arXiv subjects

Roopam Saxena

Publications and source records attributed to Roopam Saxena.

4 recordsLinked to original sources

arXiv2024

The Parameterized Complexity of Terminal Monitoring Set

In Terminal Monitoring Set (TMS), the input is an undirected graph $G=(V,E)$, together with a collection $T$ of terminal pairs and the goal is to find a subset $S$ of minimum size that hits a shortest path between every pair of terminals. We show that this problem is W[2]-hard with respect to solution size. On the positive side, we show that TMS is fixed parameter tractable with respect to solution size plus distance to cluster, solution size plus neighborhood diversity, and feedback edge number. For the weighted version of the problem, we obtain a FPT algorithm with respect to vertex cover number, and for a relaxed version of the problem, we show that it is W[1]-hard with respect to solution size plus feedback vertex number.

cs.DM↗

arXiv2022

Parameterized Complexity of Path Set Packing

In Path Set Packing, the input is an undirected graph $G$, a collection $\calp$ of simple paths in $G$, and a positive integer $k$. The problem is to decide whether there exist $k$ edge-disjoint paths in $\calp$. We study the parameterized complexity of Path Set Packing with respect to both natural and structural parameters. We show that the problem is $W[1]$-hard with respect to vertex cover number, and $W[1]$-hard respect to pathwidth plus maximum degree plus solution size. These results answer an open question raised in COCOON 2018. On the positive side, we present an FPT algorithm parameterized by feedback vertex number plus maximum degree, and present an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in $\calp$. These positive results complement the hardness of Path Set Packing with respect to any subset of the parameters used in the FPT algorithms. We also give a $4$-approximation algorithm for maximum path set packing problem which runs in FPT time when parameterized by feedback edge number.

cs.DS↗

arXiv2021

Perfectly Matched Sets in Graphs: Parameterized and Exact Computation

In an undirected graph $G=(V,E)$, we say $(A,B)$ is a pair of perfectly matched sets if $A$ and $B$ are disjoint subsets of $V$ and every vertex in $A$ (resp. $B$) has exactly one neighbor in $B$ (resp. $A$). The size of a pair of perfectly matched sets $(A,B)$ is $|A|=|B|$. The PERFECTLY MATCHED SETS problem is to decide whether a given graph $G$ has a pair of perfectly matched sets of size $k$. We show that PMS is $W[1]$-hard when parameterized by solution size $k$ even when restricted to split graphs and bipartite graphs. We observe that PMS is FPT when parameterized by clique-width, and give FPT algorithms with respect to the parameters distance to cluster, distance to co-cluster and treewidth. Complementing FPT results, we show that PMS does not admit a polynomial kernel when parameterized by vertex cover number unless $NP\subseteq coNP/poly$. We also provide an exact exponential algorithm running in time $O^*(1.966^n)$. Finally, considering graphs with structural assumptions, we show that PMS remains $NP$-hard on planar graphs.

cs.DM↗

arXiv2021

An FPT algorithm for Matching Cut and d-cut

Given a positive integer $d$, the d-CUT is the problem of deciding if an undirected graph $G=(V,E)$ has a cut $(A,B)$ such that every vertex in $A$ (resp. $B$) has at most $d$ neighbors in $B$ (resp. $A$). For $d=1$, the problem is referred to as MATCHING CUT. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for d-CUT parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However, their paper doesn't provide an explicit bound on the running time, as it indirectly relies on a MSOL formulation and Courcelle's Theorem. Motivated by this, we design and present an FPT algorithm for d-CUT for general graphs with running time $2^{O(k\log k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut. This is the first FPT algorithm for the d-CUT and MATCHING CUT with an explicit dependence on this parameter. We also observe that there is no algorithm solving MATCHING CUT in time $2^{o(k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut unless ETH fails.

cs.DS↗