The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

Topological Phases in the Single-Layer FeSe

A distinct electronic structure was observed in the single-layer FeSe which shows surprising high temperature superconductivity over 65k. Here we demonstrate that the electronic structure can be explained by the strain effect due to substrates. More importantly, we find that this electronic structure can be tuned into robust topological phases from a topologically trivial metallic phase by the spin-orbital interaction and couplings to substrates. The topological phase is robust against any perturbations that preserve the time-reversal symmetry. Our studies suggest that topological phases and topologically related properties such as Majorana Fermions can be realized in iron-based high T$_{c}$ superconductors.Comment: 9 pages, 5 figue