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.

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.

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.

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.

Equilibrium and equivariant triangulations of some small covers with minimum number of vertices

Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal 4-equivariant triangulations of 2-dimensional small covers. We discuss vertex minimal equilibrium triangulations of RP3#RP3, S-1 x RP2 and a nontrivial S-1 bundle over RP2. We construct some nice equilibrium triangulations of the real projective space RPn with 2(n) + n 1 vertices. The main tool is the theory of small covers.

Orientational relaxation in a random dipolar lattice: Wave-number and frequency dependence

In order to understand the role of translational modes in the orientational relaxation in dense dipolar liquids, we have carried out a computer ''experiment'' where a random dipolar lattice was generated by quenching only the translational motion of the molecules of an equilibrated dipolar liquid. The lattice so generated was orientationally disordered and positionally random. The detailed study of orientational relaxation in this random dipolar lattice revealed interesting differences from those of the corresponding dipolar liquid. In particular, we found that the relaxation of the collective orientational correlation functions at the intermediate wave numbers was markedly slower at the long times for the random lattice than that of the liquid. This verified the important role of the translational modes in this regime, as predicted recently by the molecular theories. The single-particle orientational correlation functions of the random lattice also decayed significantly slowly at long times, compared to those of the dipolar liquid.

Bounds on isoperimetric values of trees

Let G = (V, E) be a finite, simple and undirected graph. For S subset of V, let delta(S, G) = {(u, v) is an element of E : u is an element of S and v is an element of V - S} be the edge boundary of S. Given an integer i, 1 <= i <= vertical bar V vertical bar, let the edge isoperimetric value of G at i be defined as b(e)(i, G) = min(S subset of V:vertical bar S vertical bar=i)vertical bar delta(S, G)vertical bar. The edge isoperimetric peak of G is defined as b(e)(G) = max(1 <= j <=vertical bar V vertical bar)b(e)(j, G). Let b(v)(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi: 10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees. The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as T-d(2)), c(1)d <= b(e) (T-d(2)) <= d and c(2)d <= b(v)(T-d(2)) <= d where c(1), c(2) are constants. For a complete t-ary tree of depth d (denoted as T-d(t)) and d >= c log t where c is a constant, we show that c(1)root td <= b(e)(T-d(t)) <= td and c(2)d/root t <= b(v) (T-d(t)) <= d where c(1), c(2) are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T = (V, E, r) be a finite, connected and rooted tree - the root being the vertex r. Define a weight function w : V -> N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index eta(T) be defined as the number of distinct weights in the tree, i.e eta(T) vertical bar{w(u) : u is an element of V}vertical bar. For a positive integer k, let l(k) = vertical bar{i is an element of N : 1 <= i <= vertical bar V vertical bar, b(e)(i, G) <= k}vertical bar. We show that l(k) <= 2(2 eta+k k)

Practical Algorithms For Mean Velocity Estimation In Pulse Doppler Weather Radars Using A Small Number Of Samples

Doppler weather radars with fast scanning rates must estimate spectral moments based on a small number of echo samples. This paper concerns the estimation of mean Doppler velocity in a coherent radar using a short complex time series. Specific results are presented based on 16 samples. A wide range of signal-to-noise ratios are considered, and attention is given to ease of implementation. It is shown that FFT estimators fare poorly in low SNR and/or high spectrum-width situations. Several variants of a vector pulse-pair processor are postulated and an algorithm is developed for the resolution of phase angle ambiguity. This processor is found to be better than conventional processors at very low SNR values. A feasible approximation to the maximum entropy estimator is derived as well as a technique utilizing the maximization of the periodogram. It is found that a vector pulse-pair processor operating with four lags for clear air observation and a single lag (pulse-pair mode) for storm observation may be a good way to estimate Doppler velocities over the entire gamut of weather phenomena.

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). Let Delta = Delta(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K-n,K-n. Alon, McDiarmid and Reed observed that a'(K-p-1,K-p-1) = p for every prime p. In this paper we prove that a'(K-p,K-p) <= p + 2 = Delta + 2 when p is prime. Basavaraju, Chandran and Kummini proved that a'(K-n,K-n) >= n + 2 = Delta + 2 when n is odd, which combined with our result implies that a'(K-p,K-p) = p + 2 = Delta + 2 when p is an odd prime. Moreover we show that if we remove any edge from K-p,K-p, the resulting graph is acyclically Delta + 1 = p + 1-edge-colorable. (C) 2009 Elsevier B.V. All rights reserved.

On the Arrangement of Cliques in Chordal Graphs with respect to the Cuts

A cut (A, B) (where B = V - A) in a graph G = (V, E) is called internal if and only if there exists a vertex x in A that is not adjacent to any vertex in B and there exists a vertex y is an element of B such that it is not adjacent to any vertex in A. In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut (A, B) in a chordal graph G, there exists a clique with kappa(G) + vertices (where kappa(G) is the vertex connectivity of G) such that it is (approximately) bisected by the cut (A, B). In fact we give a stronger result: For any internal cut (A, B) of a chordal graph, and for each i, 0 <= i <= kappa(G) + 1 such that vertical bar K-i vertical bar = kappa(G) + 1, vertical bar A boolean AND K-i vertical bar = i and vertical bar B boolean AND K-i vertical bar = kappa(G) + 1 - i. An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be Omega(k(2)), where kappa(G) = k. Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least kappa(G)(kappa(G)+1)/2 where kappa(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to kappa(G). This result is tight.

On the number of two-line and three-line latin rectangles - an alternative approach

The number of two-line and three-line Latin rectangles is obtained by recursive methods in a setting slightly more general than usually considered. We show how this leads to a generalisation which is proved elsewhere.

Number and proportion of selenocucleosides in the transfer RNA of escherichia-coli

tRNA isolated from escherichia-coli grown in a medium containing [75Se] sodium selenosulfate was converted to nucleosides and analysed for selenonucleosides on a phosphocellulose column. Upon chromatography of the nucleosides on phosphocellulose column, the radioactivity resolved into three peaks. The first peak consisted of free selenium and traces of undigested nucleotides. The second peak was identified as 4-selenouridine by co-chromatographing with an authentic sample of 4-selenouridine. The identity of the third peak was not established. The second and third peaks represented 93% and 7% of the selenium present in nucleosides respectively.

Number and proportion of selenocucleosides in the transfer RNA of Image Image

tRNA isolated from . grown in a medium containing [75Se] sodium selenosulfate was converted to nucleosides and analysed for selenonucleosides on a phosphocellulose column. Upon chromatography of the nucleosides on phosphocellulose column, the radioactivity resolved into three peaks. The first peak consisted of free selenium and traces of undigested nucleotides. The second peak was identified as 4-selenouridine by co-chromatographing with an authentic sample of 4-selenouridine. The identity of the third peak was not established. The second and third peaks represented 93% and 7% of the selenium present in nucleosides respectively.