Robert Weismantel

Let $A \in \mathbb {Z}^{m \times n}$ A ∈ ℤ m × n be an integral matrix and a, b, $c \in \mathbb {Z}$ c ∈ ℤ satisfy a ≥ b ≥ c ≥ 0. The question is to recognize whether A is {a,b,c}-modular, i.e., whether the set of n × n subdeterminants of A in absolute value is {a,b,c}. We will succeed in solving this problem in polynomial time unless A possesses a duplicative relation, that is, A has nonzero n × n subdeterminants k₁ and k₂ satisfying 2 ⋅|k₁| = |k₂|. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over {a,b,c}-modular constraint matrices for any constants a, b and c.

We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. We create a framework for studying the asymptotic distribution of general functions related to integer optimization. There has been a significant amount of research focused around the extreme values that these functions can attain, however less is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.

We consider integer and linear programming problems for which the linear constraints exhibit a (recursive) block-structure: The problem decomposes into independent and efficiently solvable sub-problems if a small number of constraints is deleted. A prominent example are $n$-fold integer programming problems and their generalizations which have received considerable attention in the recent literature. The previously known algorithms for these problems are based on the augmentation framework, a tailored integer programming variant of local search. In this paper we propose a different approach. Our algorithm relies on parametric search and a new proximity bound. We show that block-structured linear programming can be solved efficiently via an adaptation of a parametric search framework by Norton, Plotkin, and Tardos in combination with Megiddo's multidimensional search technique. This also forms a subroutine of our algorithm for the integer programming case by solving a strong relaxation of it. Then we show that, for any given optimal vertex solution of this relaxation, there is an optimal integer solution within $\ell_1$-distance independent of the dimension of the problem. This in turn allows us to find an optimal integer solution efficiently. We apply our techniques to integer and linear programming with $n$-fold structure or bounded dual treedepth, two benchmark problems in this field. We obtain the first algorithms for these cases that are both near-linear in the dimension of the problem and strongly polynomial. Moreover, unlike the augmentation algorithms, our approach is highly parallelizable.

In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using mainly tools from discrete geometry, we show that some sparsity conditions on the original data matrix allow us to solve the problem in polynomial time.