Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R-1 x R-2 x ... x R-k where each R-i is a closed interval on the real line. The boxicity of a graph G,denoted as box(G), is the minimum integer k such that G is 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 x R-2 x ... x R-k where each Ri is a closed interval on the real line of the form [a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G) <= t + inverted right perpendicularlog(n - t)inverted left perpendicular - 1 and box(G) <= left perpendiculart/2right perpendicular + 1, where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds. F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, box(G) <= left perpendicularn/2right perpendicular and cub(G) <= inverted right perpendicular2n/3inverted left perpendicular, where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then box(G) <= inverted right perpendicularn/4inverted left perpendicular and this bound is tight. We also show that if G is a bipartite graph then cub(G) <= n/2 + inverted right perpendicularlog n inverted left perpendicular - 1. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to n/4. Interestingly, if boxicity is very close to n/2, then chromatic number also has to be very high. In particular, we show that if box(G) = n/2 - s, s >= 0, then chi (G) >= n/2s+2, where chi (G) is the chromatic number of G.

Solving MIN ONES 2-SAT as fast as VERTEX COVER

The problem of finding a satisfying assignment that minimizes the number of variables that are set to 1 is NP-complete even for a satisfiable 2-SAT formula. We call this problem MIN ONES 2-SAT. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k. In this paper, we present a polynomial-time reduction from MIN ONES 2-SAT to VERTEX COVER without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k - c logk-variable kernel subsuming (or, in the case of kernels, improving) the results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover. Finally we show that the optimum value of the LP relaxation of the MIN ONES 2-SAT and that of the corresponding VERTEX COVER are the same. This implies that the (recent) results of VERTEX COVER version parameterized above the optimum value of the LP relaxation of VERTEX COVER carry over to the MIN ONES 2-SAT version parameterized above the optimum of the LP relaxation of MIN ONES 2-SAT. (C) 2013 Elsevier B.V. All rights reserved.

Vertex Cover Gets Faster and Harder on Low Degree Graphs

The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric objects, often turns out to be exactly the vertex cover problem on restricted classes of graphs. In this work we explore a particular instance of such a phenomenon. We consider the problem of hitting all axis-parallel slabs induced by a point set P, and show that it is equivalent to the problem of finding a vertex cover on a graph whose edge set is the union of two Hamiltonian Paths. We show the latter problem to be NP-complete, and also give an algorithm to find a vertex cover of size at most k, on graphs of maximum degree four, whose running time is 1.2637(k) n(O(1)).

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.

Constrained Linear Spectral Unmixing Technique for Regional Land Cover Mapping Using MODIS Data

Over the last few decades, there has been a significant land cover (LC) change across the globe due to the increasing demand of the burgeoning population and urban sprawl. In order to take account of the change, there is a need for accurate and up-to-date LC maps. Mapping and monitoring of LC in India is being carried out at national level using multi-temporal IRS AWiFS data. Multispectral data such as IKONOS, Landsat-TM/ETM+, IRS-ICID LISS-III/IV, AWiFS and SPOT-5, etc. have adequate spatial resolution (similar to 1m to 56m) for LC mapping to generate 1:50,000 maps. However, for developing countries and those with large geographical extent, seasonal LC mapping is prohibitive with data from commercial sensors of limited spatial coverage. Superspectral data from the MODIS sensor are freely available, have better temporal (8 day composites) and spectral information. MODIS pixels typically contain a mixture of various LC types (due to coarse spatial resolution of 250, 500 and 1000 in), especially in more fragmented landscapes. In this context, linear spectral unmixing would be useful for mapping patchy land covers, such as those that characterise much of the Indian subcontinent. This work evaluates the existing unmixing technique for LC mapping using MODIS data, using end-members that are extracted through Pixel Purity Index (PPI), Scatter plot and N-dimensional visualisation. The abundance maps were generated for agriculture, built up, forest, plantations, waste land/others and water bodies. The assessment of the results using ground truth and a LISS-III classified map shows 86% overall accuracy, suggesting the potential for broad-scale applicability of the technique with superspectral data for natural resource planning and inventory applications. Index Terms-Remote sensing, digital

