Towards Effective Exact Algorithms for the Maximum Balanced Biclique Problem

The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP. Firstly, we introduce an Upper Bound Propagation procedure to pre-compute an upper bound involving each vertex. Then we extend an existing branch-and-bound algorithm by integrating the pre-computed upper bounds. We also present a set of new valid inequalities induced from the upper bounds to tighten an existing mathematical formulation for MBBP. Lastly, we investigate another exact algorithm scheme which enumerates a subset of balanced bicliques based on our upper bounds. Experiments show that compared to existing approaches, the proposed algorithms and formulations are more efficient in solving a set of random graphs and large real-life instances

Lacing topological orders in two dimensions: exactly solvable models for Kitaev's sixteen-fold way

A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev's sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with $\nu$ species of Majorana fermions coupled to a static $\mathbb{Z}_2$ gauge field. Here each specie of Majorana fermions gives rise to an energy band that carries a Chern number $\mathcal{C}=1$, yielding a total Chern number $\mathcal{C}=\nu$. It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when $\nu$ is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.Comment: 20 pages, 9 figure