arXiv subjects

Venkatesh Raman

Publications and source records attributed to Venkatesh Raman.

At least 19 recordsLinked to original sources

arXiv2025

Learning with Structure: Computing Consistent Subsets on Structurally-Regular Graphs

The Minimum Consistent Subset (MCS) problem arises naturally in the context of supervised clustering and instance selection. In supervised clustering, one aims to infer a meaningful partitioning of data using a small labeled subset. However, the sheer volume of training data in modern applications poses a significant computational challenge. The MCS problem formalizes this goal: given a labeled dataset $\mathcal{X}$ in a metric space, the task is to compute a smallest subset $S \subseteq \mathcal{X}$ such that every point in $\mathcal{X}$ shares its label with at least one of its nearest neighbors in $S$. Recently, the MCS problem has been extended to graph metrics, where distances are defined by shortest paths. Prior work has shown that MCS remains NP-hard even on simple graph classes like trees, though an algorithm with runtime $\mathcal{O}(2^{6c} \cdot n^6)$ is known for trees, where $c$ is the number of colors and $n$ the number of vertices. This raises the challenge of identifying graph classes that admit algorithms efficient in both $n$ and $c$. In this work, we study the Minimum Consistent Subset problem on graphs, focusing on two well-established measures: the vertex cover number ($vc$) and the neighborhood diversity ($nd$). We develop an algorithm with running time $vc^{\mathcal{O}(vc)}\cdot\text{Poly}(n,c)$, and another algorithm with runtime $nd^{\mathcal{O}(nd)}\cdot\text{Poly}(n,c)$. In the language of parameterized complexity, this implies that MCS is fixed-parameter tractable (FPT) parameterized by the vertex cover number and the neighborhood diversity. Notably, our algorithms remain efficient for arbitrarily many colors, as their complexity is polynomially dependent on the number of colors.

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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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arXiv2023

Dominator Coloring and CD Coloring in Almost Cluster Graphs

In this paper, we study two popular variants of Graph Coloring -- Dominator Coloring and CD Coloring. In both problems, we are given a graph $G$ and a natural number $\ell$ as input and the goal is to properly color the vertices with at most $\ell$ colors with specific constraints. In Dominator Coloring, we require for each $v \in V(G)$, a color $c$ such that $v$ dominates all vertices colored $c$. In CD Coloring, we require for each color $c$, a $v \in V(G)$ which dominates all vertices colored $c$. These problems, defined due to their applications in social and genetic networks, have been studied extensively in the last 15 years. While it is known that both problems are fixed-parameter tractable (FPT) when parameterized by $(t,\ell)$ where $t$ is the treewidth of $G$, we consider strictly structural parameterizations which naturally arise out of the problems' applications. We prove that Dominator Coloring is FPT when parameterized by the size of a graph's cluster vertex deletion (CVD) set and that CD Coloring is FPT parameterized by CVD set size plus the number of remaining cliques. En route, we design a simpler and faster FPT algorithms when the problems are parameterized by the size of a graph's twin cover, a special CVD set. When the parameter is the size of a graph's clique modulator, we design a randomized single-exponential time algorithm for the problems. These algorithms use an inclusion-exclusion based polynomial sieving technique and add to the growing number of applications using this powerful algebraic technique.

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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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arXiv2021

Space-Efficient FPT Algorithms

We prove algorithmic results showing that a number of natural parameterized problems are in the restricted-space parameterized classes Para-L and FPT+XL. The first class comprises problems solvable in f(k) n^{O(1)} time using g(k) + O(log n)) bits of space (k is the parameter and n is the input size; f and g are computable functions). The second class comprises problems solvable under the same time bound, but using g(k) log n bits of space instead. Earlier work on these classes has focused largely on their structural aspects and their relationships with various other classes. We complement this with Para-L and FPT+XL algorithms for a restriction of Hitting Set, some graph deletion problems where the target class has an infinite forbidden set characterization, a number of problems parameterized by vertex cover number, and Feedback Vertex Set.

