Philipp Pabst | Searcharxiv

arXiv2025

Fare Zone Assignment on Trees

Designing fare systems for public transportation networks is a challenging task. A popular approach is to partition the network into fare zones (``zoning'') and fix journey prices depending on the number of traversed zones (``pricing''). In this paper, we focus on finding revenue-optimal solutions to the zoning problem for a given subadditive pricing function. We consider tree networks with $n$ vertices, since trees already pose non-trivial algorithmic challenges. Our main results are efficient algorithms that yield a simple $\mathcal{O}(\log n)$-approximation as well as a more involved $\mathcal{O}(\log n/\log \log n)$-approxi\-ma\-tion. We show that rooted instances, in which all demand arises at a single source, can be solved exactly. We further show APX-hardness for general instances on star graphs. For paths, we prove strong NP-hardness and outline a PTAS. Moreover, we show that computing an optimal solution is in FPT or XP for several natural problem parameters.

cs.DS↗

arXiv2025

Approximating Graphic Multi-Path TSP and Graphic Ordered TSP

The path version of the Traveling Salesman Problem is one of the most well-studied variants of the ubiquitous TSP. Its generalization, the Multi-Path TSP, has recently been used in the best known algorithm for path TSP by Traub and Vygen [Cambridge University Press, 2024]. The best known approximation factor for this problem is $2.214$ by B\"{o}hm, Friggstad, M\"{o}mke and Spoerhase [SODA 2025]. In this paper we show that for the case of graphic metrics, a significantly better approximation guarantee of $2$ can be attained. Our algorithm is based on sampling paths from a decomposition of the flow corresponding to the optimal solution to the LP for the problem, and connecting the left-out vertices with doubled edges. The cost of the latter is twice the optimum in the worst case; we show how the cost of the sampled paths can be absorbed into it without increasing the approximation factor. Furthermore, we prove that any below-$2$ approximation algorithm for the special case of the problem where each source is the same as the corresponding sink yields a below-$2$ approximation algorithm for Graphic Multi-Path TSP. We also show that our ideas can be utilized to give a factor $1.791$-approximation algorithm for Ordered TSP in graphic metrics, for which the aforementioned paper [SODA 2025] and Armbruster, Mnich and N\"agele [APPROX 2024] give a $1.868$-approximation algorithm in general metrics.

cs.DS↗