software engineers in logistics

Abstract: We give an $n^{0.3+\varepsilon}$-approximation algorithm for the Steiner $k$-Forest problem, for any constant $\varepsilon>0$. As a function of $n$, this improves over the $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation of Gupta et al. [ESA'07, TALG'10] which has stood for nearly two decades for the general case, as well as the later $n^{0.448}$-approximation of Dinitz et. al [APPROX-RANDOM'14, TALG'17] for the uniform weight case. On the other hand, we show that, due to lower bounds on the Densest $k$-Subgraph problem, the $O(\sqrt k)$-approximation for Steiner $k$-Forest likely cannot be improved. Specifically, we show that for any sufficiently small $\varepsilon>0$, an $O(k^{1/2-\varepsilon})$-approximation for Steiner $k$-Forest would surpass known degree-$n^{Ω(\varepsilon^2)}$ Sum-of-Squares integrality gaps for Densest k-Subgraph and refute the corresponding dense-versus-random conjecture.

Abstract: We give a fixed parameter tractable (FPT) algorithm with running time $f(k,Δ)\cdot {|I|}^{O(1)}$ for integer linear programs with 4-block structure, parameterized by the maximum block dimension $k$ and the largest absolute matrix entry $Δ$. This result resolves a long-standing open question in parameterized complexity, and answers a conjecture by Eisenbrand and Rothvoss (2026) in the positive. This result has several implications for other block-structured integer programming models, including 3-block, mixed fracture number, and special cases of 4-block programs with large entries outside of the diagonal. Important tools for this algorithm are structural properties of generalized $n$-fold integer programs shown by Ligthart~(2026) and an algorithm by Veselov et al.~(2020) for optimizing discrete convic functions.

Abstract: We study the use of graph neural networks (GNNs) for finding approximate ground states of Ising models. Efficiently finding these ground states is of broad significance because many combinatorial optimization problems can be formulated as an Ising model with the appropriate choice of couplings and fields. Exactly solving these problems is hard but there are many good heuristic methods. A lineage of these heuristics build from mean-field approximations: one approach uses the leading eigenvector of an appropriately defined matrix, another is the min-sum algorithm, also known as the energetic cavity method. Without modification, GNNs perform worse than both of these methods. We consider small modifications to the GNN to incorporate these heuristics and find that this considerably improves performance. While the modified approach is competitive against other deep-learning approaches, we still find that simulated annealing is reliably at least as good as deep learning methods for the same computational cost.