Polynomial time and parameterized approximation algorithms for boxicity

The boxicity (cubicity) of a graph G, denoted by box(G) (respectively cub(G)), is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes (cubes) in ℝ k . The problem of computing boxicity (cubicity) is known to be inapproximable in polynomial time even for graph classes like bipartite, co-bipartite and split graphs, within an O(n 0.5 − ε ) factor for any ε > 0, unless NP = ZPP. We prove that if a graph G on n vertices has a clique on n − k vertices, then box(G) can be computed in time n22O(k2logk) . Using this fact, various FPT approximation algorithms for boxicity are derived. The parameter used is the vertex (or edge) edit distance of the input graph from certain graph families of bounded boxicity - like interval graphs and planar graphs. Using the same fact, we also derive an O(nloglogn√logn√) factor approximation algorithm for computing boxicity, which, to our knowledge, is the first o(n) factor approximation algorithm for the problem. We also present an FPT approximation algorithm for computing the cubicity of graphs, with vertex cover number as the parameter.

Automatic Data Allocation and Buffer Management for Multi-GPU Machines

Multi-GPU machines are being increasingly used in high-performance computing. Each GPU in such a machine has its own memory and does not share the address space either with the host CPU or other GPUs. Hence, applications utilizing multiple GPUs have to manually allocate and manage data on each GPU. Existing works that propose to automate data allocations for GPUs have limitations and inefficiencies in terms of allocation sizes, exploiting reuse, transfer costs, and scalability. We propose a scalable and fully automatic data allocation and buffer management scheme for affine loop nests on multi-GPU machines. We call it the Bounding-Box-based Memory Manager (BBMM). BBMM can perform at runtime, during standard set operations like union, intersection, and difference, finding subset and superset relations on hyperrectangular regions of array data (bounding boxes). It uses these operations along with some compiler assistance to identify, allocate, and manage data required by applications in terms of disjoint bounding boxes. This allows it to (1) allocate exactly or nearly as much data as is required by computations running on each GPU, (2) efficiently track buffer allocations and hence maximize data reuse across tiles and minimize data transfer overhead, and (3) and as a result, maximize utilization of the combined memory on multi-GPU machines. BBMM can work with any choice of parallelizing transformations, computation placement, and scheduling schemes, whether static or dynamic. Experiments run on a four-GPU machine with various scientific programs showed that BBMM reduces data allocations on each GPU by up to 75% compared to current allocation schemes, yields performance of at least 88% of manually written code, and allows excellent weak scaling.

Small strong epsilon nets

Let P be a set of n points in R-d. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than dn/d+1 points of P. We call a point x a strong centerpoint for a family of objects C if x is an element of P is contained in every object C is an element of C that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in R-2. We prove that a strong centerpoint exists for axis-parallel boxes in Rd and give exact bounds. We then extend this to small strong epsilon-nets in the plane. Let epsilon(S)(i) represent the smallest real number in 0, 1] such that there exists an epsilon(S)(i)-net of size i with respect to S. We prove upper and lower bounds for epsilon(S)(i) where S is the family of axis-parallel rectangles, halfspaces and disks. (C) 2014 Elsevier B.V. All rights reserved.

Checking Liveness Properties of Presburger Counter Systems Using Reachability Analysis

Counter systems are a well-known and powerful modeling notation for specifying infinite-state systems. In this paper we target the problem of checking liveness properties in counter systems. We propose two semi decision techniques towards this, both of which return a formula that encodes the set of reachable states of the system that satisfy a given liveness property. A novel aspect of our techniques is that they use reachability analysis techniques, which are well studied in the literature, as black boxes, and are hence able to compute precise answers on a much wider class of systems than previous approaches for the same problem. Secondly, they compute their results by iterative expansion or contraction, and hence permit an approximate solution to be obtained at any point. We state the formal properties of our techniques, and also provide experimental results using standard benchmarks to show the usefulness of our approaches. Finally, we sketch an extension of our liveness checking approach to check general CTL properties.

