arXiv subjects

Soumen Maity

Publications and source records attributed to Soumen Maity.

At least 19 recordsLinked to original sources

arXiv2026

Hardness and Tractability of T_{h+1}-Free Edge Deletion

We study the parameterized complexity of the T(h+1)-Free Edge Deletion problem. Given a graph G and integers k and h, the task is to delete at most k edges so that every connected component of the resulting graph has size at most h. The problem is NP-complete for every fixed h at least 3, while it is solvable in polynomial time for h at most 2. Recent work showed strong hardness barriers: the problem is W[1]-hard when parameterized by the solution size together with the size of a feedback edge set, ruling out fixed-parameter tractability for many classical structural parameters. We significantly strengthen these negative results by proving W[1]-hardness when parameterized by the vertex deletion distance to a disjoint union of paths, the vertex deletion distance to a disjoint union of stars, or the twin cover number. These results unify and extend known hardness results for treewidth, pathwidth, and feedback vertex set, and show that several restrictive parameters, including treedepth, cluster vertex deletion number, and modular width, do not yield fixed-parameter tractability when h is unbounded. On the positive side, we identify parameterizations that restore tractability. We show that the problem is fixed-parameter tractable when parameterized by cluster vertex deletion together with h, vertex deletion set into a clique and also when parameterized by neighborhood diversity together with h via an integer linear programming formulation. We further present a fixed-parameter tractable bicriteria approximation algorithm parameterized by k. Finally, we show that the problem admits fixed-parameter tractable algorithms on split graphs and interval graphs, and we establish hardness for a directed generalization even on directed acyclic graphs.

cs.DS↗

arXiv2025

Inclusive and Exclusive Vertex Splitting into Specific Graph Classes: NP Hardness and Algorithms

We study a family of graph modification problems called the F-Vertex Splitting problem. Given a graph G, the task is to determine whether G can be transformed into a graph G-prime belonging to a graph class F through a sequence of at most k vertex splits. We investigate this problem for several target graph classes, namely constellations, cycle graphs, linear forests, and bipartite graphs. We analyze both inclusive and exclusive variants of vertex splitting, as introduced by Abu-Khzam and collaborators (ISCO 2018). Our results show that the F-Vertex Splitting problem is polynomial-time solvable when F is a cycle graph or a linear forest, for both variants. In contrast, when F is a constellation or a bipartite graph, the problem is NP-complete for both variants.

cs.DS↗

arXiv2025

MaxMin Separation Problems: FPT Algorithms for $st$-Separator and Odd Cycle Transversal

In this paper, we study the parameterized complexity of the MaxMin versions of two fundamental separation problems: Maximum Minimal $st$-Separator and Maximum Minimal Odd Cycle Transversal (OCT), both parameterized by the solution size. In the Maximum Minimal $st$-Separator problem, given a graph $G$, two distinct vertices $s$ and $t$ and a positive integer $k$, the goal is to determine whether there exists a minimal $st$-separator in $G$ of size at least $k$. Similarly, the Maximum Minimal OCT problem seeks to determine if there exists a minimal set of vertices whose deletion results in a bipartite graph, and whose size is at least $k$. We demonstrate that both problems are fixed-parameter tractable parameterized by $k$. Our FPT algorithm for Maximum Minimal $st$-Separator answers the open question by Hanaka, Bodlaender, van der Zanden and Ono (TCS 2019). One unique insight from this work is the following. We use the meta-result of Lokshtanov, Ramanujan, Saurabh and Zehavi (ICALP 2018) that enables us to reduce our problems to highly unbreakable graphs. This is interesting, as an explicit use of the recursive understanding and randomized contractions framework of Chitnis, Cygan, Hajiaghayi, Pilipczuk and Pilipczuk (SICOMP 2016) to reduce to the highly unbreakable graphs setting (which is the result that Lokshtanov et al. tries to abstract out in their meta-theorem) does not seem obvious because certain ``extension'' variants of our problems are W[1]-hard.

