arXiv subjects

L. Sunil Chandran

Publications and source records attributed to L. Sunil Chandran.

At least 55 records · Page 3Linked to original sources

arXiv2015

Sublinear Approximation Algorithms for Boxicity and Related Problems

Boxicity of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. Cubicity is a variant of boxicity, where the axis parallel boxes in the intersection representation are restricted to be of unit length sides. Deciding whether boxicity (resp. cubicity) of a graph is at most k is NP-hard, even for k=2 or 3. Computing these parameters is inapproximable within $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless NP=ZPP, even for many simple graph classes. In this paper, we give a polynomial time $κ(n)$ factor approximation algorithm for computing boxicity and a $κ(n)\lceil \log \log n\rceil$ factor approximation algorithm for computing the cubicity, where $κ(n) =2\left\lceil\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right\rceil$. These o(n) factor approximation algorithms also produce the corresponding box (resp. cube) representations. As a special case, this resolves the question paused by Spinrad about polynomial time construction of o(n) dimensional box representations for boxicity 2 graphs. Other consequences of our approximation algorithm include $O(κ(n))$ factor approximation algorithms for computing the following parameters: the partial order dimension of finite posets, the interval dimension of finite posets, minimum chain cover of bipartite graphs, threshold dimension of split graphs and Ferrer's dimension of digraphs. Each of these parameters is inapproximable within an $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless NP=ZPP and the algorithms we derive seem to be the first o(n) factor approximation algorithms known for all these problems.

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arXiv2014

Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $Θ(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.

math.CO↗

arXiv2014

Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by Adiga and Chandran (Journal of Graph Theory, 65(4), 2010) that for any graph $G$, $cub(G) \le$ box$(G) \left \lceil \log_2 α\right \rceil$, where $α= α(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 χ(G) \right \rceil box(G) + χ(G) \left \lceil \log_2 α(G) \right \rceil $. In general, this result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we get, $cub(G) \le 2 (box(G) + \left \lceil \log_2 α(G) \right \rceil )$. Moreover we show that for every positive integer $k$, there exist graphs with chromatic number $k$, such that for every $ε> 0$, the value given by our upper bound is at most $(1+ε)$ times their cubicity. Thus, our upper bound is almost tight.

math.CO↗

arXiv2014

Boxicity and separation dimension

A family $\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $π(H)$. Equivalently, $π(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph $H$ is equal to the 'boxicity' of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.

math.CO↗

arXiv2014

Rainbow Colouring of Split Graphs

A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one rainbow path. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Between them, Chakraborty et al. [J. Comb. Optim., 2011] and Ananth et al. [FSTTCS, 2012] have shown that for every integer k, k \geq 2, it is NP-complete to decide whether a given graph can be rainbow coloured using k colours. A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. Chandran and Rajendraprasad have shown that the problem of deciding whether a given split graph G can be rainbow coloured using 3 colours is NP-complete and further have described a linear time algorithm to rainbow colour any split graph using at most one colour more than the optimum [COCOON, 2012]. In this article, we settle the computational complexity of the problem on split graphs and thereby discover an interesting dichotomy. Specifically, we show that the problem of deciding whether a given split graph can be rainbow coloured using k colours is NP-complete for k \in {2,3}, but can be solved in polynomial time for all other values of k.

cs.DM↗

arXiv2014

Separation dimension of sparse graphs

The separation dimension of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of permutations of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $Θ(\log n)$. In this article, we focus on sparse graphs and show that the maximum separation dimension of a $k$-degenerate graph on $n$ vertices is $O(k \log\log n)$ and that there exists a family of $2$-degenerate graphs with separation dimension $Ω(\log\log n)$. We also show that the separation dimension of the graph $G^{1/2}$ obtained by subdividing once every edge of another graph $G$ is at most $(1 + o(1)) \log\log χ(G)$ where $χ(G)$ is the chromatic number of the original graph.

