General Bounds for Incremental Maximization

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $k\in\mathbb{N}$ that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all $k$ between the incremental solution after $k$ steps and an optimum solution of cardinality $k$. We define a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general $2.618$-competitive incremental algorithm for this class of problems, and show that no algorithm can have competitive ratio below $2.18$ in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be $1.58$-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) ($b$-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) $2.313$ for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems.Comment: fixed typo

Spin-directed network model for the surface states of weak three-dimensional Z2 \mathbb{Z}^{\,}_{2} topological insulators

A two-dimensional spin-directed $\mathbb{Z}^{\,}_{2}$ network model is constructed that describes the combined effects of dimerization and disorder for the surface states of a weak three-dimensional $\mathbb{Z}^{\,}_{2}$ topological insulator. The network model consists of helical edge states of two-dimensional layers of $\mathbb{Z}^{\,}_{2}$ topological insulators which are coupled by time-reversal symmetric interlayer tunneling. It is argued that, without dimerization of interlayer couplings, the network model has no insulating phase for any disorder strength. However, a sufficiently strong dimerization induces a transition from a metallic phase to an insulating phase. The critical exponent $\nu$ for the diverging localization length at metal-insulator transition points is obtained by finite-size scaling analysis of numerical data from simulations of this network model. It is shown that the phase transition belongs to the two-dimensional symplectic universality class of Anderson transition.Comment: 36 pages and 27 figures, plus Supplemental Materia

Directed and elliptic flow in heavy ion collisions from Ebeam=90E_{\rm beam}=90 MeV/nucleon to Ec.m.=200E_{\rm c.m.}=200 GeV/nucleon

Recent data from the NA49 experiment on directed and elliptic flow for Pb+Pb reactions at CERN-SPS are compared to calculations with a hadron-string transport model, the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model. The rapidity and transverse momentum dependence of the directed and elliptic flow, i.e. $v_1$ and $v_2$, are investigated. The flow results are compared to data at three different centrality bins. Generally, a reasonable agreement between the data and the calculations is found. Furthermore, the energy excitation functions of $v_1$ and $v_2$ from $E_{\rm beam}=90A$ MeV to $E_{\rm cm}=200A$ GeV are explored within the UrQMD framework and discussed in the context of the available data. It is found that, in the energy regime below $E_{\rm beam}\leq 10A$ GeV, the inclusion of nuclear potentials is necessary to describe the data. Above $40A$ GeV beam energy, the UrQMD model starts to underestimate the elliptic flow. Around the same energy the slope of the rapidity spectra of the proton directed flow develops negative values. This effect is known as the third flow component ("antiflow") and cannot be reproduced by the transport model. These differences can possibly be explained by assuming a phase transition from hadron gas to quark gluon plasma at about $40A$ GeV.Comment: 19 pages, minor changes and modified title as published in PR