We show that the classical LP relaxation of the asymmetric traveling salesman path problem (ATSPP) has constant integrality ratio. If ρ ATSP and ρ ATSPP denote the integrality ratios for the asymmetric TSP and its path version, then ρ ATSPP ≤ 4ρ ATSP − 3.We prove an even better bound for node-weighted instances: if the integrality ratio for ATSP on node-weighted instances is ρ NW ATSP , then the integrality ratio for ATSPP on node-weighted instances is at most 2ρ NW ATSP − 1. Moreover, we show that for ATSP node-weighted instances and unweighted digraph instances are almost equivalent. From this we deduce a lower bound of 2 on the integrality ratio of unweighted digraph instances.
Autonomous driving functions (ADFs) in public traffic have to comply with complex system requirements that are based on knowledge of experts from different disciplines, e.g., lawyers, safety experts, psychologists. In this paper, we present a research preview regarding the validation of ADFs with respect to such requirements. We investigate the suitability of Traffic Sequence Charts (TSCs) for the formalization of such requirements and present a concept for monitoring system compliance during validation runs. We find TSCs, with their intuitive visual syntax over symbols from the traffic domain, to be a promising choice for the collaborative formalization of such requirements. For an example TSC, we describe the construction of a runtime monitor according to our novel concept that exploits the separation of spatial and temporal aspects in TSCs, and successfully apply the monitor on exemplary runs. The monitor continuously provides verdicts at runtime, which is particularly beneficial in ADF validation, where validation runs are expensive. The next open research questions concern the generalization of our monitor construction, the identification of the limits of TSC monitorability, and the investigation of the monitor's performance in practical applications. Perspectively, TSC runtime monitoring could provide a useful technique in other emerging application areas such as AI training, safeguarding ADFs during operation, and gathering meaningful traffic data in the field.
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