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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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arXiv2021

Structural Parameterizations of Budgeted Graph Coloring

We introduce a variant of the graph coloring problem, which we denote as {\sc Budgeted Coloring Problem} (\bcp). Given a graph $G$, an integer $c$ and an ordered list of integers $\{b_1, b_2, \ldots, b_c\}$, \bcp asks whether there exists a proper coloring of $G$ where the $i$-th color is used to color at most $b_i$ many vertices. This problem generalizes two well-studied graph coloring problems, {\sc Bounded Coloring Problem} (\bocp) and {\sc Equitable Coloring Problem} (\ecp) and as in the case of other coloring problems, it is \nph even for constant values of $c$. So we study \bcp under the paradigm of parameterized complexity. \begin{itemize} \item We show that \bcp is \fpt (fixed-parameter tractable) parameterized by the vertex cover size. This generalizes a similar result for \ecp and immediately extends to the \bocp, which was earlier not known. \item We show that \bcp is polynomial time solvable for cluster graphs generalizing a similar result for \ecp. However, we show that \bcp is \fpt, but unlikely to have polynomial kernel, when parameterized by the deletion distance to clique, contrasting the linear kernel for \ecp for the same parameter. \item While the \bocp is known to be polynomial time solvable on split graphs, we show that \bcp is \nph on split graphs. As \bocp is hard on bipartite graphs when $c>3$, the result follows for \bcp as well. We provide a dichotomy result by showing that \bcp is polynomial time solvable on bipartite graphs when $c=2$. We also show that \bcp is \nph on co-cluster graphs, contrasting the polynomial time algorithm for \ecp and \bocp. \end{itemize} Finally we present an $\mathcal{O}^*(2^{|V(G)|})$ algorithm for the \bcp, generalizing the known algorithm with a similar bound for the standard chromatic number.

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arXiv2021

Sublinear-Space Approximation Algorithms for Max r-SAT

In the Max $r$-SAT problem, the input is a CNF formula with $n$ variables where each clause is a disjunction of at most $r$ literals. The objective is to compute an assignment which satisfies as many of the clauses as possible. While there are a large number of polynomial-time approximation algorithms for this problem, we take the viewpoint of space complexity following [Biswas et al., Algorithmica 2021] and design sublinear-space approximation algorithms for the problem. We show that the classical algorithm of [Lieberherr and Specker, JACM 1981] can be implemented to run in $n^{O(1)}$ time while using $(\log{n})$ bits of space. The more advanced algorithms use linear or semi-definite programming, and seem harder to carry out in sublinear space. We show that a more recent algorithm with approximation ratio $\sqrt{2}/2$ [Chou et al., FOCS 2020], designed for the streaming model, can be implemented to run in time $n^{O(r)}$ using $O(r \log{n})$ bits of space. While known streaming algorithms for the problem approximate optimum values and use randomization, our algorithms are deterministic and can output the approximately optimal assignments in sublinear space. For instances of Max $r$-SAT with planar incidence graphs, we devise a factor-$(1 - ε)$ approximation scheme which computes assignments in time $n^{O(r / ε)}$ and uses $\max\{\sqrt{n} \log{n}, (r / ε) \log^2{n}\}$ bits of space.

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arXiv2021

Approximation in (Poly-) Logarithmic Space

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for d-Hitting Set that runs in time n^{O(d^2 + d/ε})}, uses O((d^2 + d/ε) log n) bits of space, and achieves an approximation ratio of O((d/ε) n^ε) for any positive ε\leq 1 and any natural number d. In particular, this yields a factor-O(log n) approximation algorithm which runs in time n^{O(log n)} and uses O(log^2 n) bits of space (for constant d). As a corollary, we obtain similar bounds for Vertex Cover and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for Vertex Cover on graphs with maximum degree Δ, and an algorithm for computing maximal independent sets which both run in time n^{O(Δ)} and use O(Δlog n) bits of space. For the more general d-Hitting Set problem, we devise a factor-d approximation algorithm which runs in time n^{O(d δ^2)} and uses O(d δ^2 log n) bits of space on set families where each element appears in at most δsets. For Independent Set restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses O(log n) bits of space. We also devise a factor-O(d^2) approximation algorithm for Dominating Set on d-degenerate graphs which runs in time n^{O(log n)} and uses O(log^2 n) bits of space. For d-regular graphs, we show how a known randomized factor-O(log d) approximation algorithm can be derandomized to run in time n^{O(1)} and use O(log n) bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.

