New Bounds on Augmenting Steps of Block-Structured Integer Programs

Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4-block n-fold integer programs, which are the most general class considered so far. A 4-block n-fold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the so-called Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the ??- or ?_?-norm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4-block n-fold IP has Graver elements of ?_?-norm at most ?_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and ?_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to ?_FPT(1), perhaps at least in some restricted cases. We prove that the ?_?-norm of the Graver elements of 4-block n-fold IP is upper bounded by ?_FPT(n^{s_D}), improving significantly over the previous bound ?_FPT(n^{2^{s_D}}). We also provide a matching lower bound of ?(n^{s_D}) which even holds for arbitrary non-zero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4-block n-fold in which C is a zero matrix, called 3-block n-fold IP. We show that while the ?_?-norm of its Graver elements is ?(n^{s_D}), there exists a different decomposition into lattice elements whose ?_?-norm is bounded by ?_FPT(1), which allows us to provide improved upper bounds on the ?_?-norm of Graver elements for 3-block n-fold IP. The key difference between the respective decompositions is that a Graver basis guarantees a sign-compatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3-block n-fold IP and 4-block IP, and our lower bounds strongly hint at parameterized hardness of 4-block and even 3-block n-fold IP. Furthermore, we show that 3-block n-fold IP is without loss of generality in the sense that 4-block n-fold IP can be solved in FPT oracle time by taking an algorithm for 3-block n-fold IP as an oracle

Rapid Invasion of Spartina Alterniflora in the Coastal Zone of Mainland China: Spatiotemporal Patterns and Human Prevention

Given the extensive spread and ecological consequences of exotic Spartina alterniflora (S. alterniflora) over the coast of mainland China, monitoring its spatiotemporal invasion patterns is important for the sake of coastal ecosystem management and ecological security. In this study, Landsat series images from 1990 to 2015 were used to establish multi-temporal datasets for documenting the temporal dynamics of S. alterniflora invasion. Our observations revealed that S. alterniflora had a continuous expansion with the area increasing by 50,204 ha during the considered 25 years. The largest expansion was identified in Jiangsu Province during the period of 1990-2000, and in Zhejiang Province during the periods 2000-2010 and 2010-2015. Three noticeable hotspots for S. alterniflora invasion were Yancheng of Jiangsu, Chongming of Shanghai, and Ningbo of Zhejiang, and each had a net area increase larger than 5000 ha. Moreover, an obvious shrinkage of S. alterniflora was identified in three coastal cities including the city of Cangzhou of Hebei, Dongguan, and Jiangmen of Guangdong. S. alterniflora invaded mostly into mudflats (>93%) and shrank primarily due to aquaculture (55.5%). This study sheds light on the historical spatial patterns in S. alterniflora distribution and thus is helpful for understanding its invasion mechanism and invasive species management

Portfolio Selection under Systemic Risk

This paper proposes a modified Sharpe ratio to construct optimal portfolios under systemic events. The portfolio allocation problem is solved analytically under the absence of short-selling restrictions and numerically when short-selling restrictions are imposed. This approach is made operational by embedding it in a multivariate dynamic setting using dynamic conditional correlation and copula models. We evaluate the out-of-sample performance of our portfolio empirically over the period 2007 to 2020 using ex post final wealth paths and systemic risk metrics against mean–variance, equally weighted, and global minimum variance portfolios. Our portfolio outperforms all competitors under market distress and remains competitive in noncrisis periods