arXiv subjects

Diptapriyo Majumdar

Publications and source records attributed to Diptapriyo Majumdar.

At least 19 recordsLinked to original sources

arXiv2026

On the Structural Parameterizations of 2-Club with Triangle Constraints

Given an undirected graph G = (V, E) and an integer k, the s-Club asks if Gcontains a vertex subset S of at least k vertices such that G[S] has diameter at most s. Recently, Vertex r-Triangle s-Club, and Edge r-Triangle s-Club that generalize the notion of s-Club have been studied by Garvardt et al. [TOCS-2023, IWOCA-2022] from the perspective of parameterized complexity. Given a graph G and an integer k, the Vertex r-Triangle s-Club asks if there is an s-Club S with at least k vertices such that every vertex u \in S is part of at least r triangles in G[S]. In this paper, we initiate a systematic study of Vertex r-Triangle s-Club for every integer r >= 1 from the perspective of structural parameters of the input graph. In particular, we provide FPT algorithms for Vertex r-Triangle 2-Club when parameterized by the treewidth (tw) of the input graph, and an XP algorithm when parameterized by the h-index of the input graph. Additionally, when parameterized by the feedback edge number (fes) of the input graph. We provide a kernel of O(fes) edges for Vertex r-Triangle s-Club.

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arXiv2026

A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees

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. In this paper, 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 densest as possible or as sparsest as possible (while being connected). We develop a kernel consisting of O(k^5) vertices for this problem.

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arXiv2026

A Polynomial Kernel for Vertex Deletion to the Scattered Class of Proper Interval Graph and Trees

Vertex deletion to hereditary graph class is well-studied in parameterized complexity. Vertex deletion to the scattered graph classes has gained attention in recent years. In this paper, we consider (Proper-Interval, Tree)-Vertex Deletion, the input to which is an undirected graph $G = (V, E)$ and an integer $k$. The goal is to pick a set $X \subseteq V(G)$ of at most $k$ vertices such that $G - X$ is a simple graph and every connected component of $G - X$ is a proper interval graph or a tree. When parameterized by the solution size $k$, (Proper-Interval, Tree)-Vertex Deletion has been proved to be fixed-parameter tractable by Jacob et al. [JCSS-2023, FCT-2021]. In this paper, we consider this problem from the perspective of polynomial kernelization. We provide a first nontrivial polynomial kernel for (Proper-Interval, Tree)-Vertex Deletion, with $O(k^{33})$ vertices.

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arXiv2026

Polynomial Kernels for Spanning Tree with Diversity Requirements

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 $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i) \triangle E(T_j)| \geq k$. Given a connected undirected graph $G$ and integers $p, q, k, \ell$, Leaf & Internal-Constrained Diverse Spanning Trees asks whether there are $\ell$ distinct spanning trees $T_1,\ldots,T_{\ell}$ 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 $G$, $V_{NT}\subseteq V(G)$, and three integers $p, k, \ell$, and asks if $G$ has $\ell$ spanning trees that are pairwise $k$-diverse, and each has at least $p$ leaves and conains 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 $p + q + k + \ell$ and $p + |V_{\rm NT}| + k + \ell$, respectively.

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arXiv2026

On the Polynomial Kernelizations of Finding a Shortest Path with Positive Disjunctive Constraints

We study the SHORTEST PATH problem with positive disjunctive constraints from the perspective of parameterized complexity. For positive disjunctive constraints, there are certain pair of edges such that any feasible solution must contain at least one edge from every such pair. In this paper, we initiate the study of SHORTEST PATH problem subject to some positive disjunctive constraints the classical version is known to be NP-Complete. Formally, given an undirected graph G = (V, E) with a forcing graph H = (E, F) such that the vertex set of H is same as the edge set of G. The goal is to find a set S of at most k edges from G such that S forms a vertex cover in H and there is a path from s to t in the subgraph of G induced by the edge set S. In this paper, we consider two natural parameterizations for this problem. One natural parameter is the solution size, i.e. k for which we provide a kernel with O(k^5) vertices when both G and H are general graphs. Additionally, when either G or H (but not both) belongs to some special graph classes, we provied kernelization results with O(k^3) vertices . The other natural parameter we consider is structural properties of H, i.e. the size of a vertex deletion set of H to some special graph classes. We provide some fixed-parameter tractability results for those structural parameterizations.

