arXiv subjects

Abhijin Adiga

Publications and source records attributed to Abhijin Adiga.

29 records · Page 2Linked to original sources

arXiv2019

Learning Everywhere: Pervasive Machine Learning for Effective High-Performance Computation

The convergence of HPC and data-intensive methodologies provide a promising approach to major performance improvements. This paper provides a general description of the interaction between traditional HPC and ML approaches and motivates the Learning Everywhere paradigm for HPC. We introduce the concept of effective performance that one can achieve by combining learning methodologies with simulation-based approaches, and distinguish between traditional performance as measured by benchmark scores. To support the promise of integrating HPC and learning methods, this paper examines specific examples and opportunities across a series of domains. It concludes with a series of open computer science and cyberinfrastructure questions and challenges that the Learning Everywhere paradigm presents.

cs.DC↗

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.

cs.DM↗

arXiv2015

Approximation Algorithms for Reducing the Spectral Radius to Control Epidemic Spread

The largest eigenvalue of the adjacency matrix of a network (referred to as the spectral radius) is an important metric in its own right. Further, for several models of epidemic spread on networks (e.g., the `flu-like' SIS model), it has been shown that an epidemic dies out quickly if the spectral radius of the graph is below a certain threshold that depends on the model parameters. This motivates a strategy to control epidemic spread by reducing the spectral radius of the underlying network. In this paper, we develop a suite of provable approximation algorithms for reducing the spectral radius by removing the minimum cost set of edges (modeling quarantining) or nodes (modeling vaccinations), with different time and quality tradeoffs. Our main algorithm, \textsc{GreedyWalk}, is based on the idea of hitting closed walks of a given length, and gives an $O(\log^2{n})$-approximation, where $n$ denotes the number of nodes; it also performs much better in practice compared to all prior heuristics proposed for this problem. We further present a novel sparsification method to improve its running time. In addition, we give a new primal-dual based algorithm with an even better approximation guarantee ($O(\log n)$), albeit with slower running time. We also give lower bounds on the worst-case performance of some of the popular heuristics. Finally we demonstrate the applicability of our algorithms and the properties of our solutions via extensive experiments on multiple synthetic and real networks.

cs.SI↗

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↗

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↗

arXiv2011

Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes just touch at their boundaries. Further, this construction can be realized in linear time.

math.CO↗

arXiv2011

A Constant Factor Approximation Algorithm for Boxicity of Circular Arc Graphs

Boxicity of a graph $G(V,E)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional axis parallel rectangles in $\mathbf{R}^k$. Equivalently, it is the minimum number of interval graphs on the vertex set $V$ such that the intersection of their edge sets is $E$. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below $O(n^{0.5 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. Till date, there is no well known graph class of unbounded boxicity for which even an $n^ε$-factor approximation algorithm for computing boxicity is known, for any $ε<1$. In this paper, we study the boxicity problem on Circular Arc graphs - intersection graphs of arcs of a circle. We give a $(2+\frac{1}{k})$-factor polynomial time approximation algorithm for computing the boxicity of any circular arc graph along with a corresponding box representation, where $k \ge 1$ is its boxicity. For Normal Circular Arc(NCA) graphs, with an NCA model given, this can be improved to an additive 2-factor approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity is $O(mn+n^2)$ in both these cases and in $O(mn+kn^2)= O(n^3)$ time we also get their corresponding box representations, where $n$ is the number of vertices of the graph and $m$ is its number of edges. The additive 2-factor algorithm directly works for any Proper Circular Arc graph, since computing an NCA model for it can be done in polynomial time.

cs.DS↗

arXiv2010

The Hardness of Approximating the Threshold Dimension, Boxicity and Cubicity of a Graph

