arXiv subjects

Pradeesha Ashok

Publications and source records attributed to Pradeesha Ashok.

11 recordsLinked to original sources

arXiv2026

From One Solution to Many: An Oracle-Based FPT Framework for Diverse Solutions under Generalized Diversity Measures

The problem of computing \emph{diverse} solutions has recently emerged as an important area of study, motivated by applications in fairness, robustness, and security. Instead of returning a single feasible or optimal solution, the goal is to output a \emph{collection} of meaningfully different solutions, often measured by symmetric differences. Diverse variants have been studied using sparsification, network-flow reductions, and algebraic techniques. We investigate the fixed-parameter tractability of diverse variants of an implicit set-system problem. Given parameters $k$ and $r$ and a threshold $b$, the task is to compute $r$ feasible solutions, each of size at most $k$, whose diversity under a specified objective is at least $b$. Our main contribution is an oracle-based meta-theorem. We identify a broad class of objectives, called \emph{consistently diverse}, that includes several standard measures. Assuming an \emph{exact empty-extension oracle} given a forbidden set ${\sf Forb}$, which returns a feasible solution of a prescribed size avoiding ${\sf Forb}$ or reports that none exists, we obtain a fixed-parameter tractable algorithm parameterized by $k+r$. The algorithm makes at most $(2kr)^{kr} \cdot r$ oracle calls, and in each call the oracle parameter satisfies $s+|{\sf Forb}| \leq k+2kr$. Our framework unifies and strengthens previous oracle-based approaches. Compared with Kumabe's framework (ESA 2025), which gives a doubly exponential bound on the number of oracle calls, our approach achieves the single exponential bound $2^{O(kr\log(kr))}$ and directly constructs the desired tuple of solutions. We recover fixed-parameter tractable algorithms for all problems covered by that framework, with improved oracle complexity, and obtain strong bounds for diverse variants of classical graph and matroid problems.

cs.DS↗

arXiv2024

(Independent) Roman Domination Parameterized by Distance to Cluster

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} (RDF) if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be an \emph{Independent Roman Dominating function} (IRDF), if $V_1\cup V_2$ forms an independent set, where $V_i=\{v\in V~\vert~f(v)=i\}$, for $i\in \{0,1,2\}$. The total weight of $f$ is equal to $\sum_{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Roman Domination Number} (resp. \emph{Independent Roman Domination Number}) of $G$, denoted by $γ_R(G)$ (resp. $i_R(G)$), is defined as min$\{w(f)~\vert~f$ is an RDF (resp. IRDF) of $G\}$. For a given graph $G$, the problem of computing $γ_R(G)$ (resp. $i_R(G)$) is defined as the \emph{Roman Domination problem} (resp. \emph{Independent Roman Domination problem}). In this paper, we examine structural parameterizations of the (Independent) Roman Domination problem. We propose fixed-parameter tractable (FPT) algorithms for the (Independent) Roman Domination problem in graphs that are $k$ vertices away from a cluster graph. These graphs have a set of $k$ vertices whose removal results in a cluster graph. We refer to $k$ as the distance to the cluster graph. Specifically, we prove the following results when parameterized by the deletion distance $k$ to cluster graphs: we can find the Roman Domination Number (and Independent Roman Domination Number) in time $4^kn^{O(1)}$. In terms of lower bounds, we show that the Roman Domination number can not be computed in time $2^{εk}n^{O(1)}$, for any $0<ε<1$ unless a well-known conjecture, SETH fails. In addition, we also show that the Roman Domination problem parameterized by distance to cluster, does not admit a polynomial kernel unless NP $\subseteq$ coNP$/$poly.

cs.CC↗

arXiv2023

Coloring a Dominating Set Without Conflicts: q-Subset Square Coloring

The \emph{Square Colouring} of a graph $G$ refers to colouring of vertices of a graph such that any two distinct vertices which are at distance at most two receive different colours. In this paper, we initiate the study of a related colouring problem called the \emph{subset square colouring} of graphs. Broadly, the subset square colouring of a graph studies the square colouring of a dominating set of a graph using $q$ colours. Here, the aim is to optimize the number of colours used. This also generalizes the well-studied Efficient Dominating Set problem. We show that the q-Subset Square Colouring problem is NP-hard for all values of $q$ even on bipartite graphs and chordal graphs. We further study the parameterized complexity of this problem when parameterized by a number of structural parameters. We further show bounds on the number of colours needed to subset square colour some graph classes.