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arXiv2026

On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints

Given an undirected graph G and a set A \subseteq 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) \setminus A. An A-path is called an (A,\ell)-path if the length of the path is exactly \ell. In the {\sc (A, \ell)-Path Packing} problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, \ell)-paths in G or not. We pursue this problem with respect to structural parameters. We prove that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A|. In addition, we consider the combined parameters distance to cluster (cvd) + |A| and distance to cluster (cvd) + \ell. For both these combined parameters, we provide FPT algorithms. Finally, we consider the vertex cover number (vc) as the parameter and provide a kernel with O(vc^2) vertices.

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arXiv2025

Highly Connected Steiner Subgraph -- Parameterized Algorithms and Applications to Hitting Set Problems

Given a simple connected undirected graph G = (V, E), a set X \subseteq V(G), and integers k and p, STEINER SUBGRAPH EXTENSION problem asks if there exists a set S \supseteq X with at most k vertices such that G[S] is p-edge-connected. This is a natural generalization of a well-studied problem STEINER TREE (set p=1 and X as the set of all terminals). In this paper, we initiate the study of STEINER SUBGRAPH EXTENSION from the perspective of parameterized complexity and give a fixed-parameter algorithm parameterized by k and p on graphs of bounded degeneracy. In case we remove the assumption of the input graph being bounded degenerate, then the STEINER SUBGRAPH EXTENSION problem becomes W[1]-hard. 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 singly exponential-time FPT algorithms for several vertex deletion problem studied in the literature, where the goal is to delete a smallest set of vertices such that (i) the resulting graph belongs to a specific hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.

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arXiv2025

Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

We investigate fine-grained algorithmic aspects of identification problems in graphs and set systems, with a focus on Locating-Dominating Set and Test Cover. We prove the (tight) conditional lower bounds for these problems when parameterized by treewidth and solution as. Formally, \textsc{Locating-Dominating Set} (respectively, \textsc{Test Cover}) parameterized by the treewidth of the input graph (respectively, of the natural auxiliary graph) does not admit an algorithm running in time $2^{2^{o(tw)}} \cdot poly(n)$ (respectively, $2^{2^{o(tw)}} \cdot poly(|U| + |\mathcal{F}|))$. This result augments the small list of NP-Complete problems that admit double exponential lower bounds when parameterized by treewidth. Then, we first prove that \textsc{Locating-Dominating Set} does not admit an algorithm running in time $2^{o(k^2)} \cdot poly(n)$, nor a polynomial time kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(k)}$ vertices, unless the Ð fails. To the best of our knowledge, \textsc{Locating-Dominating Set} is the first problem that admits such an algorithmic lower-bound (with a quadratic function in the exponent) when parameterized by the solution size. Finally, we prove that \textsc{Test Cover} does not admit an algorithm running in time $2^{2^{o(k)}} \cdot poly(|U| + |\mathcal{F}|)$. This is also a rare example of the problem that admits a double exponential lower bound when parameterized by the solution size. We also present algorithms whose running times match the above lower bounds.

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arXiv2025

Enumeration Kernels of Polynomial Size for Cuts of Bounded Degree

Enumeration kernelization was first proposed by Creignou et al. [TOCS 2017] and was later refined by Golovach et al. [JCSS 2022] into two different variants: fully-polynomial enumeration kernelization and polynomial-delay enumeration kernelization. In this paper, we consider the d-CUT problem from the perspective of (polynomial-delay) enumeration kenrelization. Given an undirected graph G = (V, E), a cut F = (A, B) is a d-cut of G if every $u \in A$ has at most d neighbors in B and every $v \in B$ has at most d neighbors in A. Checking the existence of a d-cut in a graph is a well-known NP-hard problem and is well-studied in parameterized complexity [Algorithmica 2021, IWOCA 2021]. This problem also generalizes a well-studied problem MATCHING CUT (set d = 1) that has been a central problem in the literature of polynomial-delay enumeration kernelization. In this paper, we study three different enumeration variants of this problem, ENUM d-CUT, ENUM MIN-d-CUT and ENUM MAX-d-CUT that intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal d-cuts respectively. We consider various structural parameters of the input, e.g. vertex cover number, neighborhood diversity, and clique partition number. When vertex cover number and neighborhood diversity are considered as parameters, we provide polynomial-delay enumeration kernelizations of polynomial size for ENUM d-CUT and ENUM MAX-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-d-CUT. When clique partition number is considered as the parameter, we provide bijective enumeration kernels for each of these three problems.