A constant factor approximation algorithm for boxicity of circular arc graphs

The boxicity (resp. cubicity) 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 (resp. cubes) in R-k. Equivalently, it is the minimum number of interval graphs (resp. unit interval graphs) on the vertex set V, such that the intersection of their edge sets is E. The problem of computing boxicity (resp. cubicity) is known to be inapproximable, even for restricted graph classes like bipartite, co-bipartite and split graphs, within an O(n(1-epsilon))-factor for any epsilon > 0 in polynomial time, unless NP = ZPP. For any well known graph class of unbounded boxicity, there is no known approximation algorithm that gives n(1-epsilon)-factor approximation algorithm for computing boxicity in polynomial time, for any epsilon > 0. In this paper, we consider the problem of approximating the boxicity (cubicity) of circular arc graphs intersection graphs of arcs of a circle. Circular arc graphs are known to have unbounded boxicity, which could be as large as Omega(n). We give a (2 + 1/k) -factor (resp. (2 + log n]/k)-factor) polynomial time approximation algorithm for computing the boxicity (resp. cubicity) of any circular arc graph, where k >= 1 is the value of the optimum solution. For normal circular arc (NCA) graphs, with an NCA model given, this can be improved to an additive two approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity (resp. cubicity) 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 (resp. cube) representations, where n is the number of vertices of the graph and m is its number of edges. Our additive two approximation algorithm directly works for any proper circular arc graph, since their NCA models can be computed in polynomial time. (C) 2014 Elsevier B.V. All rights reserved.

LOWER BOUNDS FOR BOXICITY

An axis-parallel b-dimensional box is a Cartesian product R-1 x R-2 x ... x 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 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 delta is at least n/2(n-delta-1). 2. Consider the g(n,p) model of random graphs. Let p <= 1 - 40logn/n(2.) Then with high `` probability, box(G) = Omega(np(1 - p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Omega(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and m <= c((n)(2)) edges have boxicity Omega(m/n). 3. Let G be a connected k-regular graph on n vertices. Let lambda be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is a least (kappa(2)/lambda(2)/log(1+kappa(2)/lambda(2))) (n-kappa-1/2n). 4. For any positive constant c 1, almost all balanced bipartite graphs on 2n vertices and m <= cn(2) edges have boxicity Omega(m/n).

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 dth 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 dth Cartesian power of any given finite graph is, respectively, in O(log d/ log log d) and circle dot(d/ log d). 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. (C) 2015 Elsevier Ltd. All rights reserved.

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

The boxicity (respectively cubicity) of a graph G is the least 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 (2010) that for any graph G, cub(G) <= box(G) log(2) alpha(G], where alpha(G) is the maximum size of an independent set in G. In this note we show that cub(G) <= 2 log(2) X (G)] box(G) + X (G) log(2) alpha(G)], where x (G) is the chromatic number of G. 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 obtain cub(G) <= 2(box(G) + log(2) alpha(G)] Moreover, we show that for every positive integer k, there exist graphs with chromatic number k such that for every epsilon > 0, the value given by our upper bound is at most (1 + epsilon) times their cubicity. Thus, our upper bound is almost tight. (c) 2015 Elsevier B.V. All rights reserved.

The mycobacterial iron-dependent regulator IdeR induces ferritin (bfrB) by alleviating Lsr2 repression

Emerging evidence indicates that precise regulation of iron (Fe) metabolism and maintenance of Fe homeostasis in Mycobacterium tuberculosis (Mtb) are essential for its survival and proliferation in the host. IdeR is a central transcriptional regulator of Mtb genes involved in Fe metabolism. While it is well understood how IdeR functions as a repressor, how it induces transcription of a subset of its targets is still unclear. We investigated the molecular mechanism of IdeR-mediated positive regulation of bfrB, the gene encoding the major Fe-storage protein of Mtb. We found that bfrB induction by Fe required direct interaction of IdeR with a DNA sequence containing four tandem IdeR-binding boxes located upstream of the bfrB promoter. Results of in vivo and in vitro transcription assays identified a direct repressor of bfrB, the histone-like protein Lsr2. IdeR counteracted Lsr2-mediated repression in vitro, suggesting that IdeR induces bfrB transcription by antagonizing the repressor activity of Lsr2. Together, these results elucidate the main mechanism of bfrB positive regulation by IdeR and identify Lsr2 as a new factor contributing to Fe homeostasis in mycobacteria.