arXiv subjects

Prafullkumar Tale

Publications and source records attributed to Prafullkumar Tale.

35 records · Page 2Linked to original sources

arXiv2023

Problems in NP can Admit Double-Exponential Lower Bounds when Parameterized by Treewidth or Vertex Cover

Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $\Sigma_2^p$, $\Pi_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds.

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arXiv2022

Romeo and Juliet Meeting in Forest Like Regions

The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of $k \ge 1$ agents. While the initial positions of Facilitator's agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator's goal to bring both her agents at same vertex and Divider's goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with $k$ agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents. This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when parameterized by the modular width and the number of allowed rounds. However, they left open the complexity of the problem from the perspective of other structural parameters. In particular, they explicitly asked whether the problem admits an FPT or XP-algorithm with respect to the treewidth of the input graph. We answer this question in the negative and show that Rendezvous is co-NP-hard even for graphs of constant treewidth. Further, we show that the problem is co-W[1]-hard when parameterized by the feedback vertex set number and the number of agents, and is unlikely to admit a polynomial kernel when parameterized by the vertex cover number and the number of agents. Complementing these hardness results, we show that the Rendezvous is FPT when parameterized by both the vertex cover number and the solution size. Finally, for graphs of treewidth at most two and girds, we show that the problem can be solved in polynomial time.

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arXiv2022

Domination and Cut Problems on Chordal Graphs with Bounded Leafage

The leafage of a chordal graph G is the minimum integer l such that G can be realized as an intersection graph of subtrees of a tree with l leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $2^{O(l^2)} n^{O(1)}$. We present a conceptually much simpler algorithm that runs in time $2^{O(l)} n^{O(1)}$. We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple $n^{O(l)}$-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $n^{O(1)}$-time.

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arXiv2022

Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters

For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set} if for any two vertices $u,v \in V(G)$, there exists a vertex $w \in S$ such that $d(w,u) \neq d(w,v)$. The {\sc Metric Dimension} problem takes as input a graph $G$ and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. This problem was introduced in the 1970s and is known to be \NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the problem is \W[2]-hard when parameterized by the natural parameter $k$. They also observed that it is \FPT\ when parameterized by the vertex cover number and asked about its complexity under \emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {\sc Metric Dimension} is \W[1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit~[IPEC 2019] which states that the problem is \W[1]-hard parameterized by the pathwidth. On the positive side, we show that {\sc Metric Dimension} is \FPT\ when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

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arXiv2022

The Complexity of Contracting Bipartite Graphs into Small Cycles

For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problem takes as input an undirected simple graph $G$ and determines whether $G$ can be transformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$ vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that $C_4$-Contractibility is NP-complete in general graphs. It is easy to verify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ on bipartite graphs for $\ell = 6$ and posed as open problems the status of the problem when $\ell$ is 4 or 5. In this paper, we show that both $C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite graphs.

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arXiv2022

Parameterized Complexity of Weighted Multicut in Trees

The Edge Multicut problem is a classical cut problem where given an undirected graph $G$, a set of pairs of vertices $\mathcal{P}$, and a budget $k$, the goal is to determine if there is a set $S$ of at most $k$ edges such that for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. Edge Multicut has been relatively recently shown to be fixed-parameter tractable (FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], and independently by Bousquet et al. [SICOMP 2018]. In the weighted version of the problem, called Weighted Edge Multicut one is additionally given a weight function $\mathtt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and the goal is to determine if there is a solution of size at most $k$ and weight at most $w$. Both the FPT algorithms for Edge Multicut by Marx et al. and Bousquet et al. fail to generalize to the weighted setting. In fact, the weighted problem is non-trivial even on trees and determining whether Weighted Edge Multicut on trees is FPT was explicitly posed as an open problem by Bousquet et al. [STACS 2009]. In this article, we answer this question positively by designing an algorithm which uses a very recent result by Kim et al. [STOC 2022] about directed flow augmentation as subroutine. We also study a variant of this problem where there is no bound on the size of the solution, but the parameter is a structural property of the input, for example, the number of leaves of the tree. We strengthen our results by stating them for the more general vertex deletion version.

