Random Contractions and Sampling for Hypergraph and Hedge Connectivity

We initiate the study of hedge connectivity of undirected graphs, motivated by dependent edge failures in real-world networks. In this model, edges are partitioned into groups called hedges that fail together. The hedge connectivity of a graph is the minimum number of hedges whose removal disconnects the graph. We give a polynomial-time approximation scheme and a quasi-polynomial exact algorithm for hedge connectivity. This provides strong evidence that the hedge connectivity problem is tractable, which contrasts with prior work that established the intractability of the corresponding s−t min-cut problem. Our techniques also yield new combinatorial and algorithmic results in hypergraph connectivity. Next, we study the behavior of hedge graphs under uniform random sampling of hedges. We show that unlike graphs, all cuts in the sample do not converge to their expected value in hedge graphs. Nevertheless, the min-cut of the sample does indeed concentrate around the expected value of the original min-cut. This leads to a sharp threshold on hedge survival probabilities for graph disconnection. To the best of our knowledge, this is the first network reliability analysis under dependent edge failures

Faster Algorithms for the Geometric Transportation Problem

Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric: * For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost. * For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point. * An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2

Approximate Birkhoff-James orthogonality and smoothness in the space of bounded linear operators

We study approximate Birkhoff-James orthogonality of bounded linear operators defined between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}.$ As an application of the results obtained, we characterize smoothness of a bounded linear operator $T$ under the condition that $\mathbb{K}(\mathbb{X},\mathbb{Y}),$ the space of compact linear operators is an $M-$ideal in $\mathbb{L}(\mathbb{X},\mathbb{Y}),$ the space of bounded linear operators

(3+1)(3+1) (3+1) -Dimensional topologically massive 2-form gauge theory: geometrical superfield approach

Abstract We derive the complete set of off-shell nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations corresponding to the combined “scalar” and “vector” gauge symmetry transformations for the $$(3+1)$$ (3+1) -dimensional (4D) topologically massive non-Abelian $$(B \wedge F)$$ (B∧F) theory with the help of geometrical superfield formalism. For this purpose, we use three horizontality conditions (HCs). The first HC produces the (anti-)BRST transformations for the 1-form gauge field and corresponding (anti-)ghost fields whereas the second HC yields the (anti-)BRST transformations for 2-form field and associated (anti-)ghost fields. The integrability of second HC produces third HC. The latter HC produces the (anti-)BRST symmetry transformations for the compensating auxiliary vector field and corresponding ghosts. We obtain five (anti-)BRST invariant Curci–Ferrari (CF)-type conditions which emerge very naturally as the off-shoots of superfield formalism. Out of five CF-type conditions, two are fermionic in nature. These CF-type conditions play a decisive role in providing the absolute anticommutativity of the (anti-)BRST transformations and also responsible for the derivation of coupled but equivalent (anti-)BRST invariant Lagrangian densities. Furthermore, we capture the (anti-)BRST invariance of the coupled Lagrangian densities in terms of the superfields and translation generators along the Grassmannian directions $$\theta$$ θ and $$\bar{\theta }$$ θ¯

A near-linear time algorithm for constructing a cactus representation of minimum cuts

We present an Õ(m) (near-linear) time Monte Carlo algorithm for constructing the cactus data structure, a useful representation of all the global minimum edge cuts of an undirected graph. Our algorithm represents a fundamental improvement over the best previous (quadratic time) algorithms: because there can be quadratically many min-cuts, our algorithm must avoid looking at all min-cuts during the construction, but nonetheless builds a data structure representing them all. Our result closes the gap between the (near-linear) time required to find a single min-cut and that for (implicitly) finding all the min-cuts

A Near-Linear Time Algorithm for Constructing a Cactus Representation of Minimum Cuts

We present an Õ(m) (near-linear) time Monte Carlo algorithm for constructing the cactus data structure, a useful representation of all the global minimum edge cuts of an undirected graph. Our algorithm represents a fundamental improvement over the best previous (quadratic time) algorithms: because there can be quadratically many min-cuts, our algorithm must avoid looking at all min-cuts during the construction, but nonetheless builds a data structure representing them all. Our result closes the gap between the (near-linear) time required to find a single min-cut and that for (implicitly) finding all the min-cuts.