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arXiv2025

Polynomial-Size Enumeration Kernelizations for Long Path Enumeration

Enumeration kernelization for parameterized enumeration problems was defined by Creignou et al. [Theory Comput. Syst. 2017] and was later refined by Golovach et al. [J. Comput. Syst. Sci. 2022, STACS 2021] to polynomial-delay enumeration kernelization. We consider ENUM LONG-PATH, the enumeration variant of the Long-Path problem, from the perspective of enumeration kernelization. Formally, given an undirected graph G and an integer k, the objective of ENUM LONG-PATH is to enumerate all paths of G having exactly k vertices. We consider the structural parameters vertex cover number, dissociation number, and distance to clique and provide polynomial-delay enumeration kernels of polynomial size for each of these parameters.

cs.DM↗

arXiv2024

Structural Parameterization of Locating-Dominating Set and Test Cover

We investigate structural parameterizations of two identification problems: LOCATING-DOMINATING SET and TEST COVER. In the first problem, an input is a graph $G$ on $n$ vertices and an integer $k$, and one asks if 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 set of subsets $\mathcal{F}$ of $U$ called $tests$ and an integer $k$, and one asks if there is a set $S$ of at most $k$ tests such that any two items belong to distinct subsets of tests of $S$. These two problems are "identification" analogues of DOMINATING SET and SET COVER, respectively. Chakraborty et al. [ISAAC 2024] proved that both the problems admit conditional double-exponential lower bounds and matching algorithms when parameterized by treewidth of the input graph. We continue this line of investigation and consider parameters larger than treewidth, like vertex cover number and feedback edge set number. We design a nontrivial dynamic programming scheme to solve TEST COVER in "slightly super-exponential" time $2^{O(|U|\log |U|)}(|U|+|\mathcal{F}|)^{O(1)}$ in the number $|U|$ of items and LOCATING-DOMINATING SET in time $2^{O(\textsf{vc} \log \textsf{vc})} \cdot n^{O(1)}$, where $\textsf{vc}$ is the vertex cover number and $n$ is the order of the graph. This shows that the lower bound results with respect to treewidth from Chakraborty et al. [ISAAC 2024] cannot be extended to vertex cover number. We also show that, parameterized by feedback edge set number, LOCATING-DOMINATING SET admits a linear kernel thereby answering an open question in [Cappelle et al., LAGOS 2021]. Finally, we show that neither LOCATING-DOMINATING SET nor TEST COVER is likely to admit a compression algorithm returning an input with a subquadratic number of bits, unless $\textsf{NP} \subseteq \textsf{coNP}/poly$.

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arXiv2024

Parameterized Complexity of Dominating Set Variants in Almost Cluster and Split Graphs

We consider structural parameterizations of the fundamental Dominating Set problem and its variants in the parameter ecology program. We give improved FPT algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). When parameterized by the deletion distance k to cluster graphs - we can find a minimum dominating set (DS) in 3^k n^{O(1)}-time. Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum dominating clique (DC) or a minimum efficient dominating set (EDS) or a minimum total dominating set (TDS). We also show that most of these variants of dominating set do not have polynomial sized kernel. Additionally, we show that when parameterized by the deletion distance k to split graphs - IDS can be solved in 2^k n^{O(1)}-time and EDS can be solved in 3^{k/2}n^{O(1)}.

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arXiv2024

Bi-objective Optimization in Role Mining

Role mining is a technique used to derive a role-based authorization policy from an existing policy. Given a set of users $U$, a set of permissions $P$ and a user-permission authorization relation $\mahtit{UPA}\subseteq U\times P$, a role mining algorithm seeks to compute a set of roles $R$, a user-role authorization relation $\mathit{UA}\subseteq U\times R$ and a permission-role authorization relation $\mathit{PA}\subseteq R\times P$, such that the composition of $\mathit{UA}$ and $\mathit{PA}$ is close (in some appropriate sense) to $\mathit{UPA}$. In this paper, we first introduce the Generalized Noise Role Mining problem (GNRM) -- a generalization of the MinNoise Role Mining problem -- which we believe has considerable practical relevance. Extending work of Fomin et al., we show that GNRM is fixed parameter tractable, with parameter $r + k$, where $r$ is the number of roles in the solution and $k$ is the number of discrepancies between $\mathit{UPA}$ and the relation defined by the composition of $\mathit{UA}$ and $\mathit{PA}$. We further introduce a bi-objective optimization variant of GNRM, where we wish to minimize both $r$ and $k$ subject to upper bounds $r\le \bar{r}$ and $k\le \bar{k}$, where $\bar{r}$ and $\bar{k}$ are constants. We show that the Pareto front of this bi-objective optimization problem (BO-GNRM) can be computed in fixed-parameter tractable time with parameter $\bar{r}+\bar{k}$. We then report the results of our experimental work using the integer programming solver Gurobi to solve instances of BO-GNRM. Our key findings are that (a) we obtained strong support that Gurobi's performance is fixed-parameter tractable, (b) our results suggest that our techniques may be useful for role mining in practice, based on our experiments in the context of three well-known real-world authorization policies.