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arXiv2020

Vertex Cover Reconfiguration and Beyond

In the Vertex Cover Reconfiguration (VCR) problem, given a graph $G$, positive integers $k$ and $\ell$ and two vertex covers $S$ and $T$ of $G$ of size at most $k$, we determine whether $S$ can be transformed into $T$ by a sequence of at most $\ell$ vertex additions or removals such that every operation results in a vertex cover of size at most $k$. Motivated by results establishing the W[1]-hardness of VCR when parameterized by $\ell$, we delineate the complexity of the problem restricted to various graph classes. In particular, we show that VCR remains W[1]-hard on bipartite graphs, is NP-hard, but fixed-parameter tractable on (regular) graphs of bounded degree and more generally on nowhere dense graphs and is solvable in polynomial time on trees and (with some additional restrictions) on cactus graphs.

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arXiv2020

Structural Parameterizations of Tracking Paths Problem

Given a graph $G$ with source and destination vertices $s,t\in V(G)$ respectively, \textsc{Tracking Paths} asks for a minimum set of vertices $T\subseteq V(G)$, such that the sequence of vertices encountered in each simple path from $s$ to $t$ is unique. The problem was proven \textsc{NP}-hard \cite{tr-j} and was found to admit a quadratic kernel when parameterized by the size of the desired solution \cite{quadratic}. Following recent trends, for the first time, we study \textsc{Tracking Paths} with respect to structural parameters of the input graph, parameters that measure how far the input graph is, from an easy instance. We prove that \textsc{Tracking Paths} admits fixed-parameter tractable (\textsc{FPT}) algorithms when parameterized by the size of vertex cover, and the size of cluster vertex deletion set for the input graph.

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arXiv2020

Improved Kernels for Tracking Path Problem

Tracking of moving objects is crucial to security systems and networks. Given a graph $G$, terminal vertices $s$ and $t$, and an integer $k$, the \textsc{Tracking Paths} problem asks whether there exists at most $k$ vertices, which if marked as trackers, would ensure that the sequence of trackers encountered in each s-t path is unique. It is known that the problem is NP-hard and admits a kernel (reducible to an equivalent instance) with $\mathcal{O}(k^6)$ vertices and $\mathcal{O}(k^7)$ edges, when parameterized by the size of the output (tracking set) $k$ [5]. An interesting question that remains open is whether the existing kernel can be improved. In this paper we answer this affirmatively: (i) For general graphs, we show the existence of a kernel of size $\mathcal{O}(k^2)$, (ii) For planar graphs, we improve this further by giving a kernel of size $\mathcal{O}(k)$. In addition, we also show that finding a tracking set of size at most $n-k$ for a graph on $n$ vertices is hard for the parameterized complexity class W[1], when parameterized by $k$.

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arXiv2020

Fixed-parameter tractable algorithms for Tracking Shortest Paths

We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shortest Paths asks if there exists a k-sized subset of vertices (referred to as tracking set) that intersects each shortest s-t path in a distinct set of vertices. We first generalize this problem for set systems, namely Tracking Set System, where given a family of subsets of a universe, we are required to find a subset of elements from the universe that has a unique intersection with each set in the family. Tracking Set System is shown to be fixed-parameter tractable due to its relation with a known problem, Test Cover. By a reduction to the well-studied d-hitting set problem, we give a polynomial (with respect to k) kernel for the case when the set sizes are bounded by d. This also helps solving Tracking Shortest Paths when the input graph diameter is bounded by d. While the results for Tracking Set System help to show that Tracking Shortest Paths is fixed-parameter tractable, we also give an independent algorithm by using some preprocessing rules, resulting in an improved running time.