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arXiv2022

Polynomial Kernels for Generalized Domination Problems

In this paper, we study the parameterized complexity of a generalized domination problem called the [$σ, ρ$] Dominating Set problem. This problem generalizes a large number of problems including the Minimum Dominating Set problem and its many variants. The parameterized complexity of the [$σ, ρ$] Dominating Set problem parameterized by treewidth is well studied. Here the properties of the sets $σ$ and $ρ$ that make the problem tractable are identified [1]. We consider a larger parameter and investigate the existence of polynomial sized kernels. When $σ$ and $ρ$ are finite, we identify the exact condition when the [$σ, ρ$] Dominating Set problem parameterized by vertex cover admits polynomial kernels. Our lower and upper bound results can also be extended to more general conditions and provably smaller parameters as well.

cs.CC↗

arXiv2022

Red Blue Set Cover Problem on Axis-Parallel Hyperplanes and Other Objects

Given a universe $\mathcal{U}=R \cup B$ of a finite set of red elements $R$, and a finite set of blue elements $B$ and a family $\mathcal{F}$ of subsets of $\mathcal{U}$, the \RBSC problem is to find a subset $\mathcal{F}'$ of $\mathcal{F}$ that covers all blue elements of $B$ and minimum number of red elements from $R$. We prove that the \RBSC problem is NP-hard even when $R$ and $B$ respectively are sets of red and blue points in ${\rm I\!R}^2$ and the sets in $\mathcal{F}$ are defined by axis-parallel lines i.e, every set is a maximal set of points with the same $x$ or $y$ coordinate. We then study the parameterized complexity of a generalization of this problem, where $\mathcal{U}$ is a set of points in ${\rm I\!R}^d$ and $\mathcal{F}$ is a collection of set of axis-parallel hyperplanes in ${\rm I\!R}^d$, under different parameterizations. For every parameter, we show that the problem is fixed-parameter tractable and also show the existence of a polynomial kernel. We further consider the \RBSC problem for some special types of rectangles in ${\rm I\!R}^2$.

cs.CG↗

arXiv2022

Geometric Separability using Orthogonal Objects

Given a bichromatic point set $P=\textbf{R} \cup \textbf{B}$ of red and blue points, a separator is an object of a certain type that separates $\textbf{R}$ and $\textbf{B}$. We study the geometric separability problem when the separator is a) rectangular annulus of fixed orientation b) rectangular annulus of arbitrary orientation c) square annulus of fixed orientation d) orthogonal convex polygon. In this paper, we give polynomial time algorithms to construct separators of each of the above type that also optimizes a given parameter.

cs.CG↗

arXiv2019

FPT Algorithms for Conflict-free Coloring of Graphs and Chromatic Terrain Guarding

We present fixed parameter tractable algorithms for the conflict-free coloring problem on graphs. Given a graph $G=(V,E)$, \emph{conflict-free coloring} of $G$ refers to coloring a subset of $V$ such that for every vertex $v$, there is a color that is assigned to exactly one vertex in the closed neighborhood of $v$. The \emph{k-Conflict-free Coloring} problem is to decide whether $G$ can be conflict-free colored using at most $k$ colors. This problem is NP-hard even for $k=1$ and therefore under standard complexity theoretic assumptions, FPT algorithms do not exist when parameterised by the solution size. We consider the \emph{k-Conflict-free Coloring} problem parameterised by the treewidth of the graph and show that this problem is fixed parameter tractable. We also initiate the study of \emph{Strong Conflict-free Coloring} of graphs. Given a graph $G=(V,E)$, \emph{strong conflict-free coloring} of $G$ refers to coloring a subset of $V$ such that every vertex $v$ has at least one colored vertex in its closed neighborhood and moreover all the colored vertices in $v$'s neighborhood have distinct colors. We show that this problem is in FPT when parameterised by both the treewidth and the solution size. We further apply these algorithms to get efficient algorithms for a geometric problem namely the Terrain Guarding problem, when parameterised by a structural parameter.

