Packing squares independently

Given a set of squares and a strip of bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from $\Omega(n!)$ to $O(2^n)$, with $n$ the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard, and then we propose a fully polynomial-time approximation scheme (FPTAS) to solve it. We also propose three mathematical programming formulations based on different solution representations and confirm the performance of these algorithms through computational experiments. Finally, we discuss several extensions that are relevant to practical applications.Comment: 15 page

House-price crash and macroeconomic crisis: a Hong Kong case study

The crash of house prices has become an important feature of macroeconomic crisis. We argue that the crash of house prices driven by contractionary monetary policy is not only a reaction, but also accelerates and amplifies the fluctuations of major macroeconomic variables. The impulse response of consumption to the house price shock estimated from Bayesian VAR is of same level as that of investment in Hong Kong, which is distinct from the United States. Therefore, in this paper we conduct a case study of Hong Kong in the 1997-1998 financial crisis and quantitatively analyze the mechanism by developing a general equilibrium model incorporating financial accelerator in both household and entrepreneur sectors. In addition, we introduce real estate producers in order to modify the unrealistic mechanism in existing literature. After estimating the parameters with a combination of calibration and Bayesian method, the simulated impulse responses imply that our model can explain the co-movement of house prices, consumption and investment much better than alternative ones. Moreover, the results of variance decomposition show that interest rate shock can explain most of the house price fluctuations, and a substantial fraction of fluctuations in major macroeconomic variables