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arXiv2020

Recognizing $k$-Clique Extendible Orderings

A graph is $k$-clique-extendible if there is an ordering of the vertices such that whenever two $k$-sized overlapping cliques $A$ and $B$ have $k-1$ common vertices, and these common vertices appear between the two vertices $a,b\in (A\setminus B)\cup (B\setminus A)$ in the ordering, there is an edge between $a$ and $b$, implying that $A\cup B$ is a $(k+1)$-sized clique. Such an ordering is said to be a $k$-C-E ordering. These graphs arise in applications related to modelling preference relations. Recently, it has been shown that a maximum sized clique in such a graph can be found in $n^{O(k)}$ time when the ordering is given. When $k$ is $2$, such graphs are precisely the well-known class of comparability graphs and when $k$ is $3$ they are called triangle-extendible graphs. It has been shown that triangle-extendible graphs appear as induced subgraphs of visibility graphs of simple polygons, and the complexity of recognizing them has been mentioned as an open problem in the literature. While comparability graphs (i.e. $2$-C-E graphs) can be recognized in polynomial time, we show that recognizing $k$-C-E graphs is NP-hard for any fixed $k \geq 3$ and co-NP-hard when $k$ is part of the input. While our NP-hardness reduction for $k \geq 4$ is from the betweenness problem, for $k=3$, our reduction is an intricate one from the $3$-colouring problem. We also show that the problems of determining whether a given ordering of the vertices of a graph is a $k$-C-E ordering, and that of finding an $\ell$-sized (or maximum sized) clique in a $k$-C-E graph, given a $k$-C-E ordering, are complete for the parameterized complexity classes co-W[1] and W[1] respectively, when parameterized by $k$. However we show that the former is fixed-parameter tractable when parameterized by the treewidth of the graph.

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arXiv2020

Structural Parameterizations with Modulator Oblivion

It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most $k$ vertices whose deletion results in a chordal graph, when parameterized by $k$. While this investigation fits naturally into the recent trend of what are called `structural parameterizations', here we assume that the deletion set is not given. One method to solve them is to compute a $k$-sized or an approximate ($f(k)$ sized, for a function $f$) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least $k^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ running time when we use the known parameterized or approximation algorithms for finding a $k$-sized chordal deletion set on an $n$ vertex graph. In this work, we design $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in time $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ where each bag is a union of four cliques and $\mathcal{O}(k)$ vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are adaptive (robust) in the sense that given an integer $k$, they detect whether the graph has a chordal vertex deletion set of size at most $k$ or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the Strong Exponential Time Hypothesis (SETH).

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arXiv2019

Tractability of Konig Edge Deletion Problems

A graph is said to be a Konig graph if the size of its maximum matching is equal to the size of its minimum vertex cover. The Konig Edge Deletion problem asks if in a given graph there exists a set of at most k edges whose deletion results in a Konig graph. While the vertex version of the problem (Konig vertex deletion) has been shown to be fixed-parameter tractable more than a decade ago, the fixed-parameter-tractability of the Konig Edge Deletion problem has been open since then, and has been conjectured to be W[1]-hard in several papers. In this paper, we settle the conjecture by proving it W[1]-hard. We prove that a variant of this problem, where we are given a graph G and a maximum matching M and we want a k-sized Konig edge deletion set that is disjoint from M, is fixed-parameter-tractable.

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arXiv2017

Frameworks for Designing In-place Graph Algorithms

Read-only memory model is a classical model of computation to study time-space tradeoffs of algorithms. One of the classical results on the ROM model is that any sorting algorithm that uses O(s) words of extra space requires $Ω(n^2/s)$ comparisons for $ \lg n \leq s \leq n/\lg n$ and the bound has also been recently matched by an algorithm. However, if we relax the model (from ROM), we do have sorting algorithms (say Heapsort) that can sort using $O(n \lg n)$ comparisons using $O(\lg n)$ bits of extra space, even keeping a permutation of the given input sequence at any point of time during the algorithm. We address similar questions for graph algorithms. We show that a simple natural relaxation of ROM model allows us to implement fundamental graph search methods like BFS and DFS more space efficiently than in ROM. By simply allowing elements in the adjacency list of a vertex to be permuted, we show that, on an undirected or directed connected graph $G$ having $n$ vertices and $m$ edges, the vertices of $G$ can be output in a DFS or BFS order using $O(\lg n)$ bits of extra space and $O(n^3 \lg n)$ time. Thus we obtain similar bounds for reachability and shortest path distance (both for undirected and directed graphs). With a little more (but still polynomial) time, we can also output vertices in the lex-DFS order. As reachability in directed graphs and shortest path distance are NL-complete, and lex-DFS is P-complete, our results show that our model is more powerful than ROM if L $\neq$ P. En route, we also introduce and develop algorithms for another relaxation of ROM where the adjacency lists of the vertices are circular lists and we can modify only the heads of the lists. All our algorithms are simple but quite subtle, and we believe that these models are practical enough to spur interest for other graph problems in these models.

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