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DOI: 10.4230/LIPIcs.ITCS.2026.80

FPT Approximations for Connected Maximum Coverage

Authors: Tanmay Inamdar, Satyabrata Jana, Madhumita Kundu, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set (PartialConRBDS). Given a bipartite graph G = (R∪ B,E) with red vertices R and blue vertices B, an auxiliary connectivity graph G_{conn} on R, and integers k,t, the task is to find a set S ⊆ R with |S| ≤ k such that G_{conn}[S] is connected and S dominates at least t blue vertices. This formulation captures connected variants of Maximum Coverage [Hochbaum-Rao, Inf. Proc. Lett., 2020; D'Angelo-Delfaraz, AAMAS 2025], Partial Vertex Cover, and Partial Dominating Set [Khuller et al., SODA 2014; Lamprou et al., TCS 2021] via standard encodings. Limits to parameterized tractability. PartialConRBDS is W[1]-hard parameterized by k even under strong restrictions: it remains hard when G_{conn} is a clique or a star and the incidence graph G is 3-degenerate, or when G is K_{2,2}-free. Inapproximability. For every ε > 0, there is no polynomial-time (1, 1-1/e+ε)-approximation unless 𝖯 = NP. Moreover, under ETH, no algorithm running in f(k)⋅ n^{o(k)} time achieves an g(k)-approximation for k for any computable function g(⋅), or for any ε > 0, a (1-1/e+ε)-approximation for t. Graphical special cases. Partial Connected Dominating Set is W[2]-hard parameterized by k and inherits the same ETH-based f(k)⋅ n^{o(k)} inapproximability bound as above; Partial Connected Vertex Cover is W[1]-hard parameterized by k. These hardness boundaries delineate a natural "sweet spot" for study: within appropriate structural restrictions on the incidence graph, one can still aim for fine-grained (FPT) approximations. Our algorithms. We solve PartialConRBDS exactly by reducing it to Relaxed Directed Steiner Out-Tree in time (2e)^t ⋅ n^{𝒪(1)}. For biclique-free incidences (i.e., when G excludes K_{d,d} as an induced subgraph), we obtain two complementary parameterized schemes: - An Efficient Parameterized Approximation Scheme (EPAS) running in time 2^{𝒪(k² d/ε)}⋅ n^{𝒪(1)} that either returns a connected solution of size at most k covering at least (1-ε)t blue vertices, or correctly reports that no connected size-k solution covers t; and - A Parameterized Approximation Scheme (PAS) running in time 2^{𝒪(kd(k²+log d))}⋅ n^{𝒪(1/ε)} that either returns a connected solution of size at most (1+ε)k covering at least t blue vertices, or correctly reports that no connected size-k solution covers t. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

Cite as

Tanmay Inamdar, Satyabrata Jana, Madhumita Kundu, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. FPT Approximations for Connected Maximum Coverage. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 80:1-80:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{inamdar_et_al:LIPIcs.ITCS.2026.80,
  author =	{Inamdar, Tanmay and Jana, Satyabrata and Kundu, Madhumita and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav},
  title =	{{FPT Approximations for Connected Maximum Coverage}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{80:1--80:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.80},
  URN =		{urn:nbn:de:0030-drops-253674},
  doi =		{10.4230/LIPIcs.ITCS.2026.80},
  annote =	{Keywords: Partial Dominating Set, Connectivity, Maximum Coverage, FPT Approximation, Fixed-parameter Tractability}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2025.18