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arXiv2022

Parameterized and Exact Algorithms for Class Domination Coloring

A class domination coloring (also called cd-Coloring or dominated coloring) of a graph is a proper coloring in which every color class is contained in the neighbourhood of some vertex. The minimum number of colors required for any cd-coloring of $G$, denoted by $χ_{cd}(G)$, is called the class domination chromatic number (cd-chromatic number) of $G$. In this work, we consider two problems associated with the cd-coloring of a graph in the context of exact exponential-time algorithms and parameterized complexity. (1) Given a graph $G$ on $n$ vertices, find its cd-chromatic number. (2) Given a graph $G$ and integers $k$ and $q$, can we delete at most $k$ vertices such that the cd-chromatic number of the resulting graph is at most $q$? For the first problem, we give an exact algorithm with running time $\Oh(2^n n^4 \log n)$. Also, we show that the problem is \FPT\ with respect to the number $q$ of colors as the parameter on chordal graphs. On graphs of girth at least 5, we show that the problem also admits a kernel with $\Oh(q^3)$ vertices. For the second (deletion) problem, we show \NP-hardness for each $q \geq 2$. Further, on split graphs, we show that the problem is \NP-hard if $q$ is a part of the input and \FPT\ with respect to $k$ and $q$ as combined parameters. As recognizing graphs with cd-chromatic number at most $q$ is \NP-hard in general for $q \geq 4$, the deletion problem is unlikely to be \FPT\ when parameterized by the size of the deletion set on general graphs. We show fixed parameter tractability for $q \in \{2,3\}$ using the known algorithms for finding a vertex cover and an odd cycle transversal as subroutines.

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arXiv2022

Sparsification Lower Bound for Linear Spanners in Directed Graphs

For $α\ge 1$, $β\ge 0$, and a graph $G$, a spanning subgraph $H$ of $G$ is said to be an $(α, β)$-spanner if $\dist(u, v, H) \le α\cdot \dist(u, v, G) + β$ holds for any pair of vertices $u$ and $v$. These type of spanners, called \emph{linear spanners}, generalizes \emph{additive spanners} and \emph{multiplicative spanners}. Recently, Fomin, Golovach, Lochet, Misra, Saurabh, and Sharma initiated the study of additive and multiplicative spanners for directed graphs (IPEC $2020$). In this article, we continue this line of research and prove that \textsc{Directed Linear Spanner} parameterized by the number of vertices $n$ admits no polynomial compression of size $\calO(n^{2 - ε})$ for any $ε> 0$ unless $\NP \subseteq \coNP/poly$. We show that similar results hold for \textsc{Directed Additive Spanner} and \textsc{Directed Multiplicative Spanner} problems. This sparsification lower bound holds even when the input is a directed acyclic graph and $α, β$ are \emph{any} computable functions of the distance being approximated.

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arXiv2022

Reducing the Vertex Cover Number via Edge Contractions

The CONTRACTION(vc) problem takes as input a graph $G$ on $n$ vertices and two integers $k$ and $d$, and asks whether one can contract at most $k$ edges to reduce the size of a minimum vertex cover of $G$ by at least $d$. Recently, Lima et al. [JCSS 2021] proved, among other results, that unlike most of the so-called blocker problems, CONTRACTION(vc) admits an XP algorithm running in time $f(d) \cdot n^{O(d)}$. They left open the question of whether this problem is FPT under this parameterization. In this article, we continue this line of research and prove the following results: 1. CONTRACTION(vc) is W[1]-hard parameterized by $k + d$. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time $f(k + d) \cdot n^{o(k + d)}$ for any function $f$. In particular, this answers the open question stated in Lima et al. [JCSS 2021] in the negative. 2. It is NP-hard to decide whether an instance $(G, k, d)$ of CONTRACTION(vc) is a yes-instance even when $k = d$, hence enhancing our understanding of the classical complexity of the problem. 3. CONTRACTION(vc) can be solved in time $2^{O(d)} \cdot n^{k - d + O(1)}$. This XP algorithm improves the one of Lima et al. [JCSS 2021], which uses Courcelle's theorem as a subroutine and hence, the $f(d)$-factor in the running time is non-explicit and probably very large. On the other hard, it shows that when $k=d$, the problem is FPT parameterized by $d$ (or by $k$).

