arXiv subjects

Atrayee Majumder

Publications and source records attributed to Atrayee Majumder.

5 recordsLinked to original sources

arXiv2025

Polynomial-time algorithms for PATH COVER and PATH PARTITION on trees and graphs of bounded treewidth

In the PATH COVER problem, one asks to cover the vertices of a graph using the smallest possible number of (not necessarily disjoint) paths. While the variant where the paths need to be pairwise vertex-disjoint, which we call PATH PARTITION, is extensively studied, surprisingly little is known about PATH COVER. We start filling this gap by designing a linear-time algorithm for PATH COVER on trees. We show that PATH COVER can be solved in polynomial time on graphs of bounded treewidth using a dynamic programming scheme. It runs in XP time $n^{t^{O(t)}}$ (where $n$ is the number of vertices and $t$ the treewidth of the input graph) or $κ^{t^{O(t)}}n$ if there is an upper-bound $κ$ on the solution size. A similar algorithm gives an FPT $2^{O(t\log t)}n$ algorithm for PATH PARTITION, which can be improved to (randomized) $2^{O(t)}n$ using the Cut\&Count technique. These results also apply to the variants where the paths are required to be induced (i.e. chordless) and/or edge-disjoint.

cs.DS↗

arXiv2022

Bounding threshold dimension: realizing graphic Boolean functions as the AND of majority gates

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic vectors of the cliques of $G$. Introduced in [Chv{á}tal and Hammer, Annals of Discrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$, denoted by $\dimth(G)$, is the minimum number of threshold graphs whose intersection yields $G$. Given a graph $G$ on $n$ vertices, in line with Chv{á}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is the Boolean function that has the property that $f_G(x) = 1$ if and only if $x$ is the characteristic vector of a clique in $G$. A Boolean function $f$ for which there exists a graph $G$ such that $f=f_G$ is called a \emph{graphic} Boolean function. It follows that for a graph $G$, $\dimth(G)$ is precisely the minimum number of \emph{majority} gates whose AND (or conjunction) realizes the graphic Boolean function $f_G$. The fact that there exist Boolean functions which can be realized as the AND of only exponentially many majority gates motivates us to study threshold dimension of graphs. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of its treewidth, maximum degree, degeneracy, number of vertices, size of a minimum vertex cover, etc. We also study threshold dimension of random graphs and graphs with high girth.

math.CO↗

arXiv2022

Local Boxicity and Maximum Degree

The \emph{local boxicity} of a graph $G$, denoted by $lbox(G)$, is the minimum positive integer $l$ such that $G$ can be obtained using the intersection of $k$ (, where $k \geq l$,) interval graphs where each vertex of $G$ appears as a non-universal vertex in at most $l$ of these interval graphs. Let $G$ be a graph on $n$ vertices having $m$ edges. Let $Δ$ denote the maximum degree of a vertex in $G$. We show that, (i) $lbox(G) \leq 2^{13\log^{*}Δ} Δ$. There exist graphs of maximum degree $Δ$ having a local boxicity of $Ω(\fracΔ{\logΔ})$. (ii) $lbox(G) \in O(\frac{n}{\log{n}})$. There exist graphs on $n$ vertices having a local boxicity of $Ω(\frac{n}{\log n})$. (iii) $lbox(G) \leq (2^{13\log^{*}{\sqrt{m}}} + 2 )\sqrt{m}$. There exist graphs with $m$ edges having a local boxicity of $Ω(\frac{\sqrt{m}}{\log m})$. (iv) the local boxicity of $G$ is at most its \emph{product dimension}. This connection helps us in showing that the local boxicity of the \emph{Kneser graph} $K(n,k)$ is at most $\frac{k}{2} \log{\log{n}}$. The above results can be extended to the \emph{local dimension} of a partially ordered set due to the known connection between local boxicity and local dimension. Finally, we show that the \emph{cubicity} of a graph on $n$ vertices of girth greater than $g+1$ is $O(n^{\frac{1}{\lfloor g/2\rfloor}}\log n)$.

math.CO↗

arXiv2020

Dimension of CPT posets

A collection of linear orders on $X$, say $\mathcal{L}$, is said to \emph{realize} a partially ordered set (or poset) $\mathcal{P} = (X, \preceq)$ if, for any two distinct $x,y \in X$, $x \preceq y$ if and only if $x \prec_L y$, $\forall L \in \mathcal{L}$. We call $\mathcal{L}$ a \emph{realizer} of $\mathcal{P}$. The \emph{dimension} of $\mathcal{P}$, denoted by $dim(\mathcal{P})$, is the minimum cardinality of a realizer of $\mathcal{P}$. A \emph{containment model} $M_{\mathcal{P}}$ of a poset $\mathcal{P}=(X,\preceq)$ maps every $x \in X$ to a set $M_x$ such that, for every distinct $x,y \in X,\ x \preceq y$ if and only if $M_x \varsubsetneq M_y$. We shall be using the collection $(M_x)_{x \in X}$ to identify the containment model $M_{\mathcal{P}}$. A poset $\mathcal{P}=(X,\preceq)$ is a Containment order of Paths in a Tree (CPT poset), if it admits a containment model $M_{\mathcal{P}}=(P_x)_{x \in X}$ where every $P_x$ is a path of a tree $T$, which is called the host tree of the model. We show that if a poset $\mathcal{P}$ admits a CPT model in a host tree $T$ of maximum degree $Δ$ and radius $r$, then \rogers{$dim(\mathcal{P}) \leq \lg\lg Δ+ (\frac{1}{2} + o(1))\lg\lg\lg Δ+ \lg r + \frac{1}{2} \lg\lg r + \frac{1}{2}\lg π+ 3$. This bound is asymptotically tight up to an additive factor of $\min(\frac{1}{2}\lg\lg\lg Δ, \frac{1}{2}\lg\lg r)$. Further, let $\mathcal{P}(1,2;n)$ be the poset consisting of all the $1$-element and $2$-element subsets of $[n]$ under `containment' relation and let $dim(1,2;n)$ denote its dimension. The proof of our main theorem gives a simple algorithm to construct a realizer for $\mathcal{P}(1,2;n)$ whose cardinality is only an additive factor of at most $\frac{3}{2}$ away from the optimum.

math.CO↗

arXiv2015

Understanding of Genetic Code Degeneracy and New Way of Classifying of Protein Family: A Mathematical Approach

The genetic code is the set of rules by which information encoded in genetic material (DNA or RNA sequences) is translated into proteins (amino acid sequences) by living cells. The code defines a mapping between tri-nucleotide sequences, called codons, and amino acids. Since there are 20 amino acids and 64 possible tri-nucleotide sequences, more than one among these 64 triplets can code for a single amino acid which incorporates the problem of degeneracy. This manuscript explains the underlying logic of degeneracy of genetic code based on a mathematical point of view using a parameter named Impression. Classification of protein family is also a long standing problem in the field of Bio-chemistry and Genomics. Proteins belonging to a particular class have some similar bio-chemical properties which are of utmost importance for new drug design. Using the same parameter Impression and using graph theoretic properties we have also devised a new way of classifying a protein family.

q-bio.OT↗