A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as $\cub(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-cubes. The {\it threshold dimension} of a graph $G(V,E)$ is the smallest integer $k$ such that $E$ can be covered by $k$ threshold spanning subgraphs of $G$. In this paper we will show that there exists no polynomial-time algorithm to approximate the threshold dimension of a graph on $n$ vertices with a factor of $O(n^{0.5-ε})$ for any $ε>0$, unless $NP=ZPP$. From this result we will show that there exists no polynomial-time algorithm to approximate the boxicity and the cubicity of a graph on $n$ vertices with factor $O(n^{0.5-ε})$ for any $ ε>0$, unless $NP=ZPP$. In fact all these hardness results hold even for a highly structured class of graphs namely the split graphs. We will also show that it is NP-complete to determine if a given split graph has boxicity at most 3.

math.CO↗

arXiv2010

Boxicity and Poset Dimension

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$. The {\it boxicity} of $G$, $\boxi(G)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes, i.e. each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\poset=(S,P)$ be a poset where $S$ is the ground set and $P$ is a reflexive, anti-symmetric and transitive binary relation on $S$. The dimension of $\poset$, $\dim(\poset)$ is the minimum integer $t$ such that $P$ can be expressed as the intersection of $t$ total orders. Let $G_\poset$ be the \emph{underlying comparability graph} of $\poset$, i.e. $S$ is the vertex set and two vertices are adjacent if and only if they are comparable in $\poset$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $\poset$, $\boxi(G_\poset)/(χ(G_\poset)-1) \le \dim(\poset)\le 2\boxi(G_\poset)$, where $χ(G_\poset)$ is the chromatic number of $G_\poset$ and $χ(G_\poset)\ne1$. It immediately follows that if $\poset$ is a height-2 poset, then $\boxi(G_\poset)\le \dim(\poset)\le 2\boxi(G_\poset)$ since the underlying comparability graph of a height-2 poset is a bipartite graph. The second result of the paper relates the boxicity of a graph $G$ with a natural partial order associated with the \emph{extended double cover} of $G$, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ which are copies of $V(G)$ such that corresponding to every $u\in V(G)$, there are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in $G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\poset_c$ be the natural height-2 poset associated with $G_c$ by making $A$ the set of minimal elements and $B$ the set of maximal elements. We show that $\frac{\boxi(G)}{2} \le \dim(\poset_c) \le 2\boxi(G)+4$. These results have some immediate and significant consequences. The upper bound $\dim(\poset)\le 2\boxi(G_\poset)$ allows us to derive hitherto unknown upper bounds for poset dimension such as $\dim(\poset)\le 2\tw(G_\poset)+4$, since boxicity of any graph is known to be at most its $\tw+2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $Δ$ is $O(Δ\log^2Δ)$ which is an improvement over the best known upper bound of $Δ^2+2$. (2) There exist graphs with boxicity $Ω(Δ\logΔ)$. This disproves a conjecture that the boxicity of a graph is $O(Δ)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on $n$ vertices with a factor of $O(n^{0.5-ε})$ for any $ε>0$, unless $NP=ZPP$.

math.CO↗

arXiv2009

Cubicity of interval graphs and the claw number

Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by $\cub(G)$ is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw number} $ψ(G)$ of the graph to be the largest positive integer $m$ such that $S(m)$ is an induced subgraph of $G$. It can be easily shown that the cubicity of any graph is at least $\ceil{\log_2ψ(G)}$. In this paper, we show that, for an interval graph $G$ $\ceil{\log_2ψ(G)}\le\cub(G)\le\ceil{\log_2ψ(G)}+2$. Till now we are unable to find any interval graph with $\cub(G)>\ceil{\log_2ψ(G)}$. We also show that, for an interval graph $G$, $\cub(G)\le\ceil{\log_2α}$, where $α$ is the independence number of $G$. Therefore, in the special case of $ψ(G)=α$, $\cub(G)$ is exactly $\ceil{\log_2α}$. The concept of cubicity can be generalized by considering boxes instead of cubes. A $b$-dimensional box is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval on the real line. The \emph{boxicity} of a graph, denoted $ box(G)$, is the minimum $k$ such that $G$ is the intersection graph of $k$-dimensional boxes. It is clear that $ box(G)\le\cub(G)$. From the above result, it follows that for any graph $G$, $\cub(G)\le box(G)\ceil{\log_2α}$.

math.CO↗