On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms

We give a lower bound on the iteration complexity of a natural class of Lagrangean-relaxation algorithms for approximately solving packing/covering linear programs. We show that, given an input with $m$ random 0/1-constraints on $n$ variables, with high probability, any such algorithm requires $\Omega(\rho \log(m)/\epsilon^2)$ iterations to compute a $(1+\epsilon)$-approximate solution, where $\rho$ is the width of the input. The bound is tight for a range of the parameters $(m,n,\rho,\epsilon)$. The algorithms in the class include Dantzig-Wolfe decomposition, Benders' decomposition, Lagrangean relaxation as developed by Held and Karp [1971] for lower-bounding TSP, and many others (e.g. by Plotkin, Shmoys, and Tardos [1988] and Grigoriadis and Khachiyan [1996]). To prove the bound, we use a discrepancy argument to show an analogous lower bound on the support size of $(1+\epsilon)$-approximate mixed strategies for random two-player zero-sum 0/1-matrix games

Reduced magnetohydrodynamic theory of oblique plasmoid instabilities

The three-dimensional nature of plasmoid instabilities is studied using the reduced magnetohydrodynamic equations. For a Harris equilibrium with guide field, represented by \vc{B}_o = B_{po} \tanh (x/\lambda) \hat{y} + B_{zo} \hat{z}, a spectrum of modes are unstable at multiple resonant surfaces in the current sheet, rather than just the null surface of the polodial field $B_{yo} (x) = B_{po} \tanh (x/\lambda)$, which is the only resonant surface in 2D or in the absence of a guide field. Here $B_{po}$ is the asymptotic value of the equilibrium poloidal field, $B_{zo}$ is the constant equilibrium guide field, and $\lambda$ is the current sheet width. Plasmoids on each resonant surface have a unique angle of obliquity $\theta \equiv \arctan(k_z/k_y)$. The resonant surface location for angle $\theta$ is x_s = - \lambda \arctanh (\tan \theta B_{zo}/B_{po}), and the existence of a resonant surface requires $|\theta| < \arctan (B_{po} / B_{zo})$. The most unstable angle is oblique, i.e. $\theta \neq 0$ and $x_s \neq 0$, in the constant-$\psi$ regime, but parallel, i.e. $\theta = 0$ and $x_s = 0$, in the nonconstant-$\psi$ regime. For a fixed angle of obliquity, the most unstable wavenumber lies at the intersection of the constant-$\psi$ and nonconstant-$\psi$ regimes. The growth rate of this mode is $\gamma_{\textrm{max}}/\Gamma_o \simeq S_L^{1/4} (1-\mu^4)^{1/2}$, in which $\Gamma_o = V_A/L$, $V_A$ is the Alfv\'{e}n speed, $L$ is the current sheet length, and $S_L$ is the Lundquist number. The number of plasmoids scales as $N \sim S_L^{3/8} (1-\mu^2)^{-1/4} (1 + \mu^2)^{3/4}$.Comment: 9 pages, 8 figures, to be published in Physics of Plasma

D∗DπD^*D\pi and B∗BπB^*B\pi couplings in QCD

We calculate the $D^*D\pi$ and $B^*B\pi$ couplings using QCD sum rules on the light-cone. In this approach, the large-distance dynamics is incorporated in a set of pion wave functions. We take into account two-particle and three-particle wave functions of twist 2, 3 and 4. The resulting values of the coupling constants are $g_{D^*D\pi}= 12.5\pm 1$ and $g_{B^*B\pi}= 29\pm 3$. From this we predict the partial width \Gamma (D^{*+} \ra D^0 \pi^+ )=32 \pm 5~ keV . We also discuss the soft-pion limit of the sum rules which is equivalent to the external axial field approach employed in earlier calculations. Furthermore, using $g_{B^*B\pi}$ and $g_{D^*D\pi}$ the pole dominance model for the B \ra \pi and D\ra \pi semileptonic form factors is compared with the direct calculation of these form factors in the same framework of light-cone sum rules.Comment: 27 pages (LATEX) +3 figures enclosed as .uu file MPI-PhT/94-62 , CEBAF-TH-94-22, LMU 15/9