Satellite-derived direct radiative effect of aerosols dependent on cloud cover

Aerosols from biomass burning can alter the radiative balance of the Earth by reflecting and absorbing solar radiation(1). Whether aerosols exert a net cooling or a net warming effect will depend on the aerosol type and the albedo of the underlying surface(2). Here, we use a satellite-based approach to quantify the direct, top-of-atmosphere radiative effect of aerosol layers advected over the partly cloudy boundary layer of the southeastern Atlantic Ocean during July-October of 2006 and 2007. We show that the warming effect of aerosols increases with underlying cloud coverage. This relationship is nearly linear, making it possible to define a critical cloud fraction at which the aerosols switch from exerting a net cooling to a net warming effect. For this region and time period, the critical cloud fraction is about 0.4, and is strongly sensitive to the amount of solar radiation the aerosols absorb and the albedo of the underlying clouds. We estimate that the regional-mean warming effect of aerosols is three times higher when large-scale spatial covariation between cloud cover and aerosols is taken into account. These results demonstrate the importance of cloud prediction for the accurate quantification of aerosol direct effects.

Vertex-Fused Metallaborane Clusters: Synthesis, Characterization and Electronic Structure of [(eta(5)-C5Me5Mo)(3)MoB9H18]

The reaction of the [(eta(5)-C5Me5)MoCl4] complex with [LiBH4 - TH F] in toluene at - 70 degrees C, followed by pyrolysis at 110 degrees C, afforded dark brown [(eta(5)-C5Me5Mo)(3)MoB9H18], 2, in parallel with the known [(eta(5)-C5Me5Mo)(2)B5H9], 1. Compound 2 has been characterized in solution by H-1, B-11, and C-13 NMR spectroscopy and elemental analysis, and the structural types were unequivocally established by crystallographic studies. The title compound represents a novel class of vertex-fused clusters in which a Mo atom has been fused in a perpendicular fashion between two molybdaborane clusters. Electronic structure calculations employing density functional theory yield geometries in agreement with the structure determinations, and on grounds of density functional theory calculations, we have analyzed the bonding patterns in the structure,

Uniqueness of Walkup's 9-vertex 3-dimensional Klein bottle

Via a computer search, Altshuler and Steinberg found that there are 1296+1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold View the MathML source triangulates the twisted S2-bundle over S1. It was first constructed by Walkup. In this paper, we present a computer-free proof of the uniqueness of this non-sphere combinatorial 3-manifold. As opposed to the computer-generated proof, ours does not require wading through all the 9-vertex 3-spheres. As a preliminary result, we also show that any 9-vertex combinatorial 3-manifold is equivalent by proper bistellar moves to a 9-vertex neighbourly 3-manifold.

Flow of a maxwell liquid fluid between two rotating coaxial cones having the same vertex

We consider the secondary flows arising in the motion of a Maxwell fluid between two rotating coaxial cones having the same vertex. We find that in any meridian plane passing through the common axis of the cones, the flow field is divided into two regions. Such a division of flow field was first reported by Bhatnagar and Rathna.

An approximate algorithm for the minimal vertex nested polygon problem

Given two simple polygons, the Minimal Vertex Nested Polygon Problem is one of finding a polygon nested between the given polygons having the minimum number of vertices. In this paper, we suggest efficient approximate algorithms for interesting special cases of the above using the shortest-path finding graph algorithms.

