arXiv subjects
Publications and source records attributed to Shi Li.
In this note, we show that the integrality gap of the $k$-Directed-Component- Relaxation($k$-DCR) LP for the Steiner tree problem, introduced by Byrka, Grandoni, Rothvob and Sanita (STOC 2010), is at most $\ln(4)<1.39$. The proof is constructive: we can efficiently find a Steiner tree whose cost is at most $\ln(4)$ times the cost of the optimal fractional $k$-restricted Steiner tree given by the $k$-DCR LP.
The notion of vertex sparsification is introduced in \cite{M}, where it was shown that for any graph $G = (V, E)$ and a subset of $k$ terminals $K \subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ on just the terminal set so that simultaneously for all cuts $(A, K-A)$, the value of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same as the value of the corresponding cut in $H$. We give the first super-constant lower bounds for how well a cut-sparsifier $H$ can simultaneously approximate all minimum cuts in $G$. We prove a lower bound of $Ω(\log^{1/4} k)$ -- this is polynomially-related to the known upper bound of $O(\log k/\log \log k)$. This is an exponential improvement on the $Ω(\log \log k)$ bound given in \cite{LM} which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers. Despite this negative result, we show that for many natural problems, we do not need to incur a multiplicative penalty for our reduction. We obtain optimal $O(\log k)$-competitive Steiner oblivious routing schemes, which generalize the results in \cite{R}. We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most $O(\log k)$ times the integrality gap restricted to trees. This result helps to explain the ubiquity of the $O(\log k)$ guarantees for such problems. Lastly, we use our ideas to give an efficient construction for vertex-sparsifiers that match the current best existential results -- this was previously open. Our algorithm makes novel use of Earth-mover constraints.
The method of anomaly cancellation to derive Hawking radiation initiated by Robinson and Wilczek is applied to 2+1 dimensional stationary black holes. Using the dimensional reduction technique, we find that the near-horizon physics for the fermionic field in the background of the general 2+1 dimensional stationary black hole can be approximated by an infinite collection of two component fermionic fields in 1+1 dimensional spacetime background coupled with dilaton field and U(1) gauge field. By restoring the gauge invariance and the general coordinate covariance for the reduced two dimensional theory, Hawking flux and temperature of black hole are obtained. We apply this method to two types of black holes in three dimensional spacetime, which are BTZ black hole in Einstein gravity and a rotating black hole in Bergshoeff-Hohm-Townsend (BHT) massive gravity.
Given a metric space $(X,d_X)$, the earth mover distance between two distributions over $X$ is defined as the minimum cost of a bipartite matching between the two distributions. The doubling dimension of a metric $(X, d_X)$ is the smallest value $α$ such that every ball in $X$ can be covered by $2^α$ ball of half the radius. We study efficient algorithms for approximating earth mover distance over metrics with bounded doubling dimension. Given a metric $(X, d_X)$, with $|X| = n$, we can use $\tilde O(n^2)$ preprocessing time to create a data structure of size $\tilde O(n^{1 + \e})$, such that subsequently queried EMDs can be $O(α_X/\e)$-approximated in $\tilde O(n)$ time. We also show a weaker form of sketching scheme, which we call "encoding scheme". Given $(X, d_X)$, by using $\tilde O(n^2)$ preprocessing time, every subsequent distribution $μ$ over $X$ can be encoded into $F(μ)$ in $\tilde O(n^{1 + \e})$ time. Given $F(μ)$ and $F(ν)$, the EMD between $μ$ and $ν$ can be $O(α_X/\e)$-approximated in $\tilde O(n^\e)$ time.
Two different hot isostatic pressing cycles (HIPing) were investigated to synthesize bulk MgB2 samples: a "standard" cycle where a low vessel pressure is maintained while heating to the process temperature with a subsequent simultaneous pressure and temperature decrease and a new method - dense material cooling under pressure (DMCUP). The latter method allowed the synthesis of dense samples with diameters up to 20 mm and thicknesses up to 10 mm from commercial MgB2 powder. Optimal conditions for the DMCUP method with glass encapsulation (maximum pressure 200 MPa, maximum temperature 1000 C over 200 min, and cooling under pressure) resulted in a dense material with a sharp superconducting transition at 38.5 K. This method employs a pressure which is one order of magnitude less than previously reported for pressure assisted sintering of dense material and can be scaled to larger sample sizes and complex shapes. The data for density, microhardness, fracture toughness and sound speed as well as superconducting properties for bulk magnesium diboride are presented. Ball milling the powder enhances sintering and results in a more homogeneous final microstructure.
We present electrical transport, magnetization, and specific heat measurements on bulk MgB2 samples (T_{c} = 38.5 K) synthesized under 200 MPa pressure using a process based on hot isostatic pressing with cooling under pressure. The samples are fully dense and display excellent superconducting properties, including a narrow superconducting transition width (ΔT_{c} = 0.75 K), a high upper critical field H_{c2} (H_{c2}(0) ~ 155 kOe), and a critical current density J_{c} that is the largest yet measured for bulk samples of MgB2 (J_{c}(0) ~ 1.4 MA/cm^{2}). Specific heat measurements yielded a jump ΔC at T_{c} of 92 mJ/mol K. These superconducting properties are comparable to those obtained with techniques that are not so well suited to industrial scale fabrication.
The universal behavior of the continuous superconducting (glass) phase transition is studied in a YBa$_2$Cu$_3$O$_{7-δ}$ thin film. A new analysis technique for extracting critical exponents is developed, using current-voltage ($J$-$E$) measurements. The method places narrow limits on the glass scaling exponents, $z_g$ and $ν_g$, as well as the glass transition temperature, $T_g(H)$. The universal values $z_g = 3.8\pm 0.2$ and $ν_g = 1.8\pm 0.1$ are obtained. Using these values, the data are scaled for fields in the range 1-9~T. Collapse of the scaling forms at different fields describes the universal scaling function. The field dependence of the universal collapse is interpreted in terms of the 3D~XY multicritical scaling theory.
Conductivity measurements reflect vortex solid melting in YBCO films. Field-independent glass exponents nu_g=1.9 and z_g=4.0 describe the transition T_g(H) for 0<H<26 T. At low fields, 3D XY exponents nu_{XY}=0.63 and z_{XY}=1.25 are also observed, with z_{XY} smaller than expected. These compete with glass scaling according to multicritical theory. A predicted power-law form of T_g(H) is observed for 0.5T_c<T_g<T_c. For T_g<0.5T_c, 3D XY scaling fails, but a lowest Landau level analysis becomes possible, obtaining T_{c2}(H) with positive curvature.