constant d is a mock-forest where each component has at most d cycles.
Cite as

Diptapriyo Majumdar. Structural Parameterizations of Feedback Vertex Set. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{majumdar:LIPIcs.IPEC.2016.21,
  author =	{Majumdar, Diptapriyo},
  title =	{{Structural Parameterizations of Feedback Vertex Set}},
  booktitle =	{11th International Symposium on Parameterized and Exact Computation (IPEC 2016)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-023-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{63},
  editor =	{Guo, Jiong and Hermelin, Danny},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.21},
  URN =		{urn:nbn:de:0030-drops-69337},
  doi =		{10.4230/LIPIcs.IPEC.2016.21},
  annote =	{Keywords: Parameterized Complexity, Kernelization, Feedback Vertex Set, Structural Parameterization}
}
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