Consistency Cuts for Dantzig-Wolfe Reformulations

A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In “Consistency Cuts for Dantzig-Wolfe Reformulation,” Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general.

Vaccination Schedule under Conditions of Limited Vaccine Production Rate

The paper is devoted to optimal vaccination scheduling during a pandemic to minimize the probability of infection. The recent COVID-19 pandemic showed that the international community is not properly prepared to manage a crisis of this scale. Just after the vaccines had been approved by medical agencies, the policymakers needed to decide on the distribution strategy. To successfully fight the pandemic, the key is to find the equilibrium between the vaccine distribution schedule and the available supplies caused by limited production capacity. This is why society needs to be divided into stratified groups whose access to vaccines is prioritized. Herein, we present the problem of distributing protective actions (i.e., vaccines) and formulate two mixed-integer programs to solve it. The problem of distributing protective actions (PDPA) aims at finding an optimal schedule for a given set of social groups with a constant probability of infection. The problem of distributing protective actions with a herd immunity threshold (PDPAHIT) also includes a variable probability of infection, i.e., the situation when herd immunity is obtained. The results of computational experiments are reported and the potential of the models is illustrated with examples.

A computational status update for exact rational mixed integer programming

The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.

Notes on {a,b,c}-Modular Matrices

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.

About the complexity of two-stage stochastic IPs

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form $$\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}$$ max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the constraint matrix $${\mathcal {A}} \in {\mathbb {Z}}^{r n \times s +nt}$$ A ∈ Z r n × s + n t consists roughly of n repetitions of a matrix $$A \in {\mathbb {Z}}^{r \times s}$$ A ∈ Z r × s on the vertical line and n repetitions of a matrix $$B \in {\mathbb {Z}}^{r \times t}$$ B ∈ Z r × t on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and $$\Delta $$ Δ , where $$\Delta $$ Δ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time $$f(r,s,\Delta ) \cdot \mathrm {poly}(n,t)$$ f ( r , s , Δ ) · poly ( n , t ) , where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.