Boxicity and Poset Dimension

Let G be a simple, undirected, finite graph with vertex set V(G) and edge set E(C). A k-dimensional box is a Cartesian product of closed intervals a(1), b(1)] x a(2), b(2)] x ... x a(k), b(k)]. The boxicity of G, box(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 P = (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 P, dim(P) is the minimum integer l such that P can be expressed as the intersection of t total orders. Let G(P) be the underlying comparability graph of P. 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 P, box(G(P))/(chi(G(P)) - 1) <= dim(P) <= 2box(G(P)), where chi(G(P)) is the chromatic number of G(P) and chi(G(P)) not equal 1. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with its extended double cover, denoted as G(c). Let P-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 box(G)/2 <= dim(P-c) <= 2box(G) + 4. These results have some immediate and significant consequences. The upper bound dim(P) <= 2box(G(P)) allows us to derive hitherto unknown upper bounds for poset dimension. In the other direction, using the already known bounds for partial order dimension we get the following: (I) The boxicity of any graph with maximum degree Delta is O(Delta log(2) Delta) which is an improvement over the best known upper bound of Delta(2) + 2. (2) There exist graphs with boxicity Omega(Delta log Delta). This disproves a conjecture that the boxicity of a graph is O(Delta). (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-epsilon)) for any epsilon > 0, unless NP=ZPP.

Mining Land Cover Information Using Multilayer Perceptron and Decision Tree from MODIS Data

Land cover (LC) changes play a major role in global as well as at regional scale patterns of the climate and biogeochemistry of the Earth system. LC information presents critical insights in understanding of Earth surface phenomena, particularly useful when obtained synoptically from remote sensing data. However, for developing countries and those with large geographical extent, regular LC mapping is prohibitive with data from commercial sensors (high cost factor) of limited spatial coverage (low temporal resolution and band swath). In this context, free MODIS data with good spectro-temporal resolution meet the purpose. LC mapping from these data has continuously evolved with advances in classification algorithms. This paper presents a comparative study of two robust data mining techniques, the multilayer perceptron (MLP) and decision tree (DT) on different products of MODIS data corresponding to Kolar district, Karnataka, India. The MODIS classified images when compared at three different spatial scales (at district level, taluk level and pixel level) shows that MLP based classification on minimum noise fraction components on MODIS 36 bands provide the most accurate LC mapping with 86% accuracy, while DT on MODIS 36 bands principal components leads to less accurate classification (69%).

On kuhnels 9-vertex complex projective plane

We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C P2. The original proof of existence, due to Kuhnel, as well as the original proof of uniqueness, due to Kuhnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kuhnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.

A note on surface water waves for finite depth in the presence of an ice-cover

A class of I boundary value problems involving propagation of two-dimensional surface water waves, associated with water of uniform finite depth, against a plane vertical wave maker is investigated under the assumption that the surface is covered by a thin sheet of ice. It is assumed that the ice-cover behaves like a thin isotropic elastic plate. Then the problems under consideration lead to those of solving the two-dimensional Laplace equation in a semi-infinite strip, under Neumann boundary conditions on the vertical boundary as well as on one of the horizontal boundaries, representing the bottom of the fluid region, and a condition involving upto fifth order derivatives of the unknown function on the top horizontal ice-covered boundary, along with the two appropriate edge-conditions, at the ice-covered corner, ensuring the uniqueness of the solutions. The mixed boundary value problems are solved completely, by exploiting the regularity property of the Fourier cosine transform.

From the Icosahedron to Natural Triangulations of CP(2) and S(2) x S(2)

We present two constructions in this paper: (a) a 10-vertex triangulation CP(10)(2) of the complex projective plane CP(2) as a subcomplex of the join of the standard sphere (S(4)(2)) and the standard real projective plane (RP(6)(2), the decahedron), its automorphism group is A(4); (b) a 12-vertex triangulation (S(2) x S(2))(12) of S(2) x S(2) with automorphism group 2S(5), the Schur double cover of the symmetric group S(5). It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S(2) x S(2). Both constructions have surprising and intimate relationships with the icosahedron. It is well known that CP(2) has S(2) x S(2) as a two-fold branched cover; we construct the triangulation CP(10)(2) of CP(2) by presenting a simplicial realization of this covering map S(2) x S(2) -> CP(2). The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S(2) x S(2), different from the triangulation alluded to in (b). This gives a new proof that Kuhnel's CP(9)(2) triangulates CP(2). It is also shown that CP(10)(2) and (S(2) x S(2))(12) induce the standard piecewise linear structure on CP(2) and S(2) x S(2) respectively.