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arXiv2021

A Framework for Parameterized Subexponential Algorithms for Generalized Cycle Hitting Problems on Planar Graphs

Subexponential parameterized algorithms are known for a wide range of natural problems on planar graphs, but the techniques are usually highly problem specific. The goal of this paper is to introduce a framework for obtaining $n^{O(\sqrt{k})}$ time algorithms for a family of graph modification problems that includes problems that can be seen as generalized cycle hitting problems. Our starting point is the Node Unique Label Cover problem (that is, given a CSP instance where each constraint is a permutation of values on two variables, the task is to delete $k$ variables to make the instance satisfiable). We introduce a variant of the problem where $k$ vertices have to be deleted such that every 2-connected component of the remaining instance is satisfiable. Then we extend the problem with cardinality constraints that restrict the number of times a certain value can be used (globally or within a 2-connected component of the solution). We show that there is an $n^{O(\sqrt{k})}$ time algorithm on planar graphs for any problem that can be formulated this way, which includes a large number of well-studied problems, for example, Odd Cycle Transversal, Subset Feedback Vertex Set, Group Feedback Vertex Set, Subset Group Feedback Vertex Set, Vertex Multiway Cut, and Component Order Connectivity. For those problems that admit appropriate (quasi)polynomial kernels (that increase the parameter only linearly and preserve planarity), our results immediately imply $2^{O(\sqrt{k}\cdot\operatorname{polylog}(k))}n^{O(1)}$ time parameterized algorithms on planar graphs. In particular, we use or adapt known kernelization results to obtain $2^{O(\sqrt{k}\cdot \operatorname{polylog}(k))} n^{O(1)}$ time (randomized) algorithms for Vertex Multiway Cut, Group Feedback Vertex Set, and Subset Feedback Vertex Set.

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arXiv2021

$α$-approximate Reductions: a Novel Source of Heuristics for Better Approximation Algorithms

Lokshtanov et al.~[STOC 2017] introduced \emph{lossy kernelization} as a mathematical framework for quantifying the effectiveness of preprocessing algorithms in preserving approximation ratios. \emph{$α$-approximate reduction rules} are a central notion of this framework. We propose that carefully crafted $α$-approximate reduction rules can yield improved approximation ratios in practice, while being easy to implement as well. This is distinctly different from the (theoretical) purpose for which Lokshtanov et al. designed $α$-approximate Reduction Rules. As evidence in support of this proposal we present a new 2-approximate reduction rule for the \textsc{Dominating Set} problem. This rule, when combined with an approximation algorithm for \textsc{Dominating Set}, yields significantly better approximation ratios on a variety of benchmark instances as compared to the latter algorithm alone. The central thesis of this work is that $α$-approximate reduction rules can be used as a tool for designing approximation algorithms which perform better in practice. To the best of our knowledge, ours is the first exploration of the use of $α$-approximate reduction rules as a design technique for practical approximation algorithms. We believe that this technique could be useful in coming up with improved approximation algorithms for other optimization problems as well.