Bridging Treewidth and Clique-Width via Cograph-Modular-Treewidth

Authors: Václav Blažej, Satyabrata Jana, M. S. Ramanujan, and Peter Strulo

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

Abstract

Many classical graph problems - such as Max Cut, Chromatic Number, Edge Dominating Set, and Hamiltonian Cycle - are polynomial-time solvable on cographs, fixed-parameter tractable (FPT) when parameterized by treewidth, but W[1]-hard when parameterized by clique-width. In contrast, Graph Isomorphism is FPT parameterized by treewidth, but for clique-width it is known to be in XP; whether it is FPT or W[1]-hard is open. This reveals a sharp tractability gap between treewidth and clique-width. In this work, we propose a new structural graph parameter, 𝒞-modular-treewidth, which lies between treewidth and clique-width. The parameter leverages modular decomposition and restricts modules to induce graphs from a fixed class 𝒞 (e.g., cographs or edgeless graphs). By exploiting true and false twins - a hallmark of cograph-like structure - our parameter allows the design of efficient algorithms for several hard problems beyond the reach of treewidth-based methods. In this work, we show that 𝒞-modular-treewidth enables efficient solutions under suitable choices of 𝒞, opening a new pathway in the parameterized complexity landscape between treewidth and clique-width. In particular we show that - When parameterized by cograph-modular-treewidth, Isomorphism admits an FPT algorithm, whereas Chromatic Number remains W[1]-hard. - When parameterized by independent-modular-treewidth, Hamiltonian Cycle and Edge Dominating Set remain W[1]-hard.

Cite as

Václav Blažej, Satyabrata Jana, M. S. Ramanujan, and Peter Strulo. Bridging Treewidth and Clique-Width via Cograph-Modular-Treewidth. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blazej_et_al:LIPIcs.IPEC.2025.18,
  author =	{Bla\v{z}ej, V\'{a}clav and Jana, Satyabrata and Ramanujan, M. S. and Strulo, Peter},
  title =	{{Bridging Treewidth and Clique-Width via Cograph-Modular-Treewidth}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{18:1--18:18},
  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.2025.18},
  URN =		{urn:nbn:de:0030-drops-251507},
  doi =		{10.4230/LIPIcs.IPEC.2025.18},
  annote =	{Keywords: Treewidth, Clique-width, Cograph, FPT, W\lbrack1\rbrack-hard}
}

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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).

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.2025.42

Parameterized Reunion with Achromatic Number

Authors: Satyabrata Jana, Souvik Saha, Saket Saurabh, and Anannya Upasana

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

Abstract

In this paper, we study the Achromatic Number problem. Given a graph G and an integer k, the task is to determine whether there exists a proper coloring of G, using at least k colors, in which every pair of distinct colors appears on the endpoints of some edge. It was established early on that the problem is fixed-parameter tractable (FPT)- even before the formal development of parameterized complexity. In fact, Farber, Hahn, Hell, and Miller [JCTB, 1986] devised an algorithm with a running time of 𝒪(f(k) ⋅ |E(G)|). Although the exact form of f(k) was not specified, it appears to be at least doubly exponential in k. In our work, we first present an algorithm with an explicit dependence on k, and then introduce another algorithm that is parameterized by the vertex cover number of the graph. More formally, we show the following. - Achromatic Number is solvable in time 2^𝒪(k⁵)+𝒪(|E(G)|). - Achromatic Number admits a polynomial kernel when the input is restricted to a d-degenerate graph and a more efficient kernel on trees. - We also study the parameterized complexity of the problem with respect to Vertex Cover and show that it admits an FPT algorithm running in time 2^𝒪(𝓁²) ⋅ n^𝒪(1), where 𝓁 is the size of a vertex cover.

Cite as

Satyabrata Jana, Souvik Saha, Saket Saurabh, and Anannya Upasana. Parameterized Reunion with Achromatic Number. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jana_et_al:LIPIcs.ISAAC.2025.42,
  author =	{Jana, Satyabrata and Saha, Souvik and Saurabh, Saket and Upasana, Anannya},
  title =	{{Parameterized Reunion with Achromatic Number}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{42:1--42:20},
  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.2025.42},
  URN =		{urn:nbn:de:0030-drops-249502},
  doi =		{10.4230/LIPIcs.ISAAC.2025.42},
  annote =	{Keywords: Achromatic number, Coloring, Fixed-parameter tractability, Kernelization, Lower bound, W-hardness}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.39

Sampling with a Black Box: Faster Parameterized Approximation Algorithms for Vertex Deletion Problems

