arXiv subjects

Torben Schürenberg

Publications and source records attributed to Torben Schürenberg.

5 recordsLinked to original sources

arXiv2026

Blindfolded pursuit with delays of your choice

We study pursuit-evasion games on graphs with a single pursuer and an invisible evader. The pursuer may assign integer travel times to the edges of the graph and specify a finite sequence of vertices to query, one per time step. The evader then chooses a walk over the same time horizon, aiming to elude all queries. With unit travel times, the setting in which the evader must move at every time step is known as the hunter and rabbit game, while the variant in which the evader can wait at a vertex can be phrased as a firefighting game: the vertices of a burning graph must be extinguished, and any vertex left burning reignites its neighbors. For both settings, we show that the power to choose travel times allows a single pursuer to succeed in polynomial time on any graph. This contrasts with unweighted graphs, where the number of hunters or firefighters needed can grow linearly in the number of vertices. If the evader, in addition to waiting, may start at an earlier time unknown to the pursuer, we show that the pursuer still wins on any graph given exponential time.

cs.GT↗

arXiv2026

Hamilton paths and cycles in flip graphs of (almost-)perfect matchings

We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almost-perfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs. We prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. In the geometric setting, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.

math.CO↗

arXiv2026

Complexity of Firefighting on Graphs

We consider a pursuit-evasion game that describes the process of extinguishing a fire burning on the nodes of an undirected graph. We denote the minimum number of firefighters required by ffn(G) and provide almost sharp bounds to this graph parameter for complete binary trees. We show that deciding whether ffn(G) <= m for given G and m is NP-hard. Furthermore, we show that shortest strategies can have superpolynomial length, leaving open whether the problem is in NP. We provide a construction that allows for transferring these results to a well-established Cops and Robbers variant called the "Hunter and Rabbit game".

cs.CC↗

arXiv2025

On the Price of Anarchy in Packet Routing Games with FIFO

We investigate packet routing games in which network users selfishly route themselves through a network over discrete time, aiming to reach the destination as quickly as possible. Conflicts due to limited capacities are resolved by the first-in, first-out (FIFO) principle. Building upon the line of research on packet routing games initiated by Werth et al., we derive the first non-trivial bounds for packet routing games with FIFO. Specifically, we show that the price of anarchy is at most 2 for the important and well-motivated class of uniformly fastest route equilibria introduced by Scarsini et al. on any linear multigraph. We complement our results with a series of instances on linear multigraphs, where the price of stability converges to at least $\frac{e}{e-1}$. Furthermore, our instances provide a lower bound for the price of anarchy of continuous Nash flows over time on linear multigraphs which establishes the first lower bound of $\frac{e}{e-1}$ on a graph class where the monotonicity conjecture is proven by Correa et al.

cs.GT↗

arXiv2024

On the Min-Max Star Partitioning Number

In this paper, we introduce a novel star partitioning problem for simple connected graphs $G=(V,E)$. The goal is to find a partition of the edges into stars that minimizes the maximum number of stars a node is contained in while simultaneously satisfying node-specific capacities. We design and analyze an efficient polynomial time algorithm with a runtime of $\mathcal{O}(|E|^2)$ that determines an optimal partition. Moreover, we explicitly provide a closed form of an optimal value for some graph classes. We generalize our algorithm to find even an optimal star partition of linear hypergraphs, multigraphs, and graphs with self-loop. We use flow techniques to design an algorithm for the star partitioning problem with an improved runtime of $\mathcal{O}(\log(Δ) \cdot |E| \cdot \min\{|V|^{\frac{2}{3}},|E|^{\frac{1}{2}}\})$, where $Δ$ is maximum node degree in $G$. In contrast to the unweighted setting, we show that a node-weighted decision variant of this problem is \texttt{strongly NP-complete} even without capacity constraints. Furthermore, we provide an extensive comparison to the problem of minimizing the minimum indegree satisfying node capacity constraints.

math.CO↗