arXiv subjects

Paul Paschmanns

Publications and source records attributed to Paul Paschmanns.

2 recordsLinked to original sources

arXiv2026

The Bidirected Cut Relaxation for Steiner Tree: Better Integrality Gap Bounds and the Limits of Moat Growing

The Steiner Tree problem asks for the cheapest way of connecting a given subset of the vertices in an undirected graph. One of the most prominent linear programming relaxations for Steiner Tree is the Bidirected Cut Relaxation (BCR). Determining the integrality gap of this relaxation is a long-standing open question. For several decades, the best known upper bound was 2, which is achievable by standard techniques. Only very recently, Byrka, Grandoni, and Traub [FOCS 2024] showed that the integrality gap of BCR is strictly below 2. We prove that the integrality gap of BCR is at most 1.898, improving significantly on the previous bound of 1.9988. For the important special case where a terminal minimum spanning tree is an optimal Steiner tree, we show that the integrality gap is at most 12/7, by providing a tight analysis of the dual-growth procedure by Byrka et al. To obtain the general bound of 1.898 on the integrality gap, we generalize their dual growth procedure to a broad class of moat-growing algorithms. Moreover, we prove that no such moat-growing algorithm yields dual solutions certifying an integrality gap below 12/7. Finally, we observe an interesting connection to the Hypergraphic Relaxation.

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arXiv2025

The Days On Days Off Scheduling Problem

Personnel scheduling problems have received considerable academic attention due to their relevance in various real-world applications. These problems involve preparing feasible schedules for an organization's employees and often account for factors such as qualifications of workers and holiday requests, resulting in complex constraints. While certain versions of the personnel rostering problem are widely acknowledged as NP-hard, there is limited theoretical analysis specific to many of its variants. Many studies simply assert the NP-hardness of the general problem without investigating whether the specific cases they address inherit this computational complexity. In this paper, we examine a variant of the personnel scheduling problems, which involves scheduling a homogeneous workforce subject to constraints concerning both the total number and the number of consecutive work days and days off. This problem was claimed to be NP-complete by [Brunner+2013]. In this paper, we prove its NP-completeness and investigate how the combination of constraints contributes to this complexity. Furthermore, we analyze various special cases that arise from the omission of certain parameters, classifying them as either NP-complete or polynomial-time solvable. For the latter, we provide easy-to-implement and efficient algorithms to not only determine feasibility, but also compute a corresponding schedule.

cs.DS↗