Authors: Barış Can Esmer and Ariel Kulik

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

In this paper, we present Sampling with a Black Box, a unified framework for the design of parameterized approximation algorithms for vertex deletion problems (e.g., Vertex Cover, Feedback Vertex Set, etc.). The framework relies on two components: - A Sampling Step. A polynomial-time randomized algorithm that, given a graph G, returns a random vertex v such that the optimum of G⧵ {v} is smaller by 1 than the optimum of G, with some prescribed probability q. We show that such algorithms exist for multiple vertex deletion problems. - A Black Box algorithm which is either an exact parameterized algorithm, a polynomial-time approximation algorithm, or a parameterized-approximation algorithm. The framework combines these two components together. The sampling step is applied iteratively to remove vertices from the input graph, and then the solution is extended using the black box algorithm. The process is repeated sufficiently many times so that the target approximation ratio is attained with a constant probability. We use the technique to derive parameterized approximation algorithms for several vertex deletion problems, including Feedback Vertex Set, d-Hitting Set and 𝓁-Path Vertex Cover. In particular, for every approximation ratio 1 < β < 2, we attain a parameterized β-approximation for Feedback Vertex Set, which is faster than the parameterized β-approximation of [Jana, Lokshtanov, Mandal, Rai and Saurabh, MFCS 23']. Furthermore, our algorithms are always faster than the algorithms attained using Fidelity Preserving Transformations [Fellows, Kulik, Rosamond, and Shachnai, JCSS 18'].

Cite as

Barış Can Esmer and Ariel Kulik. Sampling with a Black Box: Faster Parameterized Approximation Algorithms for Vertex Deletion Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{canesmer_et_al:LIPIcs.ICALP.2025.39,
  author =	{Can Esmer, Bar{\i}\c{s} and Kulik, Ariel},
  title =	{{Sampling with a Black Box: Faster Parameterized Approximation Algorithms for Vertex Deletion Problems}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{39:1--39:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.39},
  URN =		{urn:nbn:de:0030-drops-234165},
  doi =		{10.4230/LIPIcs.ICALP.2025.39},
  annote =	{Keywords: Parameterized Approximation Algorithms, Random Sampling}
}

Document

DOI: 10.4230/LIPIcs.FORC.2025.4

Hardness and Approximation Algorithms for Balanced Districting Problems

Authors: Prathamesh Dharangutte, Jie Gao, Shang-En Huang, and Fang-Yi Yu

Published in: LIPIcs, Volume 329, 6th Symposium on Foundations of Responsible Computing (FORC 2025)

Abstract

We introduce and study the problem of balanced districting, where given an undirected graph with vertices carrying two types of weights (different population, resource types, etc) the goal is to maximize the total weights covered in vertex disjoint districts such that each district is a star or (in general) a connected induced subgraph with the two weights to be balanced. This problem is strongly motivated by political redistricting, where contiguity, population balance, and compactness are essential. We provide hardness and approximation algorithms for this problem. In particular, we show NP-hardness for an approximation better than n^{1/2-δ} for any constant δ > 0 in general graphs even when the districts are star graphs, as well as NP-hardness on complete graphs, tree graphs, planar graphs and other restricted settings. On the other hand, we develop an algorithm for balanced star districting that gives an O(√n)-approximation on any graph (which is basically tight considering matching hardness of approximation results), an O(log n) approximation on planar graphs with extensions to minor-free graphs. Our algorithm uses a modified Whack-a-Mole algorithm [Bhattacharya, Kiss, and Saranurak, SODA 2023] to find a sparse solution of a fractional packing linear program (despite exponentially many variables) which requires a new design of a separation oracle specific for our balanced districting problem. To turn the fractional solution to a feasible integer solution, we adopt the randomized rounding algorithm by [Chan and Har-Peled, SoCG 2009]. To get a good approximation ratio of the rounding procedure, a crucial element in the analysis is the balanced scattering separators for planar graphs and minor-free graphs - separators that can be partitioned into a small number of k-hop independent sets for some constant k - which may find independent interest in solving other packing style problems. Further, our algorithm is versatile - the very same algorithm can be analyzed in different ways on various graph classes, which leads to class-dependent approximation ratios. We also provide a FPTAS algorithm for complete graphs and tree graphs, as well as greedy algorithms and approximation ratios when the district cardinality is bounded, the graph has bounded degree or the weights are binary. We refer the readers to the full version of the paper for complete set of results and proofs.