cs.CC↗

arXiv2024

Parameterized Algorithms for Editing to Uniform Cluster Graph

We study the parameterized complexity of transforming graphs into Uniform Cluster graphs, where each component is an equal-sized clique. We consider Uniform Cluster Vertex Deletion (UCVD), Uniform Cluster Edge Deletion (UCED), Uniform Cluster Edge Addition (UCEA), Uniform Cluster Edge Editing (UCEE), Uniform Cluster Exclusive Vertex Splitting (UCEVS), and Uniform Cluster Inclusive Vertex Splitting (UCIVS). For UCVD, we provide a vertex kernel of size $\mathcal{O}(k^{3})$ and an FPT algorithm with running time $2^{k} \cdot n^{\mathcal{O}(1)}$, improving the known $3^{k} \cdot n^{\mathcal{O}(1)}$ algorithm. For edge-based variants, we obtain a $\mathcal{O}(k^{2})$ vertex kernel for UCEE and linear vertex kernels for UCED and UCEA, improving the best-known results. Additionally, we present a $1.47^{k} \cdot n^{\mathcal{O}(1)}$ algorithm for UCED, improving upon the previous $2^{k} \cdot n^{\mathcal{O}(1)}$ bound. We develop a sub-exponential algorithm for UCED on everywhere dense graphs by reducing it to $d$-Way Cut. Lastly, we study vertex splitting operations and provide vertex kernels of size $4k$ for both UCIVS and UCEVS.

cs.DS↗

arXiv2023

An Improved Exact Algorithm for Knot-Free Vertex Deletion

A knot $K$ in a directed graph $D$ is a strongly connected component of size at least two such that there is no arc $(u,v)$ with $u \in V(K)$ and $v\notin V(K)$. Given a directed graph $D=(V,E)$, we study Knot-Free Vertex Deletion (KFVD), where the goal is to remove the minimum number of vertices such that the resulting graph contains no knots. This problem naturally emerges from its application in deadlock resolution since knots are deadlocks in the OR-model of distributed computation. The fastest known exact algorithm in literature for KFVD runs in time $\mathcal{O}^\star(1.576^n)$. In this paper, we present an improved exact algorithm running in time $\mathcal{O}^\star(1.4549^n)$, where $n$ is the number of vertices in $D$. We also prove that the number of inclusion wise minimal knot-free vertex deletion sets is $\mathcal{O}^\star(1.4549^n)$ and construct a family of graphs with $Ω(1.4422^n)$ minimal knot-free vertex deletion sets

cs.DS↗

arXiv2023

Parameterized Complexity of Locally Minimal Defensive Alliances

A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours is in $S$. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive alliance of maximum size. We also look at connected version of defensive alliance. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance is NP-complete, even when restricted to planar graphs, (2) a randomized FPT algorithm for Exact Connected Locally Minimal Defensive Alliance parameterized by solution size, (3) Locally Minimal Defensive Alliance is fixed-parameter tractable (FPT) when parametrized by neighbourhood diversity, (4) Locally Minimal Defensive Alliance parameterized by treewidth is W[1]-hard and thus not FPT (unless FPT=W[1]), (5) Locally Minimal Defensive Alliance can be solved in polynomial time for graphs of bounded treewidth.

cs.CC↗

arXiv2022

Parameterized Algorithms for Locally Minimal Defensive Alliance

A set $D$ of vertices of a graph is a \emph{defensive alliance} if, for each element of $D$, the majority of its neighbours are in $D$. We consider the notion of local minimality in this paper. We are interested in finding a locally minimal defensive alliance of maximum size. In Locally Minimal Defensive Alliance problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a locally minimal defensive alliance of size at least $k$. This problem is known to be NP-hard, but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance restricted to the graphs of minimum degree at least 2 is fixed-parameter tractable (FPT) when parameterized by the combined parameters solution size $k$, and maximum degree $Δ$ of the input graph, (2) Locally Minimal Defensive Alliance on the graphs of minimum degree at least 2, admits a kernel with at most $k^{k^{\mathcal{O}(k)}}$ vertices. In particular, the problem parameterized by $k$ restricted to $C_3$-free and $C_4$-free graphs of minimum degree at least 2, admits a kernel with at most $k^{\mathcal{O}(k)}$ vertices. Moreover, we prove that the problem on planar graphs of minimum degree at least 2, admits an FPT algorithm with running time $\mathcal{O}^{*}(k^{2^{\mathcal{O}(\sqrt{k})}})$. Finally, we prove that (4) Locally Minimal Defensive Alliance Extension is NP-complete.

cs.DS↗

arXiv2022

Offensive Alliances in Graphs

A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the classical and parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on natural parameter as well as parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; we thereby resolve an open question stated by Bernhard Bliem and Stefan Woltran (2018) concerning the complexity of Offensive Alliance parameterized by treewidth, (2) unless ETH fails, Offensive Alliance problem cannot be solved in time $\mathcal{O}^{*}(2^{o(k \log k)})$ where $k$ is the solution size, (3) Offensive Alliance problem does not admit a polynomial kernel parameterized by solution size and vertex cover of the input graph. On the positive side we prove that (4) Offensive Alliance can be solved in time $\mathcal{O}^{*}(\tt{vc(G)}^{\mathcal{O}(\tt{vc(G)})})$ where $\tt{vc(G)}$ is the vertex cover number of the input graph. In terms of classical complexity, we prove that (5) Offensive Alliance problem cannot be solved in time $2^{o(n)}$ even when restricted to bipartite graphs, unless ETH fails, (6) Offensive Alliance problem cannot be solved in time $2^{o(\sqrt{n})}$ even when restricted to apex graphs, unless ETH fails. We also prove that (7) Offensive Alliance problem is NP-complete even when restricted to bipartite, chordal, split and circle graphs.

