arXiv subjects

N S Narayanaswamy

Publications and source records attributed to N S Narayanaswamy.

7 recordsLinked to original sources

arXiv2024

Parameterized Algorithms for the Steiner Arborescence Problem on a Hypercube

Motivated by a phylogeny reconstruction problem in evolutionary biology, we study the minimum Steiner arborescence problem on directed hypercubes (MSA-DH). Given $m$, representing the directed hypercube $\vec{Q}_m$, and a set of terminals $R$, the problem asks to find a Steiner arborescence that spans $R$ with minimum cost. As $m$ implicitly represents $\vec{Q}_m$ comprising $2^{m}$ vertices, the running time analyses of traditional Steiner tree algorithms on general graphs does not give a clear understanding of the actual complexity of this problem. We present algorithms that exploit the structure of the hypercube and run in time polynomial in $|R|$ and $m$. We explore the MSA-DH problem on three natural parameters - $R$, and two above-guarantee parameters, number of Steiner nodes $p$ and penalty $q$. For above-guarantee parameters, the parameterized MSA-DH problem takes $p \geq 0$ or $q\geq 0$ as input, and outputs a Steiner arborescence with at most $|R| + p - 1$ or $m + q$ edges respectively. We present the following results ($\tilde{\mathcal{O}}$ hides the polynomial factors): 1. An exact algorithm that runs in $\tilde{\mathcal{O}}(3^{|R|})$ time. 2. A randomized algorithm that runs in $\tilde{\mathcal{O}}(9^q)$ time with success probability $\geq 4^{-q}$. 3. An exact algorithm that runs in $\tilde{\mathcal{O}}(36^q)$ time. 4. A $(1+q)$-approximation algorithm that runs in $\tilde{\mathcal{O}}(1.25284^q)$ time. 5. An $\mathcal{O}\left(p\ell_{\mathrm{max}} \right)$-additive approximation algorithm that runs in $\tilde{\mathcal{O}}(\ell_{\mathrm{max}}^{p+2})$ time, where $\ell_{\mathrm{max}}$ is the maximum distance of any terminal from the root.

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arXiv2024

Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage

We improve the worst-case information theoretic lower bound of Munro and Wu (ISAAC 2018) for $n-$vertex unlabeled chordal graphs when vertex leafage is bounded and leafage is unbounded. The class of unlabeled $k-$vertex leafage chordal graphs that consists of all chordal graphs with vertex leafage at most $k$ and unbounded leafage, denoted $\mathcal{G}_k$, is introduced for the first time. For $k>0$ in $o(n/\log n)$, we obtain a lower bound of $((k-1)n \log n -kn \log k - O(\log n))-$bits on the size of any data structure that encodes a graph in $\mathcal{G}_k$. Further, for every $k-$vertex leafage chordal graph $G$ such that for $k>1$ in $o(n^c), c >0$, we present a $((k-1)n \log n + o(kn \log n))-$bit succinct data structure, constructed using the succinct data structure for path graphs with $kn/2$ vertices. Our data structure supports adjacency query in $O(k \log n)$ time and using additional $2n \log n$ bits, an $O(k^2 d_v \log n + \log^2 n)$ time neighbourhood query where $d_v$ is degree of $v \in V$.

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arXiv2024

Succinct Data Structure for Graphs with $d$-Dimensional $t$-Representation

Erdős and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4πt)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

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arXiv2023

Succinct Data Structure for Path Graphs

We consider the problem of designing a succinct data structure for {\it path graphs} (which are a proper subclass of chordal graphs and a proper superclass of interval graphs) on $n$ vertices while supporting degree, adjacency, and neighborhood queries efficiently. We provide the following two solutions for this problem: - an $n \log n+o(n \log n)$-bit succinct data structure that supports adjacency query in $O(\log n)$ time, neighborhood query in $O(d \log n)$ time and finally, degree query in $\min\{O(\log^2 n), O(d \log n)\}$ where $d$ is the degree of the queried vertex. - an $O(n \log^2 n)$-bit space-efficient data structure that supports adjacency and degree queries in $O(1)$ time, and the neighborhood query in $O(d)$ time where $d$ is the degree of the queried vertex. Central to our data structures is the usage of the classical heavy path decomposition by Sleator and Tarjan~\cite{ST}, followed by a careful bookkeeping using an orthogonal range search data structure using wavelet trees~\cite{Makinen2007} among others, which maybe of independent interest for designing succinct data structures for other graph classes.

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arXiv2020

The Connected Domination Number of Grids

Closed form expressions for the domination number of an $n \times m$ grid have attracted significant attention, and an exact expression has been obtained in 2011 by Gonçalves et al. In this paper, we present our results on obtaining new lower bounds on the connected domination number of an $n \times m$ grid. The problem has been solved for grids with up to $4$ rows and with $6$ rows by Tolouse et al and the best currently known lower bound for arbitrary $m,n$ is $\lceil\frac{mn}{3}\rceil$. Fujie came up with a general construction for a connected dominating set of an $n \times m$ grid of size $\min \left\{2n+(m-4)+\lfloor\frac{m-4}{3}\rfloor(n-2), 2m+(n-4)+\lfloor\frac{n-4}{3}\rfloor(m-2) \right\}$ . In this paper, we investigate whether this construction is indeed optimum. We prove a new lower bound of $\left\lceil\frac{mn+2\lceil\frac{\min \{m,n\}}{3}\rceil}{3} \right\rceil$ for arbitrary $m,n \geq 4$.

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arXiv2020

A polynomial time 12-approximation algorithm for restricted Santa Claus problem

In this paper, we consider the restricted case of the problem and improve the current best approximation ratio by presenting a polynomial time 12-approximation algorithm using linear programming and semi-definite programming. Our algorithm starts by solving the configuration LP and uses the optimum value to get a 12-gap instance. This is then followed by the well-known clustering technique of Bansal and Sviridenko\cite{bansal}. We then apply the analysis of Asadpour \textit{et al.} \cite{AFS,AFS2} to show that the clustered instance has an integer solution which is at least $\frac{1}{6}$ times the best possible value, which was computed by solving the configuration LP. To find this solution, we formulate a problem called the Extended Assignment Problem, and formulate it as an LP. We then, show that the associated polytope is integral and gives us an fractional solution of value at least $\frac{1}{6}$ times the optimum. From this solution we find a solution to a new quadratic program that we introduce to select one machine from each cluster, and then we show that the resulting instance has an Assignment LP fractional solution of value at least $\frac{1}{6}$ times the optimum. We then use the well known rounding technique due to Bezakova and Dani \cite{bezakova} on the 12-gap instance to get our 12-approximate solution.

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arXiv2016

A Characterization for the Existence of Connected $f$-Factors of $\textit{ Large}$ Minimum Degree

It is well known that when $f(v)$ is a constant for each vertex $v$, the connected $f$-factor problem is NP-Complete. In this note we consider the case when $f(v) \geq \lceil \frac{n}{2.5}\rceil$ for each vertex $v$, where $n$ is the number of vertices. We present a diameter based characterization of graphs having a connected $f$-factor (for such $f$). We show that if a graph $G$ has a connected $f$-factor and an $f$-factor with 2 connected components, then it has a connected $f$-factor of diameter at least 3. This result yields a polynomial time algorithm which first executes the Tutte's $f$-factor algorithm, and if the output has 2 connected components, our algorithm searches for a connected $f$-factor of diameter at least 3.

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