Cite as

Prathamesh Dharangutte, Jie Gao, Shang-En Huang, and Fang-Yi Yu. Hardness and Approximation Algorithms for Balanced Districting Problems. In 6th Symposium on Foundations of Responsible Computing (FORC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 329, pp. 4:1-4:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dharangutte_et_al:LIPIcs.FORC.2025.4,
  author =	{Dharangutte, Prathamesh and Gao, Jie and Huang, Shang-En and Yu, Fang-Yi},
  title =	{{Hardness and Approximation Algorithms for Balanced Districting Problems}},
  booktitle =	{6th Symposium on Foundations of Responsible Computing (FORC 2025)},
  pages =	{4:1--4:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-367-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{329},
  editor =	{Bun, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2025.4},
  URN =		{urn:nbn:de:0030-drops-231310},
  doi =		{10.4230/LIPIcs.FORC.2025.4},
  annote =	{Keywords: Approximation algorithms, algorithmic fairness}
}

Document

DOI: 10.4230/LIPIcs.STACS.2025.14

Multivariate Exploration of Metric Dilation

Authors: Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract

Let G be a weighted graph embedded in a metric space (M, d_M). The vertices of G correspond to the points in M, with the weight of each edge uv being the distance d_M(u,v) between their respective points in M. The dilation (or stretch) of G is defined as the minimum factor t such that, for any pair of vertices u,v, the distance between u and v - represented by the weight of a shortest u,v-path - is at most t⋅ d_M(u,v). We study Dilation t-Augmentation, where the objective is, given a metric M, a graph G, and numerical values k and t, to determine whether G can be transformed into a graph with dilation t by adding at most k edges. Our primary focus is on the scenario where the metric M is the shortest path metric of an unweighted graph Γ. Even in this specific case, Dilation t-Augmentation remains computationally challenging. In particular, the problem is W[2]-hard parameterized by k when Γ is a complete graph, already for t = 2. Our main contribution lies in providing new insights into the impact of combinations of various parameters on the computational complexity of the problem. We establish the following. - The parameterized dichotomy of the problem with respect to dilation t, when the graph G is sparse: Parameterized by k, the problem is FPT for graphs excluding a biclique K_{d,d} as a subgraph for t ≤ 2 and the problem is W[1]-hard for t ≥ 3 even if G is a forest consisting of disjoint stars. - The problem is FPT parameterized by the combined parameter k+t+Δ, where Δ is the maximum degree of the graph G or Γ.

Cite as

Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh. Multivariate Exploration of Metric Dilation. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{banik_et_al:LIPIcs.STACS.2025.14,
  author =	{Banik, Aritra and Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Jana, Satyabrata and Saurabh, Saket},
  title =	{{Multivariate Exploration of Metric Dilation}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine 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.2025.14},
  URN =		{urn:nbn:de:0030-drops-228395},
  doi =		{10.4230/LIPIcs.STACS.2025.14},
  annote =	{Keywords: Metric dilation, geometric spanner, fixed-parameter tractability}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2024.4

On the Parameterized Complexity of Eulerian Strong Component Arc Deletion

Authors: Václav Blažej, Satyabrata Jana, M. S. Ramanujan, and Peter Strulo

Published in: LIPIcs, Volume 321, 19th International Symposium on Parameterized and Exact Computation (IPEC 2024)

Abstract

In this paper, we study the Eulerian Strong Component Arc Deletion problem, where the input is a directed multigraph and the goal is to delete the minimum number of arcs to ensure every strongly connected component of the resulting digraph is Eulerian. This problem is a natural extension of the Directed Feedback Arc Set problem and is also known to be motivated by certain scenarios arising in the study of housing markets. The complexity of the problem, when parameterized by solution size (i.e., size of the deletion set), has remained unresolved and has been highlighted in several papers. In this work, we answer this question by ruling out (subject to the usual complexity assumptions) a fixed-parameter tractable (FPT) algorithm for this parameter and conduct a broad analysis of the problem with respect to other natural parameterizations. We prove both positive and negative results. Among these, we demonstrate that the problem is also hard (W[1]-hard or even para-NP-hard) when parameterized by either treewidth or maximum degree alone. Complementing our lower bounds, we establish that the problem is in XP when parameterized by treewidth and FPT when parameterized either by both treewidth and maximum degree or by both treewidth and solution size. We show that these algorithms have near-optimal asymptotic dependence on the treewidth assuming the Exponential Time Hypothesis.

