arXiv subjects

Tapas Kumar Mishra

Publications and source records attributed to Tapas Kumar Mishra.

23 records · Page 2Linked to original sources

arXiv2017

System of unbiased representatives for a collection of bicolorings

Let $\mathcal{B}$ denote a set of bicolorings of $[n]$, where each bicoloring is a mapping of the points in $[n]$ to $\{-1,+1\}$. For each $B \in \mathcal{B}$, let $Y_B=(B(1),\ldots,B(n))$. For each $A \subseteq [n]$, let $X_A \in \{0,1\}^n$ denote the incidence vector of $A$. A non-empty set $A$ is said to be an `unbiased representative' for a bicoloring $B \in \mathcal{B}$ if $\left\langle X_A,Y_B\right\rangle =0$. Given a set $\mathcal{B}$ of bicolorings, we study the minimum cardinality of a family $\mathcal{A}$ consisting of subsets of $[n]$ such that every bicoloring in $\mathcal{B}$ has an unbiased representative in $\mathcal{A}$.

math.CO↗

arXiv2017

Induced bisecting families for hypergraphs

Two $n$-dimensional vectors $A$ and $B$, $A,B \in \mathbb{R}^n$, are said to be \emph{trivially orthogonal} if in every coordinate $i \in [n]$, at least one of $A(i)$ or $B(i)$ is zero. Given the $n$-dimensional Hamming cube $\{0,1\}^n$, we study the minimum cardinality of a set $\mathcal{V}$ of $n$-dimensional $\{-1,0,1\}$ vectors, each containing exactly $d$ non-zero entries, such that every `possible' point $A \in \{0,1\}^n$ in the Hamming cube has some $V \in \mathcal{V}$ which is orthogonal, but not trivially orthogonal, to $A$. We give asymptotically tight lower and (constructive) upper bounds for such a set $\mathcal{V}$ except for the even values of $d \in Ω(n^{0.5+ε})$, for any $ε$, $0< ε\leq 0.5$.

math.CO↗

arXiv2016

Bicoloring covers for graphs and hypergraphs

Let the {\it bicoloring cover number $χ^c(G)$} for a hypergraph $G(V,E)$ be the minimum number of bicolorings of vertices of $G$ such that every hyperedge $e\in E$ of $G$ is properly bicolored in at least one of the $χ^c(G)$ bicolorings. We investigate the relationship between $χ^c(G)$, matchings, hitting sets, $α(G)$(independence number) and $χ(G)$ (chromatic number). We design a factor $O(\frac{\log n}{\log \log n-\log \log \log n})$ approximation algorithm for computing a bicoloring cover. We define a new parameter for hypergraphs - "cover independence number $γ(G)$" and prove that $\log \frac{|V|}{γ(G)}$ and $\frac{|V|}{2γ(G)}$ are lower bounds for $χ^c(G)$ and $χ(G)$, respectively. We show that $χ^c(G)$ can be approximated by a polynomial time algorithm achieving approximation ratio $\frac{1}{1-t}$, if $γ(G)=n^t$, where $t<1$. We also construct a particular class of hypergraphs $G(V,E)$ called {\it cover friendly} hypergraphs where the ratio of $α(G)$ to $γ(G)$ can be arbitrarily large.We prove that for any $t\geq 1$, there exists a $k$-uniform hypergraph $G$ such that the {\it clique number} $ω(G)=k$ and $χ^c(G) > t$. Let $m(k,x)$ denote the minimum number of hyperedges %in a $k$-uniform hypergraph $G$ such that some $k$-uniform hypergraph $G$ with $m(k,x)$ hyperedges does not have a bicoloring cover of size $x$. We show that $ 2^{(k-1)x-1} < m(k,x) \leq x \cdot k^2 \cdot 2^{(k+1)x+2}$. Let the {\it dependency $d(G)$} of $G$ be the maximum number of hyperedge neighbors of any hyperedge in $G$. We propose an algorithm for computing a bicoloring cover of size $x$ for $G$ if $d(G) \leq(\frac{2^{x(k-1)}}{e}-1)$ using $nx+kx\frac{m}{d}$ random bits.

cs.DM↗

arXiv2015

Strong $(r,p)$ Cover for Hypergraphs

We introduce the notion of the { \it strong $(r,p)$ cover} number $χ^c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $χ^c(G,k,r,p)$ denotes the minimum number of $r$-colorings of vertices in $V$ such that each hyperedge in $E$ contains at least $min(p,k)$ vertices of distinct colors in at least one of the $χ^c(G,k,r,p)$ $r$-colorings. We derive the exact values of $χ^c(K_n^k,k,r,p)$ for small values of $n$, $k$, $r$ and $p$, where $K_n^k$ denotes the complete $k$-uniform hypergraph of $n$ vertices. We study the variation of $χ^c(G,k,r,p)$ with respect to changes in $k$, $r$, $p$ and $n$; we show that $χ^c(G,k,r,p)$ is at least (i) $χ^c(G,k,r-1,p-1)$, and, (ii) $χ^c(G',k-1,r,p-1)$, where $G'$ is any $(n-1)$-vertex induced sub-hypergraph of $G$. We establish a general upper bound for $χ^c(K_n^k,k,r,p)$ for complete $k$-uniform hypergraphs using a divide-and-conquer strategy for arbitrary values of $k$, $r$ and $p$. We also relate $χ^c(G,k,r,p)$ to the number $|E|$ of hyperedges, and the maximum {\it hyperedge degree (dependency)} $d(G)$, as follows. We show that $χ^c(G,k,r,p)\leq x$ for integer $x>0$, if $|E|\leq \frac{1}{2}({\frac{r^k}{(t-1)^k \binom{r}{t-1}}})^x $, for any $k$-uniform hypergraph. We prove that a { \it strong $(r,p)$ cover} of size $x$ can be computed in randomized polynomial time if $d(G)\leq \frac{1}{e}({\frac{r^k}{(p-1)^k \binom{r}{p-1}}})^x-1$.

cs.DM↗

arXiv2015

An extremal problem in proper $(r,p)$-coloring of hypergraphs

Let $G(V,E)$ be a $k$-uniform hypergraph. A hyperedge $e \in E$ is said to be properly $(r,p)$ colored by an $r$-coloring of vertices in $V$ if $e$ contains vertices of at least $p$ distinct colors in the $r$-coloring. An $r$-coloring of vertices in $V$ is called a {\it strong $(r,p)$ coloring} if every hyperedge $e \in E$ is properly $(r,p)$ colored by the $r$-coloring. We study the maximum number of hyperedges that can be properly $(r,p)$ colored by a single $r$-coloring and the structures that maximizes number of properly $(r,p)$ colored hyperedges.

cs.DM↗