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arXiv2020

On the Parameterized Complexity of \textsc{Maximum Degree Contraction} Problem

In the \textsc{Maximum Degree Contraction} problem, input is a graph $G$ on $n$ vertices, and integers $k, d$, and the objective is to check whether $G$ can be transformed into a graph of maximum degree at most $d$, using at most $k$ edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time $n^{\mathcal{O}(k)}$. As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (Ð). Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm of Parameterized Complexity and proved, among other things, that it admits an \FPT\ algorithm running in time $(d + k)^{2k} \cdot n^{\mathcal{O}(1)} = 2^{\mathcal{O}(k \log (k+d) )} \cdot n^{\mathcal{O}(1)}$, and remains \NP-hard for every constant $d \ge 2$ (Acta Informatica $(2014)$). We present a different \FPT\ algorithm that runs in time $2^{\mathcal{O}(dk)} \cdot n^{\mathcal{O}(1)}$. In particular, our algorithm runs in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$, for every fixed $d$. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by $k + d$. We answer this question in the negative and prove that it does not admit a polynomial compression unless $\NP \subseteq \coNP/poly$.

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arXiv2020

Parameterized Complexity of Maximum Edge Colorable Subgraph

A graph $H$ is {\em $p$-edge colorable} if there is a coloring $ψ: E(H) \rightarrow \{1,2,\dots,p\}$, such that for distinct $uv, vw \in E(H)$, we have $ψ(uv) \neq ψ(vw)$. The {\sc Maximum Edge-Colorable Subgraph} problem takes as input a graph $G$ and integers $l$ and $p$, and the objective is to find a subgraph $H$ of $G$ and a $p$-edge-coloring of $H$, such that $|E(H)| \geq l$. We study the above problem from the viewpoint of Parameterized Complexity. We obtain \FPT\ algorithms when parameterized by: $(1)$ the vertex cover number of $G$, by using {\sc Integer Linear Programming}, and $(2)$ $l$, a randomized algorithm via a reduction to \textsc{Rainbow Matching}, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters $p+k$, where $k$ is one of the following: $(1)$ the solution size, $l$, $(2)$ the vertex cover number of $G$, and $(3)$ $l - {\mm}(G)$, where ${\mm}(G)$ is the size of a maximum matching in $G$; we show that the (decision version of the) problem admits a kernel with $\mathcal{O}(k \cdot p)$ vertices. Furthermore, we show that there is no kernel of size $\mathcal{O}(k^{1-ε} \cdot f(p))$, for any $ε> 0$ and computable function $f$, unless $\NP \subseteq \CONPpoly$.

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arXiv2020

On the Parameterized Complexity Of Grid Contraction

For a family of graphs $\mathcal{G}$, the $\mathcal{G}$-\textsc{Contraction} problem takes as an input a graph $G$ and an integer $k$, and the goal is to decide if there exists $F \subseteq E(G)$ of size at most $k$ such that $G/F$ belongs to $\mathcal{G}$. Here, $G/F$ is the graph obtained from $G$ by contracting all the edges in $F$. In this article, we initiate the study of \textsc{Grid Contraction} from the parameterized complexity point of view. We present a fixed parameter tractable algorithm, running in time $c^k \cdot |V(G)|^{\mathcal{O}(1)}$, for this problem. We complement this result by proving that unless Ð fails, there is no algorithm for \textsc{Grid Contraction} with running time $c^{o(k)} \cdot |V(G)|^{\mathcal{O}(1)}$. We also present a polynomial kernel for this problem.

