Finding the Minimum-Weight k-Path

Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k M). If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of \tilde{O}(2^k \poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k) M n^3) and a $(1+\varepsilon)$-approximate solution of running time \tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201

Spin-charge separation in ultra-cold quantum gases

We investigate the physical properties of quasi-1D quantum gases of fermion atoms confined in harmonic traps. Using the fact that for a homogeneous gas, the low energy properties are exactly described by a Luttinger model, we analyze the nature and manifestations of the spin-charge separation. Finally we discuss the necessary physical conditions and experimental limitations confronting possible experimental implementations.Comment: 4 pages, revtex4, 2 eps figure

Charge transfer fluctuation, d−d-wave superconductivity, and the B1gB_{1g} Raman phonon in the Cuprates: A detailed analysis

The Raman spectrum of the $B_{1g}$ phonon in the superconducting cuprate materials is investigated theoretically in detail in both the normal and superconducting phases, and is contrasted with that of the $A_{1g}$ phonon. A mechanism involving the charge transfer fluctuation between the two oxygen ions in the CuO$_2$ plane coupled to the crystal field perpendicular to the plane is discussed and the resulting electron-phonon coupling is evaluated. Depending on the symmetry of the phonon the weight of different parts of the Fermi surface in the coupling is different. This provides the opportunity to obtain information on the superconducting gap function at certain parts of the Fermi surface. The lineshape of the phonon is then analyzed in detail both in the normal and superconducting states. The Fano lineshape is calculated in the normal state and the change of the linewidth with temperature below T$_{c}$ is investigated for a $d_{x^{2}-y^{2}}$ pairing symmetry. Excellent agreement is obtained for the $B_{1g}$ phonon lineshape in YBa$_{2}$Cu$_{3}$O$_{7}$. These experiments, however, can not distinguish between $d_{x^{2}-y^{2}}$ and a highly anisotropic $s$-wave pairing.Comment: Revtex, 21 pages + 4 postscript figures appended, tp

Bounded Counter Languages

We show that deterministic finite automata equipped with $k$ two-way heads are equivalent to deterministic machines with a single two-way input head and $k-1$ linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*... a_m^*$ for a fixed sequence of symbols $a_1, a_2,..., a_m$. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters

On rr-Simple kk-Path

An $r$-simple $k$-path is a {path} in the graph of length $k$ that passes through each vertex at most $r$ times. The $r$-SIMPLE $k$-PATH problem, given a graph $G$ as input, asks whether there exists an $r$-simple $k$-path in $G$. We first show that this problem is NP-Complete. We then show that there is a graph $G$ that contains an $r$-simple $k$-path and no simple path of length greater than $4\log k/\log r$. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex. We then give a randomized algorithm that runs in time $$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$ that solves the $r$-SIMPLE $k$-PATH on a graph with $n$ vertices with one-sided error. We also show that a randomized algorithm with running time $\mathrm{poly}(n)\cdot 2^{(c/2)k/ r}$ with $c<1$ gives a randomized algorithm with running time \poly(n)\cdot 2^{cn} for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an $O(\log r)$ factor

Kernel Bounds for Structural Parameterizations of Pathwidth

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.Comment: This paper contains the proofs omitted from the extended abstract published in the proceedings of Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201

Quantum Critical Scaling in a Moderately Doped Antiferromagnet

Using high temperature expansions for the equal time correlator $S(q)$ and static susceptibility $\chi(q)$ for the t-J model, we present evidence for quantum critical (QC), $z\!=\!1$, behavior at intermediate temperatures in a broad range of $t/J$ ratio, doping, and temperatures. We find that the dynamical susceptibility is very close to the universal scaling function computable for the asymptotic QC regime, and that the dominant energy scale is temperature. Our results are in excellent agreement with measurements of the spin-echo decay rate, $1/T_{\rm 2G}$, in La$_2$CuO$_4$, and provide qualitative understanding of both $1/T_1$ and $1/T_{\rm 2G}$ nuclear relaxation rates in doped cuprates.Comment: 11 pages, REVTeX v3.0, PostScript file for 3 figures is attached, UIUC-P-93-07-068. In this revised version, we calculate the scaling functions and thus present new and more direct evidence in favor of our original conclusion

First-Principles Calculations of Hyperfine Interactions in La_2CuO_4

We present the results of first-principles cluster calculations of the electronic structure of La_2CuO_4. Several clusters containing up to nine copper atoms embedded in a background potential were investigated. Spin-polarized calculations were performed both at the Hartree-Fock level and with density functional methods with generalized gradient corrections to the local density approximation. The distinct results for the electronic structure obtained with these two methods are discussed. The dependence of the electric-field gradients at the Cu and the O sites on the cluster size is studied and the results are compared to experiments. The magnetic hyperfine coupling parameters are carefully examined. Special attention is given to a quantitative determination of on-site and transferred hyperfine fields. We provide a detailed analysis that compares the hyperfine fields obtained for various cluster sizes with results from additional calculations of spin states with different multiplicities. From this we conclude that hyperfine couplings are mainly transferred from nearest neighbor Cu^{2+} ions and that contributions from further distant neighbors are marginal. The mechanisms giving rise to transfer of spin density are worked out. Assuming conventional values for the spin-orbit coupling, the total calculated hyperfine interaction parameters are compared to informations from experiments.Comment: 23 pages, 9 figure

On the low temperature properties and specific anisotropy of pure anisotropically paired superconductors

Dependences of low temperature behavior and anisotropy of various physical quantities for pure unconventional superconductors upon a particular form of momentum direction dependence for the superconducting order parameter (within the framework of the same symmetry type of superconducting pairing) are considered. A special attention is drawn to the possibility of different multiplicities of the nodes of the order parameter under their fixed positions on the Fermi surface, which are governed by symmetry. The problem of an unambiguous identification of a type of superconducting pairing on the basis of corresponding experimental results is discussed. Quasiparticle density of states at low energy for both homogeneous and mixed states, the low temperature dependences of the specific heat, penetration depth and thermal conductivity, the I-V curves of SS and NS tunnel junctions at low voltages are examined. A specific anisotropy of the boundary conditions for unconventional superconducting order parameter near $T_c$ for the case of specular reflection from the boundary is also investigated.Comment: 20 page