cs.DS↗

arXiv2022

Parameterized Complexity of Upper Edge Domination

In this paper we study a maximization version of the classical Edge Dominating Set (EDS) problem, namely, the Upper EDS problem, in the realm of Parameterized Complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal edge dominating set of size at least $k$. We obtain the following results for Upper EDS. We prove that Upper EDS admits a kernel with at most $4k^2-2$ vertices. We also design a fixed-parameter tractable (FPT) algorithm for Upper EDS running in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$.

cs.DS↗

arXiv2022

Maximum Minimal Feedback Vertex Set: A Parameterized Perspective

In this paper we study a maximization version of the classical Feedback Vertex Set (FVS) problem, namely, the Max Min FVS problem, in the realm of parameterized complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal feedback vertex set of size at least $k$. We obtain following results for Max Min FVS. 1) We first design a fixed parameter tractable (FPT) algorithm for Max Min FVS running in time $10^kn^{\mathcal{O}(1)}$. 2) Next, we consider the problem parameterized by the vertex cover number of the input graph (denoted by $\mathsf{vc}(G)$), and design an algorithm with running time $2^{\mathcal{O}(\mathsf{vc}(G)\log \mathsf{vc}(G))}n^{\mathcal{O}(1)}$. We complement this result by showing that the problem parameterized by $\mathsf{vc}(G)$ does not admit a polynomial compression unless coNP $\subseteq$ NP/poly. 3) Finally, we give an FPT-approximation scheme (fpt-AS) parameterized by $\mathsf{vc}(G)$. That is, we design an algorithm that for every $ε>0$, runs in time $2^{\mathcal{O}\left(\frac{\mathsf{vc}(G)}ε\right)} n^{\mathcal{O}(1)}$ and returns a minimal feedback vertex set of size at least $(1-ε){\sf opt}$.

cs.DS↗

arXiv2022

Globally Minimal Defensive Alliances: A Parameterized Perspective

A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider the notion of global minimality in this paper. We are interested in globally minimal defensive alliance of maximum size. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the Globally Minimal Defensive Alliance problem is FPT parameterized by the neighbourhood diversity of the input graph. The result for neighborhood diversity implies that the problem is FPT parameterized by vertex cover number also. We prove that the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP $\subseteq$ NP/poly. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treewidth and treedepth. We also proved that, given a vertex $r \in V(G)$, deciding if $G$ has a globally minimal defensive alliance of any size containing vertex $r$ is NP-complete.

cs.CC↗

arXiv2022

The Harmless Set Problem

Given a graph $G = (V,E)$, a threshold function $t~ :~ V \rightarrow \mathbb{N}$ and an integer $k$, we study the Harmless Set problem, where the goal is to find a subset of vertices $S \subseteq V$ of size at least $k$ such that every vertex $v\in V$ has less than $t(v)$ neighbors in $S$. We enhance our understanding of the problem from the viewpoint of parameterized complexity. Our focus lies on parameters that measure the structural properties of the input instance. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number. We also show that the Harmless Set problem with majority thresholds is W[1]-hard when parameterized by the treewidth of the input graph. We prove that the Harmless Set problem can be solved in polynomial time on graph with bounded cliquewidth. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to neighbourhood diversity, twin cover and vertex integrity of the input graph. We show that the problem parameterized by the solution size is fixed parameter tractable on planar graphs. We thereby resolve two open questions stated in C. Bazgan and M. Chopin (2014) concerning the complexity of {\sc Harmless Set} parameterized by the treewidth of the input graph and on planar graphs with respect to the solution size.

cs.CC↗

arXiv2021

Defensive Alliances in Graphs

A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours are in $S$. We study the parameterized complexity of the Defensive Alliance problem, where the aim is to find a minimum size defensive alliance. Our main results are the following: (1) The Defensive Alliance problem has been studied extensively during the last twenty years, but the question whether it is FPT when parameterized by feedback vertex set has still remained open. We prove that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; (2) the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP $\subseteq$ NP/poly, (3) it does not admit $2^{o(n)}$ algorithm under ETH, and (4) the Defensive Alliance problem on circle graphs is NP-complete.

cs.CC↗

arXiv2021

On Structural Parameterizations of the Offensive Alliance Problem

The Offensive Alliance problem has been studied extensively during the last twenty years. A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph.