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arXiv2020

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that {\em contracts edges} is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting $k$ edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this paper, we study the \textsc{$\cal F$-Contraction} problem, where $\cal F$ is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph $G$ and an integer $k$, \textsc{ $\cal F$-Contraction} asks whether there exists $X \subseteq E(G)$ such that $G/X \in \cal F$ and $|X| \leq k$. Here, $G/X$ is the graph obtained from $G$ by contracting edges in $X$. We obtain the following results for the \textsc{ $\cal F$-Contraction} problem. $(1)$ We show that \textsc{Clique Contraction} admits a polynomial-size approximate kernelization scheme (\textsf{PSAKS}). $(2)$ We give a $(2+ε)$-approximate polynomial kernel for \textsc{Split Contraction} (which also implies a factor $(2+ε)$-\FPT-approximation algorithm for \textsc{ Split Contraction}). Furthermore, we show that, assuming \textsf{ Gap-ETH}, there is no $\left(\frac{5}{4}-δ\right)$-\FPT-approximation algorithm for \textsc{Split Contraction}. Here, $ε, δ>0$ are fixed constants. $(3)$ \textsc{Chordal Contraction} is known to be \WTH. We complement this result by observing that the existing \textsf{W[2]-hardness} reduction can be adapted to show that, assuming \FPT $\neq$ \textsf{W[1]}, there is no $F(k)$-\FPT-approximation algorithm for \textsc{Chordal Contraction}. Here, $F(k)$ is an arbitrary function depending on $k$ alone.

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arXiv2019

Subset Feedback Vertex Set in Chordal and Split Graphs

In the \textsc{Subset Feedback Vertex Set (Subset-FVS)} problem the input is a graph $G$, a subset $T$ of vertices of $G$ called the `terminal' vertices, and an integer $k$. The task is to determine whether there exists a subset of vertices of cardinality at most $k$ which together intersect all cycles which pass through the terminals. \textsc{Subset-FVS} generalizes several well studied problems including \textsc{Feedback Vertex Set} and \textsc{Multiway Cut}. This problem is known to be \NP-Complete even in split graphs. Cygan et al. proved that \textsc{Subset-FVS} is fixed parameter tractable (\FPT) in general graphs when parameterized by $k$ [SIAM J. Discrete Math (2013)]. In split graphs a simple observation reduces the problem to an equivalent instance of the $3$-\textsc{Hitting Set} problem with same solution size. This directly implies, for \textsc{Subset-FVS} \emph{restricted to split graphs}, (i) an \FPT algorithm which solves the problem in $\OhStar(2.076^k)$ time \footnote{The $\OhStar()$ notation hides polynomial factors.}% for \textsc{Subset-FVS} in Chordal % Graphs [Wahlström, Ph.D. Thesis], and (ii) a kernel of size $\mathcal{O}(k^3)$. We improve both these results for \textsc{Subset-FVS} on split graphs; we derive (i) a kernel of size $\mathcal{O}(k^2)$ which is the best possible unless $\NP \subseteq \coNP/{\sf poly}$, and (ii) an algorithm which solves the problem in time $\mathcal{O}^*(2^k)$. Our algorithm, in fact, solves \textsc{Subset-FVS} on the more general class of \emph{chordal graphs}, also in $\mathcal{O}^*(2^k)$ time.

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arXiv2017

On the Parameterized Complexity of Contraction to Generalization of Trees

For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes as an input a graph $G$ and an integer $k$, and the goal is to decide if there exists $S \subseteq E(G)$ of size at most $k$ such that $G/S$ belongs to $\cal F$. Here, $G/S$ is the graph obtained from $G$ by contracting all the edges in $S$. Heggernes et al.~[Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied $\cal F$-Contraction when $\cal F$ is a simple family of graphs such as trees and paths. In this paper, we study the $\mathcal{F}$-Contraction problem, where $\cal F$ generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let $\mathbb{T}_\ell$ be the family of graphs such that each graph in $\mathbb{T}_\ell$ can be made into a tree by deleting at most $\ell$ edges. Thus, the problem we study is $\mathbb{T}_\ell$-Contraction. We design an FPT algorithm for $\mathbb{T}_\ell$-Contraction running in time $\mathcal{O}((2\sqrt(\ell))^{\mathcal{O}(k + \ell)} \cdot n^{\mathcal{O}(1)})$. Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by $k$. Inspired by the negative result for the kernelization, we design a lossy kernel for $\mathbb{T}_\ell$-Contraction of size $ \mathcal{O}([k(k + 2\ell)] ^{(\lceil {\fracα{α-1}\rceil + 1)}})$.

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