arXiv subjects

R. Subashini

Publications and source records attributed to R. Subashini.

3 recordsLinked to original sources

arXiv2022

Cutting a tree with Subgraph Complementation is hard, except for some small trees

For a graph property $\Pi$, Subgraph Complementation to $\Pi$ is the problem to find whether there is a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph satisfying the property $\Pi$. We prove that the problem of Subgraph Complementation to $T$-free graphs is NP-Complete, for $T$ being a tree, except for 41 trees of at most 13 vertices (a graph is $T$-free if it does not contain any induced copies of $T$). This result, along with the 4 known polynomial-time solvable cases (when $T$ is a path on at most 4 vertices), leaves behind 37 open cases. Further, we prove that these hard problems do not admit any subexponential-time algorithms, assuming the Exponential Time Hypothesis. As an additional result, we obtain that Subgraph Complementation to paw-free graphs can be solved in polynomial-time.

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arXiv2021

On subgraph complementation to H-free graphs

For a class $\mathcal{G}$ of graphs, the problem SUBGRAPH COMPLEMENT TO $\mathcal{G}$ asks whether one can find a subset $S$ of vertices of the input graph $G$ such that complementing the subgraph induced by $S$ in $G$ results in a graph in $\mathcal{G}$. We investigate the complexity of the problem when $\mathcal{G}$ is $H$-free for $H$ being a complete graph, a star, a path, or a cycle. We obtain the following results: - When $H$ is a $K_t$ (a complete graph on $t$ vertices) for any fixed $t\geq 1$, the problem is solvable in polynomial-time. This applies even when $\mathcal{G}$ is a subclass of $K_t$-free graphs recognizable in polynomial-time, for example, the class of $(t-2)$-degenerate graphs. - When $H$ is a $K_{1,t}$ (a star graph on $t+1$ vertices), we obtain that the problem is NP-complete for every $t\geq 5$. This, along with known results, leaves only two unresolved cases - $K_{1,3}$ and $K_{1,4}$. - When $H$ is a $P_t$ (a path on $t$ vertices), we obtain that the problem is NP-complete for every $t\geq 7$, leaving behind only two unresolved cases - $P_5$ and $P_6$. - When $H$ is a $C_t$ (a cycle on $t$ vertices), we obtain that the problem is NP-complete for every $t\geq 8$, leaving behind four unresolved cases - $C_4, C_5, C_6,$ and $C_7$. Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time $2^{o(|V(G)|)}$), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for $\mathcal{G}$ are applicable for $\overline{\mathcal{G}}$, thereby obtaining similar results for $H$ being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).

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arXiv2013

$d$-COS-R is FPT via Interval Deletion

A binary matrix $M$ has the Consecutive Ones Property (COP) if there exists a permutation of columns that arranges the ones consecutively in all the rows. Given a matrix, the $d$-COS-R problem is to determine if there exists a set of at most $d$ rows whose deletion results in a matrix with COP. We consider the parameterized complexity of this problem with respect to the number $d$ of rows to be deleted as the parameter. The closely related Interval Deletion problem has recently shown to be FPT [Y. Cao and D. Marx, Interval Deletion is Fixed-Parameter Tractable, arXiv:1211.5933 [cs.DS],2012]. In this work, we describe a recursive depth-bounded search tree algorithm in which the problems at the leaf-level are solved as instances of Interval Deletion. The running time of the algorithm is dominated by the running time of Interval Deletion, and therefore we show that $d$-COS-R is fixed-parameter tractable and has a run-time of $O^*(10^d)$.

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