Search

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 $\mathcal{A}$., Comment: 14 pages, 1 figure

Let $n$ be any positive integer and $\mathcal{F}$ be a family of subsets of $[n]$. A family $\mathcal{F}'$ is said to be $D$-\emph{secting} for $\mathcal{F}$ if for every $A \in \mathcal{F}$, there exists a subset $A' \in \mathcal{F}'$ such that $|A \cap A'| - |A \cap ([n] \setminus A')|=i$, where $i \in D$, $D \subseteq \{-n,-n+1,\ldots,0,\ldots,n\}$. A $D$-\emph{secting} family $\mathcal{F}'$ of $\mathcal{F}$, where $D=\{-1,0,1\}$, is a \emph{bisecting} family ensuring the existence of a subset $A' \in \mathcal{F}'$ such that $|A \cap A'| \in \{\lceil \frac{|A|}{2}\rceil,\lfloor \frac{|A|}{2}\rfloor\}$, for each $A \in \mathcal{F}$. In this paper, we study $D$-secting families for $\mathcal{F}$ with restrictions on $D$, and the cardinalities of $\mathcal{F}$ and the subsets of $\mathcal{F}$., Comment: 15 pages, text overlap with https://www.ams.org/journals/tran/1987-300-01/S0002-9947-1987-0871675-6/S0002-9947-1987-0871675-6.pdf in Theorem 7

During 2008-2015, twenty-two introductory workshops on graph and geometric algorithms were organized for teachers and students (undergraduate, post-graduate and doctoral) of engineering colleges and universities at different states and union territories of India. The lectures were meant to provide exposure to the field of graph and geometric algorithms and to motivate the participants towards research. Fifty-eight professors from TIFR, IITs, IISc, IMSc, CMI, ISI Kolkata, and other institutes and universities delivered invited lectures on different topics in the design and analysis of algorithms, discrete applied mathematics, computer graphics, computer vision, and robotics. The first four workshops were funded by TIFR, BRNS and IIT Kharagpur, and the remaining workshops were funded by the NBHM. In this paper, we present the salient features of these workshops, and state our observations on the national impact of these workshops.

We introduce the notion of the { \it strong $(r,p)$ cover} number $\chi^c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $\chi^c(G,k,r,p)$ 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 $\chi^c(G,k,r,p)$ $r$-colorings. We derive the exact values of $\chi^c(K_n^k,k,r,p)$ 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 $\chi^c(G,k,r,p)$ with respect to changes in $k$, $r$, $p$ and $n$; we show that $\chi^c(G,k,r,p)$ is at least (i) $\chi^c(G,k,r-1,p-1)$, and, (ii) $\chi^c(G',k-1,r,p-1)$, where $G'$ is any $(n-1)$-vertex induced sub-hypergraph of $G$. We establish a general upper bound for $\chi^c(K_n^k,k,r,p)$ for complete $k$-uniform hypergraphs using a divide-and-conquer strategy for arbitrary values of $k$, $r$ and $p$. We also relate $\chi^c(G,k,r,p)$ to the number $|E|$ of hyperedges, and the maximum {\it hyperedge degree (dependency)} $d(G)$, as follows. We show that $\chi^c(G,k,r,p)\leq x$ 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 $d(G)\leq \frac{1}{e}({\frac{r^k}{(p-1)^k \binom{r}{p-1}}})^x-1$.

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.

Let the {\it bicoloring cover number $\chi^c(G)$} 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 $\chi^c(G)$ bicolorings. We investigate the relationship between $\chi^c(G)$, matchings, hitting sets, $\alpha(G)$(independence number) and $\chi(G)$ (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 $\gamma(G)$" and prove that $\log \frac{|V|}{\gamma(G)}$ and $\frac{|V|}{2\gamma(G)}$ are lower bounds for $\chi^c(G)$ and $\chi(G)$, respectively. We show that $\chi^c(G)$ can be approximated by a polynomial time algorithm achieving approximation ratio $\frac{1}{1-t}$, if $\gamma(G)=n^t$, where $t<1$. We also construct a particular class of hypergraphs $G(V,E)$ called {\it cover friendly} hypergraphs where the ratio of $\alpha(G)$ to $\gamma(G)$ 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} $\omega(G)=k$ and $\chi^c(G) > t$. 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., Comment: 20 pages, 3 figures

We develop graph theoretic methods for analysing maximally entangled pure states distributed between a number of different parties. We introduce a technique called {\it bicolored merging}, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication (LOCC). We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among $n$ agents. Further, we discuss a new graph theoretic metric on a class of multi-partite states, and its implications., Comment: To appear in the Journal of Mathematical Physics, 33 pages, 16 figures

It is known that the region $V(s)$ of a simple polygon $P$, directly visible (illuminable) from an internal point $s$, is simply connected. Aronov et al. \cite{addpp981} established that the region $V_1(s)$ of a simple polygon visible from an internal point $s$ due to at most one diffuse reflection on the boundary of the polygon $P$, is also simply connected. In this paper we establish that the region $V_2(s)$, visible from $s$ due to at most two diffuse reflections may be multiply connected; we demonstrate the construction of an $n$-sided simple polygon with a point $s$ inside it so that and the region of $P$ visible from $s$ after at most two diffuse reflections is multiple connected., Comment: 10 pages, 3 figures