arXiv subjects

Akanksha Agrawal

Publications and source records attributed to Akanksha Agrawal.

25 records · Page 2Linked to original sources

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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arXiv2017

Polylogarithmic Approximation Algorithms for Weighted-$\mathcal{F}$-Deletion Problems

For a family of graphs $\cal F$, the canonical Weighted $\cal F$ Vertex Deletion problem is defined as follows: given an $n$-vertex undirected graph $G$ and a weight function $w: V(G)\rightarrow\mathbb{R}$, find a minimum weight subset $S\subseteq V(G)$ such that $G-S$ belongs to $\cal F$. We devise a recursive scheme to obtain $O(\log^{O(1)}n)$-approximation algorithms for such problems, building upon the classic technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to problems where an optimum solution $S$, together with a well-structured set $X$, form a balanced separator of $G$. We obtain the first $O(\log^{O(1)}n)$-approximation algorithms for the following problems. * We give an $O(\log^2n)$-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. * We give an $O(\log^3n)$-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth 1. Our methods also allow us to obtain in a clean fashion a $O(\log^{1.5}n)$-approximation algorithm for the Weighted $\cal F$ Vertex Deletion problem when $\cal F$ is a minor closed family excluding at least one planar graph. For the unweighted version of the problem constant factor approximation algorithms are were known~[Fomin et al., FOCS~2012], while for the weighted version considered here an $O(\log n \log\log n)$-approximation algorithm follows from~[Bansal et al., SODA~2017]. We believe that our recursive scheme can be applied to obtain $O(\log^{O(1)}n)$-approximation algorithms for many other problems as well.

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arXiv2017

Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the $O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.

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arXiv2016

Simultaneous Feedback Edge Set: A Parameterized Perspective

In this paper we consider Simultaneous Feedback Edge Set (Sim-FES) problem. In this problem, the input is an $n$-vertex graph $G$, an integer $k$ and a coloring function ${\sf col}: E(G) \rightarrow 2^{[α]}$ and the objective is to check whether there is an edge subset $S$ of cardinality at most $k$ in $G$ such that for all $i \in [α]$, $G_i - S$ is acyclic. Here, $G_i=(V(G), \{e\in E(G) \mid i \in {\sf col}(e)\})$ and $[α]=\{1,\ldots,α\}$. When $α=1$, the problem is polynomial time solvable. We show that for $α=3$ Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic graphs. The same reduction shows that the problem does not admit an algorithm of running time $O(2^{o(k)}n^{O(1)})$ unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time $O(2^{ωkα+α\log k} n^{O(1)})$, where $ω$ is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when $α=2$. We also give a kernel for Sim-FES with $(kα)^{O(α)}$ vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph $G$, an integer $q$ and, a coloring function ${\sf col}: E(G) \rightarrow 2^{[α]}$. The question is whether there is a edge subset $F$ of cardinality at least $q$ in $G$ such that for all $i\in [α]$, $G[F_i]$ is acyclic. Here, $F_i=\{e \in F \mid i \in \textsf{col}(e)\}$. We give an FPT algorithm for running in time $O(2^{ωq α}n^{O(1)})$.

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arXiv2015

A faster FPT Algorithm and a smaller Kernel for Block Graph Vertex Deletion

A graph $G$ is called a \emph{block graph} if each maximal $2$-connected component of $G$ is a clique. In this paper we study the Block Graph Vertex Deletion from the perspective of fixed parameter tractable (FPT) and kernelization algorithm. In particular, an input to Block Graph Vertex Deletion consists of a graph $G$ and a positive integer $k$ and the objective to check whether there exists a subset $S \subseteq V(G)$ of size at most $k$ such that the graph induced on $V(G)\setminus S$ is a block graph. In this paper we give an FPT algorithm with running time $4^kn^{O(1)}$ and a polynomial kernel of size $O(k^4)$ for Block Graph Vertex Deletion. The running time of our FPT algorithm improves over the previous best algorithm for the problem that ran in time $10^kn^{O(1)}$ and the size of our kernel reduces over the previously known kernel of size $O(k^9)$. Our results are based on a novel connection between Block Graph Vertex Deletion and the classical {\sc Feedback Vertex Set} problem in graphs without induced $C_4$ and $K_4-e$. To achieve our results we also obtain an algorithm for {\sc Weighted Feedback Vertex Set} running in time $3.618^kn^{O(1)}$ and improving over the running time of previously known algorithm with running time $5^kn^{O(1)}$.

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arXiv2015

Simultaneous Feedback Vertex Set: A Parameterized Perspective

Given a family of graphs $\mathcal{F}$, a graph $G$, and a positive integer $k$, the $\mathcal{F}$-Deletion problem asks whether we can delete at most $k$ vertices from $G$ to obtain a graph in $\mathcal{F}$. $\mathcal{F}$-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph $G = (V, \cup_{i=1}^α E_{i})$, where the edge set of $G$ is partitioned into $α$ color classes, is called an $α$-edge-colored graph. A natural extension of the $\mathcal{F}$-Deletion problem to edge-colored graphs is the $α$-Simultaneous $\mathcal{F}$-Deletion problem. In the latter problem, we are given an $α$-edge-colored graph $G$ and the goal is to find a set $S$ of at most $k$ vertices such that each graph $G_i \setminus S$, where $G_i = (V, E_i)$ and $1 \leq i \leq α$, is in $\mathcal{F}$. In this work, we study $α$-Simultaneous $\mathcal{F}$-Deletion for $\mathcal{F}$ being the family of forests. In other words, we focus on the $α$-Simultaneous Feedback Vertex Set ($α$-SimFVS) problem. Algorithmically, we show that, like its classical counterpart, $α$-SimFVS parameterized by $k$ is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant $α$. In particular, we give an algorithm running in $2^{O(αk)}n^{O(1)}$ time and a kernel with $O(αk^{3(α+ 1)})$ vertices. The running time of our algorithm implies that $α$-SimFVS is FPT even when $α\in o(\log n)$. We complement this positive result by showing that for $α\in O(\log n)$, where $n$ is the number of vertices in the input graph, $α$-SimFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).

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arXiv2014

Vertex Cover Gets Faster and Harder on Low Degree Graphs

The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric objects, often turns out to be exactly the vertex cover problem on restricted classes of graphs. In this work we explore a particular instance of such a phenomenon. We consider the problem of hitting all axis-parallel slabs induced by a point set P, and show that it is equivalent to the problem of finding a vertex cover on a graph whose edge set is the union of two Hamiltonian Paths. We show the latter problem to be NP-complete, and we also give an algorithm to find a vertex cover of size at most k, on graphs of maximum degree four, whose running time is 1.2637^k n^O(1).

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