math.CO↗

arXiv2014

Parameterized and Approximation Algorithms for Boxicity

Boxicity of a graph $G(V,$ $E)$, denoted by $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of axis parallel boxes in $\mathbb{R}^k$. The problem of computing boxicity is inapproximable even for graph classes like bipartite, co-bipartite and split graphs within $O(n^{1 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. We give FPT approximation algorithms for computing the boxicity of graphs, where the parameter used is the vertex or edge edit distance of the given graph from families of graphs of bounded boxicity. This can be seen as a generalization of the parameterizations discussed in \cite{Adiga2}. Extending the same idea in one of our algorithms, we also get an $O\left(\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right)$ factor approximation algorithm for computing boxicity and an $O\left(\frac{n {(\log \log n)}^{\frac{3}{2}}}{\sqrt{\log n}}\right)$ factor approximation algorithm for computing the cubicity. These seem to be the first $o(n)$ factor approximation algorithms known for both boxicity and cubicity. As a consequence of this result, a $o(n)$ factor approximation algorithm for computing the partial order dimension of finite posets and a $o(n)$ factor approximation algorithm for computing the threshold dimension of split graphs would follow.

cs.DS↗

arXiv2014

On Additive Combinatorics of Permutations of \mathbb{Z}_n

Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is not a permutation. Let the sizes be denoted by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we prove $s(n)\geq (nϕ(n))/2^k$ where $k$ is the number of distinct prime divisors of $n$. When $n$ is an odd prime we prove $s(n)\leq \frac{e^2}π n ((n-1)/e)^\frac{n-1}{2}$. For the second problem, we prove $2^{(n-1)/2}.(\frac{n-1}{2})!\leq t(n)\leq 2^k.(n-1)!/ϕ(n)$ when $n$ is odd.

math.CO↗

arXiv2014

2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl, in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.

cs.DM↗

arXiv2013

Boxicity and Cubicity of Product Graphs

The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in $R^k$. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of $d$, of the boxicity and the cubicity of the $d$-th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the $d$-th Cartesian power of any given finite graph is in $O(\log d / \log\log d)$ and $θ(d / \log d)$, respectively. On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products.

math.CO↗

arXiv2012

Pairwise Suitable Family of Permutations and Boxicity

A family F of permutations of the vertices of a hypergraph H is called "pairwise suitable" for H if, for every pair of disjoint edges in H, there exists a permutation in F in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for H is called the "separation dimension" of H and is denoted by π(H). Equivalently, π(H) is the smallest natural number k so that the vertices of H can be embedded in R^k such that any two disjoint edges of H can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph H is equal to the "boxicity" of the line graph of H. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.

math.CO↗

arXiv2012

Isoperimetric Sequences for Infinite Complete Binary Trees, Meta-Fibonacci Sequences and Signed Almost Binary Partitions

In this paper we demonstrate connections between three seemingly unrelated concepts. (1) The discrete isoperimetric problem in the infinite binary tree with all the leaves at the same level, $ {\mathcal T}_{\infty}$: The $n$-th edge isoperimetric number $δ(n)$ is defined to be $\min_{|S|=n, S \subset V({\mathcal T}_{\infty})} |(S,\bar{S})|$, where $(S,\bar{S})$ is the set of edges in the cut defined by $S$. (2) Signed almost binary partitions: This is the special case of the coin-changing problem where the coins are drawn from the set ${\pm (2^d - 1): $d$ is a positive integer}$. The quantity of interest is $τ(n)$, the minimum number of coins necessary to make change for $n$ cents. (3) Certain Meta-Fibonacci sequences: The Tanny sequence is defined by $T(n)=T(n{-}1{-}T(n{-}1))+T(n{-}2{-}T(n{-}2))$ and the Conolly sequence is defined by $C(n)=C(n{-}C(n{-}1))+C(n{-}1{-}C(n{-}2))$, where the initial conditions are $T(1) = C(1) = T(2) = C(2) = 1$. These are well-known "meta-Fibonacci" sequences. The main result that ties these three together is the following: $$ δ(n) = τ(n) = n+ 2 + 2 \min_{1 \le k \le n} (C(k) - T(n-k) - k).$$ Apart from this, we prove several other results which bring out the interconnections between the above three concepts.

