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Register a new userDual Power Assignment via Second Hamiltonian Cycle
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Added by
Karim Abu-Affash
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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A power assignment is an assignment of transmission power to each of the wireless nodes of a wireless network, so that the induced graph satisfies some desired properties. The cost of a power assignment is the sum of the assigned powers. In this paper, we consider the dual power assignment problem, in which each wireless node is assigned a high- or low-power level, so that the induced graph is strongly connected and the cost of the assignment is minimized. We improve the best known approximation ratio from $frac{pi^2}{6}-frac{1}{36}+epsilonthickapprox 1.617$ to $frac{11}{7}thickapprox 1.571$. Moreover, we show that the algorithm of Khuller et al. for the strongly connected spanning subgraph problem, which achieves an approximation ratio of $1.61$, is $1.522$-approximation algorithm for symmetric directed graphs. The innovation of this paper is in achieving these results via utilizing interesting properties for the existence of a second Hamiltonian cycle.
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We investigate the problem of determining the Hamiltonian of a locally interacting open-quantum system. To do so, we construct model estimators based on inverting a set of stationary, or dynamical, Heisenberg-Langevin equations of motion which rely on a polynomial number of measurements and parameters. We validate our Hamiltonian assignment methods by numerically simulating one-dimensional XX-interacting spin chains coupled to thermal reservoirs. We study Hamiltonian learning in the presence of systematic noise and find that, in certain time dependent cases, the Hamiltonian estimator accuracy increases when relaxing the environments physicality constraints.
We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in $RR^n$.
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Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $pi$ of the vertex set of $G$ into the plane. We prove that a wheel graph $W_n$ admits a drawing $pi$ such that, if one wants to eliminate edge crossings by shifting vertices to new positions in the plane, then at most $(2+o(1))sqrt n$ of all $n$ vertices can stay fixed. Moreover, such a drawing $pi$ exists even if it is presupposed that the vertices occupy any prescribed set of points in the plane. Similar questions are discussed for other families of planar graphs.
A Dubins path is a shortest path with bounded curvature. The seminal result in non-holonomic motion planning is that (in the absence of obstacles) a Dubins path consists either from a circular arc followed by a segment followed by another arc, or from three circular arcs [Dubins, 1957]. Dubins original proof uses advanced calculus; later, Dubins result was reproved using control theory techniques [Reeds and Shepp, 1990], [Sussmann and Tang, 1991], [Boissonnat, Cerezo, and Leblond, 1994]. We introduce and study a discrete analogue of curvature-constrained motion. We show that shortest bounded-curvature polygonal paths have the same structure as Dubins paths. The properties of Dubins paths follow from our results as a limiting case---this gives a new, discrete proof of Dubins result.
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