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arXiv2015

Multivariate Complexity Analysis of Geometric {\sc Red Blue Set Cover}

We investigate the parameterized complexity of GENERALIZED RED BLUE SET COVER (Gen-RBSC), a generalization of the classic SET COVER problem and the more recently studied RED BLUE SET COVER problem. Given a universe $U$ containing $b$ blue elements and $r$ red elements, positive integers $k_\ell$ and $k_r$, and a family $\F$ of $\ell$ sets over $U$, the \srbsc\ problem is to decide whether there is a subfamily $\F'\subseteq \F$ of size at most $k_\ell$ that covers all blue elements, but at most $k_r$ of the red elements. This generalizes SET COVER and thus in full generality it is intractable in the parameterized setting. In this paper, we study a geometric version of this problem, called Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for an array of parameters, namely, $k_\ell, k_r, r, b$, and $\ell$, and all possible combinations of them. For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). In particular, on the algorithmic side, our study shows that a combination of $k_\ell$ and $k_r$ gives rise to a nontrivial algorithm for Gen-RBSC-lines. On the hardness side, we show that the problem is para-NP-hard when parameterized by $k_r$, and W[1]-hard when parameterized by $k_\ell$. Finally, for the combination of parameters for which Gen-RBSC-lines admits FPT algorithms, we ask for the existence of polynomial kernels. We are able to provide a complete kernelization dichotomy by either showing that the problem admits a polynomial kernel or that it does not contain a polynomial kernel unless $\CoNP \subseteq \NP/\mbox{poly}$.

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arXiv2015

On Strong Centerpoints

Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $\mathcal{F}$ be a family of geometric objects. We call a point $x \in P$ a strong centerpoint of $P$ w.r.t $\mathcal{F}$ if $x$ is contained in all $F \in \mathcal{F}$ that contains more than $cn$ points from $P$, where $c$ is a fixed constant. A strong centerpoint does not exist even when $\mathcal{F}$ is the family of halfspaces in the plane. We prove the existence of strong centerpoints with exact constants for convex polytopes defined by a fixed set of orientations. We also prove the existence of strong centerpoints for abstract set systems with bounded intersection.

cs.CG↗

arXiv2014

Selection Lemmas for various geometric objects

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set $P$. This question has been widely explored for simplices in $\mathbb{R}^d$, with tight bounds in $\mathbb{R}^2$. In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from $P$. We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in $\mathbb{R}^d$. In the second selection lemma, we consider an arbitrary $m$ sized subset of the set of all objects induced by $P$. We study this problem for axis-parallel rectangles and show that there exists an point in the plane that is contained in $\frac{m^3}{24n^4}$ rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir when $m$ is almost quadratic.

cs.CG↗

arXiv2012

Small Strong Epsilon Nets

Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of objects $\mathcal{C}$ if $x \in P$ is contained in every object $C \in \mathcal{C}$ that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in $\mathbb{R}^2$. We prove that a strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$ and give exact bounds. We then extend this to small strong $ε$-nets in the plane and prove upper and lower bounds for $ε_i^\mathcal{S}$ where $\mathcal{S}$ is the family of axis-parallel rectangles, halfspaces and disks. Here $ε_i^\mathcal{S}$ represents the smallest real number in $[0,1]$ such that there exists an $ε_i^\mathcal{S}$-net of size i with respect to $\mathcal{S}$.

cs.CG↗