cs.CC↗

arXiv2021

Edge Deletion to Restrict the Size of an Epidemic

Given a graph $G=(V,E)$, a set $\mathcal{F}$ of forbidden subgraphs, we study $\mathcal{F}$-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any $F\in \mathcal{F}$ as a subgraph. For the parameter treewidth, the question of whether the problem is FPT has remained open. Here we give a negative answer by showing that the problem is W[1]-hard when parameterized by the treewidth, which rules out FPT algorithms under common assumption. Thus we give a solution to the conjecture posted by Jessica Enright and Kitty Meeks in [Algorithmica 80 (2018) 1857-1889]. We also prove that the $\mathcal{F}$-Free Edge Deletion problem is W[2]-hard when parameterized by the solution size $k$, feedback vertex set number or pathwidth of the input graph. A special case of particular interest is the situation in which $\mathcal{F}$ is the set $\mathcal{T}_{h+1}$ of all trees on $h+1$ vertices, so that we delete edges in order to obtain a graph in which every component contains at most $h$ vertices. This is desirable from the point of view of restricting the spread of disease in transmission network. We prove that the $\mathcal{T}_{h+1}$-Free Edge Deletion problem is fixed-parameter tractable (FPT) when parameterized by the vertex cover number. We also prove that it admits a kernel with $O(hk)$ vertices and $O(h^2k)$ edges, when parameterized by combined parameters $h$ and the solution size $k$.

cs.DS↗

arXiv2020

The Satisfactory Partition Problem

The Satisfactory Partition problem consists in deciding if the set of vertices of a given undirected graph can be partitioned into two nonempty parts such that each vertex has at least as many neighbours in its part as in the other part. This problem was introduced by Gerber and Kobler [European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors, but its parameterized complexity remains open until now. It is known that the Satisfactory Partition problem, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is FPT when parameterized by the neighbourhood diversity of the input graph, (2) it can be solved in $O(n^{8 {\tt cw}})$ where ${\tt cw}$ is the clique-width,(3) a generalized version of the problem is W[1]-hard when parameterized by the treewidth.

cs.DS↗

arXiv2015

Improved Strength Four Covering Arrays with Three Symbols

A covering array $t$-$CA(n,k,g)$, of size $n$, strength $t$, degree $k$, and order $g$, is a $k\times n$ array on $g$ symbols such that every $t\times n$ sub-array contains every $t\times 1$ column on $g$ symbols at least once. Covering arrays have been studied for their applications to software testing, hardware testing, drug screening, and in areas where interactions of multiple parameters are to be tested. In this paper, we present an algebraic construction that improves many of the best known upper bounds on $n$ for covering arrays 4-$CA(n,k,g)$ with $g=3$. The $coverage$ $measure$ $μ_t(A)$ of a testing array $A$ is defined by the ratio between the number of distinct $t$-tuples contained in the column vectors of $A$ and the total number of $t$-tuples. A covering array is a testing array with full coverage. The $covering$ $arrays$ $with$ $budget$ $constraints$ $problem$ is the problem of constructing a testing array of size at most $n$ having largest possible coverage measure, given values of $k,g$ and $n$. This paper presents several strength four testing arrays with high coverage. The construction here is a generalisation of the construction methods used by Chateauneuf, Colbourn and Kreher, and Meagher and Stevens.

cs.DM↗

arXiv2015

Covering Arrays on Product Graphs

Two vectors $x,y$ in $\mathbb{Z}_g^n$ are $ qualitatively$ $ independent$ if for all pairs $(a,b)\in \mathbb{Z}_g\times \mathbb{Z}_g$, there exists $i\in \{1,2,\ldots,n\}$ such that $(x_i,y_i)=(a,b)$. A covering array on a graph $G$, denoted by $CA(n,G,g)$, is a $|V(G)|\times n$ array on $\mathbb{Z}_g$ with the property that any two rows which correspond to adjacent vertices in $G$ are qualitatively independent. The number of columns in such array is called its $size$. Given a graph $G$, a covering array on $G$ with minimum size is called $optimal$. Our primary concern in this paper is with constructions that make optimal covering arrays on large graphs those are obtained from product of smaller graphs. We consider four most extensively studied graph products in literature and give upper and lower bounds on the the size of covering arrays on graph products. We find families of graphs for which the size of covering array on the Cartesian product achieves the lower bound. Finally, we present a polynomial time approximation algorithm with approximation ratio $\log(\frac{V}{2^{k-1}})$ for constructing covering array on graph $G=(V,E)$ with $k>1$ prime factors with respect to the Cartesian product.

cs.DM↗