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 ℓ₁- or ℓ_∞-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 O<sub>FPT</sub>(n<sup>2sD</sup> ), leading to an algorithm with a similar runtime; here, sD is the number of rows of matrix D and O<sub>FPT</sub> 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 O<sub>FPT</sub>(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 O_FPT(n^sD ), improving significantly over the previous bound O_FPT(n^2sD ). We also provide a matching lower bound of Ω(n^sD ) 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 Ω(n<sup>sD</sup> ), there exists a different decomposition into lattice elements whose `∞-norm is bounded by O_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.