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arXiv2023

Deletion to Scattered Graph Classes II -- Improved FPT Algorithms for Deletion to Pairs of Graph Classes

Let $Π$ be a hereditary graph class. The problem of deletion to $Π$, takes as input a graph $G$ and asks for a minimum number (or a fixed integer $k$) of vertices to be deleted from $G$ so that the resulting graph belongs to $Π$. This is a well-studied problem in paradigms including approximation and parameterized complexity. Recently, the study of a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes, and the goal is to determine whether $k$ vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be FPT as long as the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. While this was shown using some black box theorems, faster algorithms were shown when each of the hereditary graph classes has a finite forbidden set. In this paper, we do a deep dive on pairs of specific graph classes ($Π_1, Π_2$) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms. We design a general FPT algorithm and approximation algorithm for pairs of graph classes (possibly having infinite forbidden sets) satisfying certain conditions. These algorithms cover several pairs of popular graph classes. Our algorithm makes non-trivial use of the branching technique and as a black box, FPT algorithms for deletion to individual graph classes.

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arXiv2023

Expansion Lemma -- Variations and Applications to Polynomial-Time Preprocessing

In parameterized complexity, it is well-known that a parameterized problem is fixed-parameter tractable if and only if it has a kernel - an instance equivalent to the input instance, whose size is just a function of the parameter. The size of the kernel can be exponential or worse, resulting in a quest for fixed-parameter tractable problems with a polynomial-sized kernel. The developments in machinery to show lower bounds for the sizes of the kernel gave rise to the question of the asymptotically optimum size for the kernel of fixed-parameter tractable problems. In this article, we survey a tool called expansion lemma that helps in reducing the size of the kernel. Its early origin is in the form of Crown Decomposition for obtaining linear kernel for the Vertex Cover problem and the specific lemma was identified as the tool behind an optimal kernel with O(k^2) vertices and edges for the UNDIRECTED FEEDBACK VERTEX SET problem. Since then, several variations and extensions of the tool have been discovered. We survey them along with their applications in this article.

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arXiv2022

An Improved Time-Efficient Approximate Kernelization for Connected Treedepth Deletion Set

We study the CONNECTED η-TREEDEPTH DELETION problem where the input instance is an undireted graph G = (V, E) and an integer k. The objective is to decide if G has a set S \subseteq V(G) of at most k vertices such that G - S has treedepth at most ηand G[S] is connected. As this problem naturally generalizes the well-known CONNECTED VERTEX COVER, when parameterized by solution size k, the CONNECTED η-TREEDEPTH DELETION does not admit polynomial kernel unless NP \subseteq coNP/poly. This motivates us to design an approximate kernel of polynomial size for this problem. In this paper, we show that for every 0 < ε<= 1, CONNECTED η-TREEDEPTH DELETION SET admits a (1+ε)-approximate kernel with O(k^{2^{η+ 1/ε}}) vertices, i.e. a polynomial-sized approximate kernelization scheme (PSAKS).

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arXiv2022

p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms

We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p \geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless $NP \subseteq coNP/poly$, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an $O(2^{O(pk)}n^{O(1)})$-time algorithm for the $p$-Edge-Connected VC and an $O(2^{O(k^2)}n^{O(1)})$-time algorithm for the $p$-Vertex-Connected VC. Finally, we describe a $2(p+1)$-approximation algorithm for the $p$-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning $p$-vertex/edge-connected subgraph of a $p$-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).

cs.DS↗

arXiv2021

Deletion to Scattered Graph Classes I -- case of finite number of graph classes

Graph-modification problems, where we modify a graph by adding or deleting vertices or edges or contracting edges to obtain a graph in a {\it simpler} 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 a small number of vertices to make the resulting graph to belong to a given non-trivial hereditary graph class, captures several well-studied problems including {\sc Vertex Cover}, {\sc Feedback Vertex Set}, {\sc Odd Cycle Transveral}, {\sc Cluster Vertex Deletion}, and {\sc Perfect Deletion}. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. We initiate a study of a natural variation of the problem of deletion to {\it scattered graph classes}. We want 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. As our main result, we show that this problem is fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the finite number of graph classes is known to be FPT and the properties that a graph belongs to any of the classes is expressible in Counting Monodic Second Order (CMSO) logic. While this is shown using some black box theorems in parameterized complexity, we give a faster FPT algorithm when each of the graph classes has a finite forbidden set.

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