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The art gallery problem enquires about the least number of guards sufficient to ensure that an art gallery, represented by a simple polygon $P$, is fully guarded. Most standard versions of this problem are known to be NP-hard. In 1987, Ghosh provided a deterministic $\mathcal{O}(\log n)$-approximation algorithm for the case of vertex guards and edge guards in simple polygons. In the same paper, Ghosh also conjectured the existence of constant ratio approximation algorithms for these problems. We present here three polynomial-time algorithms with a constant approximation ratio for guarding an $n$-sided simple polygon $P$ using vertex guards. (i) The first algorithm, that has an approximation ratio of 18, guards all vertices of $P$ in $\mathcal{O}(n^4)$ time. (ii) The second algorithm, that has the same approximation ratio of 18, guards the entire boundary of $P$ in $\mathcal{O}(n^5)$ time. (iii) The third algorithm, that has an approximation ratio of 27, guards all interior and boundary points of $P$ in $\mathcal{O}(n^5)$ time. Further, these algorithms can be modified to obtain similar approximation ratios while using edge guards. The significance of our results lies in the fact that these results settle the conjecture by Ghosh regarding the existence of constant-factor approximation algorithms for this problem, which has been open since 1987 despite several attempts by researchers. Our approximation algorithms exploit several deep visibility structures of simple polygons which are interesting in their own right., Comment: 39 pages, 31 figures

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

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 $d \in \Omega(n^{0.5+\epsilon})$, for any $\epsilon$, $0< \epsilon \leq 0.5$., Comment: 9 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.

Given a graph $G$, a non-negative integer $k$, and a weight function that maps each vertex in $G$ to a positive real number, the \emph{Maximum Weighted Budgeted Independent Set (MWBIS) problem} is about finding a maximum weighted independent set in $G$ of cardinality at most $k$. A special case of MWBIS, when the weight assigned to each vertex is equal to its degree in $G$, is called the \emph{Maximum Independent Vertex Coverage (MIVC)} problem. In other words, the MIVC problem is about finding an independent set of cardinality at most $k$ with maximum coverage. Since it is a generalization of the well-known Maximum Weighted Independent Set (MWIS) problem, MWBIS too does not have any constant factor polynomial time approximation algorithm assuming $P \neq NP$. In this paper, we study MWBIS in the context of bipartite graphs. We show that, unlike MWIS, the MIVC (and thereby the MWBIS) problem in bipartite graphs is NP-hard. Then, we show that the MWBIS problem admits a $\frac{1}{2}$-factor approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation., Comment: 12 pages

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