Cite as

Václav Blažej, Satyabrata Jana, M. S. Ramanujan, and Peter Strulo. On the Parameterized Complexity of Eulerian Strong Component Arc Deletion. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blazej_et_al:LIPIcs.IPEC.2024.4,
  author =	{Bla\v{z}ej, V\'{a}clav and Jana, Satyabrata and Ramanujan, M. S. and Strulo, Peter},
  title =	{{On the Parameterized Complexity of Eulerian Strong Component Arc Deletion}},
  booktitle =	{19th International Symposium on Parameterized and Exact Computation (IPEC 2024)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-353-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{321},
  editor =	{Bonnet, \'{E}douard 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.IPEC.2024.4},
  URN =		{urn:nbn:de:0030-drops-222306},
  doi =		{10.4230/LIPIcs.IPEC.2024.4},
  annote =	{Keywords: Parameterized complexity, Eulerian graphs, Treewidth}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2024.17

Subset Feedback Vertex Set in Tournaments as Fast as Without the Subset

Authors: Satyabrata Jana, Lawqueen Kanesh, Madhumita Kundu, and Saket Saurabh

Published in: LIPIcs, Volume 321, 19th International Symposium on Parameterized and Exact Computation (IPEC 2024)

Abstract

In the Feedback Vertex Set in Tournaments (FVST) problem, we are given a tournament T and a positive integer k. The objective is to determine whether there exists a vertex set X ⊆ V(T) of size at most k such that T-X is a directed acyclic graph. This problem is known to be equivalent to the problem of hitting all directed triangles, thereby using the best-known algorithm for the 3-Hitting Set problem results in an algorithm for FVST with a running time of 2.076^k ⋅ n^{𝒪(1)} [Wahlström, Ph.D. Thesis]. Kumar and Lokshtanov [STACS 2016] designed a more efficient algorithm with a running time of 1.6181^k ⋅ n^{𝒪(1)}. A generalization of FVST, called Subset-FVST, includes an additional subset S ⊆ V(T) in the input. The goal for Subset-FVST is to find a vertex set X ⊆ V(T) of size at most k such that T-X contains no directed cycles that pass through any vertex in S. This generalized problem can also be represented as a 3-Hitting Set problem, leading to a running time of 2.076^k ⋅ n^{𝒪(1)}. Bai and Xiao [Theoretical Computer Science 2023] improved this and obtained an algorithm with running time 2^{k + o(k)} ⋅ n^{𝒪(1)}. In our work, we extend the algorithm of Kumar and Lokshtanov [STACS 2016] to solve Subset-FVST, obtaining an algorithm with a running time {𝒪}(1.6181^k + n^{{𝒪}(1)}), matching the running time for FVST.

Cite as

Satyabrata Jana, Lawqueen Kanesh, Madhumita Kundu, and Saket Saurabh. Subset Feedback Vertex Set in Tournaments as Fast as Without the Subset. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jana_et_al:LIPIcs.IPEC.2024.17,
  author =	{Jana, Satyabrata and Kanesh, Lawqueen and Kundu, Madhumita and Saurabh, Saket},
  title =	{{Subset Feedback Vertex Set in Tournaments as Fast as Without the Subset}},
  booktitle =	{19th International Symposium on Parameterized and Exact Computation (IPEC 2024)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-353-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{321},
  editor =	{Bonnet, \'{E}douard 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.IPEC.2024.17},
  URN =		{urn:nbn:de:0030-drops-222438},
  doi =		{10.4230/LIPIcs.IPEC.2024.17},
  annote =	{Keywords: Parameterized algorithms, Feedback vertex set, Tournaments, Fixed parameter tractable, Graph partitions}
}

Document

DOI: 10.4230/LIPIcs.ESA.2024.17

Cuts in Graphs with Matroid Constraints

Authors: Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract

Vertex (s, t)-Cut and Vertex Multiway Cut are two fundamental graph separation problems in algorithmic graph theory. We study matroidal generalizations of these problems, where in addition to the usual input, we are given a representation R ∈ 𝔽^{r × n} of a linear matroid ℳ = (V(G), ℐ) of rank r in the input, and the goal is to determine whether there exists a vertex subset S ⊆ V(G) that has the required cut properties, as well as is independent in the matroid ℳ. We refer to these problems as Independent Vertex (s, t){-cut}, and Independent Multiway Cut, respectively. We show that these problems are fixed-parameter tractable (FPT) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid ℳ). These results are obtained by exploiting the recent technique of flow augmentation [Kim et al. STOC '22], combined with a dynamic programming algorithm on flow-paths á la [Feige and Mahdian, STOC '06] that maintains a representative family of solutions w.r.t. the given matroid [Marx, TCS '06; Fomin et al., JACM]. As a corollary, we also obtain FPT algorithms for the independent version of Odd Cycle Transversal. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.