math.CO↗

arXiv2012

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.

math.CO↗

arXiv2012

Product Dimension of Forests and Bounded Treewidth Graphs

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n vertices with a treewidth at most t has a product dimension at most (t+2)(logn+1). We also show that every k-degenerate graph on n vertices has a product dimension at most \ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by Eaton and Rodl.

math.CO↗

arXiv2012

Rainbow Colouring of Split and Threshold Graphs

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. In this article, we show the following: 1. The problem of deciding whether a graph can be rainbow coloured using 3 colours remains NP-complete even when restricted to the class of split graphs. However, any split graph can be rainbow coloured in linear time using at most one more colour than the optimum. 2. For every integer k larger than 2, the problem of deciding whether a graph can be rainbow coloured using k colours remains NP-complete even when restricted to the class of chordal graphs. 3. For every positive integer k, threshold graphs with rainbow connection number k can be characterised based on their degree sequence alone. Further, we can optimally rainbow colour a threshold graph in linear time.

cs.DM↗

arXiv2012

Lower Bounds for Boxicity

An axis-parallel $b$-dimensional box is a Cartesian product $R_1\times R_2\times...\times R_b$ where $R_i$ is a closed interval of the form $[a_i,b_i]$ on the real line. For a graph $G$, its \emph{boxicity} box(G) is the minimum dimension $b$, such that $G$ is representable as the intersection graph of boxes in $b$-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: (1) The boxicity of a graph on $n$ vertices with no universal vertices and minimum degree $δ$ is at least $n/2(n-δ-1)$. (2) Consider the $\mathcal{G}(n,p)$ model of random graphs. Let $ p \le 1- \frac{40 \log n}{n^2}$. Then, for $G \in \mathcal{G}(n,p)$, almost surely $box(G)=Ω(np(1-p))$. On setting $p=1/2$ we immediately infer that almost all graphs have boxicity $Ω(n)$. (3) Spectral lower bounds for the boxicity of $k$-regular graphs. (4) The boxicity of random $k$-regular graphs on $n$ vertices (where $k$ is fixed) is $Ω(k/\log k)$. (5) There exists a positive constant$c$ such that almost all balanced bipartite graphs on $2n$ vertices with exactly $m$ edges have boxicity at least $c m/n$, for $ m\le c' n^2/3$ for any positive constant $c' < 1$.

math.CO↗

arXiv2012

Cubicity, Degeneracy, and Crossing Number

A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as $\cubi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/ 0607092, 2006.] that, for a graph $G$ with maximum degree $Δ$, $\cubi(G)\leq \lceil 4(Δ+1)\log n\rceil$. In this paper, we show that, for a $k$-degenerate graph $G$, $\cubi(G) \leq (k+2) \lceil 2e \log n \rceil$. Since $k$ is at most $Δ$ and can be much lower, this clearly is a stronger result. This bound is tight. We also give an efficient deterministic algorithm that runs in $O(n^2k)$ time to output a $8k(\lceil 2.42 \log n\rceil + 1)$ dimensional cube representation for $G$. An important consequence of the above result is that if the crossing number of a graph $G$ is $t$, then $\boxi(G)$ is $O(t^{1/4}{\lceil\log t\rceil}^{3/4})$ . This bound is tight up to a factor of $O((\log t)^{1/4})$. We also show that, if $G$ has $n$ vertices, then $\cubi(G)$ is $O(\log n + t^{1/4}\log t)$. Using our bound for the cubicity of $k$-degenerate graphs we show that cubicity of almost all graphs in $\mathcal{G}(n,m)$ model is $O(d_{av}\log n)$, where $d_{av}$ denotes the average degree of the graph under consideration.

math.CO↗

arXiv2012

Bipartite powers of k-chordal graphs

Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G^m is chordal then so is G^{m+2}. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. Given a bipartite graph G and an odd positive integer m, we define the graph G^{[m]} to be a bipartite graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power of G. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is odd.

math.CO↗