Cite as

Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, and Saket Saurabh. Cuts in Graphs with Matroid Constraints. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{banik_et_al:LIPIcs.ESA.2024.17,
  author =	{Banik, Aritra and Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Jana, Satyabrata and Saurabh, Saket},
  title =	{{Cuts in Graphs with Matroid Constraints}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.17},
  URN =		{urn:nbn:de:0030-drops-210887},
  doi =		{10.4230/LIPIcs.ESA.2024.17},
  annote =	{Keywords: s-t-cut, multiway Cut, matroid, odd cycle transversal, feedback vertex set, fixed-parameter tractability}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2023.5

Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity

Authors: Sriram Bhyravarapu, Satyabrata Jana, Saket Saurabh, and Roohani Sharma

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Abstract

We consider the question of polynomial kernelization of a generalization of the classical Vertex Cover problem parameterized by a parameter that is provably smaller than the solution size. In particular, we focus on the c-Component Order Connectivity problem (c-COC) where given an undirected graph G and a non-negative integer t, the objective is to test whether there exists a set S of size at most t such that every component of G-S contains at most c vertices. Such a set S is called a c-coc set. It is known that c-COC admits a kernel with {O}(ct) vertices. Observe that for c = 1, this corresponds to the Vertex Cover problem. We study the c-Component Order Connectivity problem parameterized by the size of a d-coc set (c-COC/d-COC), where c,d ∈ ℕ with c ≤ d. In particular, the input is an undirected graph G, a positive integer t and a set M of at most k vertices of G, such that the size of each connected component in G - M is at most d. The question is to find a set S of vertices of size at most t, such that the size of each connected component in G - S is at most c. In this paper, we give a kernel for c-COC/d-COC with O(k^{d-c+1}) vertices and O(k^{d-c+2}) edges. Our result exhibits that the difference in d and c, and not their absolute values, determines the exact degree of the polynomial in the kernel size. When c = d = 1, the c-COC/d-COC problem is exactly the Vertex Cover problem parameterized by the solution size, which has a kernel with O(k) vertices and O(k²) edges, and this is asymptotically tight [Dell & Melkebeek, JACM 2014]. We also show that the dependence of d-c in the exponent of the kernel size cannot be avoided under reasonable complexity assumptions.

Cite as

Sriram Bhyravarapu, Satyabrata Jana, Saket Saurabh, and Roohani Sharma. Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bhyravarapu_et_al:LIPIcs.IPEC.2023.5,
  author =	{Bhyravarapu, Sriram and Jana, Satyabrata and Saurabh, Saket and Sharma, Roohani},
  title =	{{Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.5},
  URN =		{urn:nbn:de:0030-drops-194241},
  doi =		{10.4230/LIPIcs.IPEC.2023.5},
  annote =	{Keywords: Kernelization, Component Order Connectivity, Vertex Cover, Structural Parameterizations}
}

Document

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.IPEC.2022.2

Parameterized Complexity of Perfectly Matched Sets

Authors: Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Abstract

For an undirected graph G, a pair of vertex disjoint subsets (A, B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (= |B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(√k)⋅ n^O(1), and ii) K_{b,b}-free graphs. We obtain a linear kernel for planar graphs and k^𝒪(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.

Cite as

Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu. Parameterized Complexity of Perfectly Matched Sets. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{agrawal_et_al:LIPIcs.IPEC.2022.2,
  author =	{Agrawal, Akanksha and Bhattacharjee, Sutanay and Jana, Satyabrata and Sahu, Abhishek},
  title =	{{Parameterized Complexity of Perfectly Matched Sets}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{2:1--2:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.2},
  URN =		{urn:nbn:de:0030-drops-173580},
  doi =		{10.4230/LIPIcs.IPEC.2022.2},
  annote =	{Keywords: Perfectly Matched Sets, Parameterized Complexity, Apex-minor-free graphs, d-degenerate graphs, Planar graphs, Interval Graphs}
}

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