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Pascal Kunz

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3 papers

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IJCAI Conference 2023 Conference Paper

In Which Graph Structures Can We Efficiently Find Temporally Disjoint Paths and Walks?

Pascal Kunz
Hendrik Molter
Meirav Zehavi

A temporal graph has an edge set that may change over discrete time steps, and a temporal path (or walk) must traverse edges that appear at increasing time steps. Accordingly, two temporal paths (or walks) are temporally disjoint if they do not visit any vertex at the same time. The study of the computational complexity of finding temporally disjoint paths or walks in temporal graphs has recently been initiated by Klobas et al. . This problem is motivated by applications in multi-agent path finding (MAPF), which include robotics, warehouse management, aircraft management, and traffic routing. We extend Klobas et al. ’s research by providing parameterized hardness results for very restricted cases, with a focus on structural parameters of the so-called underlying graph. On the positive side, we identify sufficiently simple cases where we can solve the problem efficiently. Our results reveal some surprising differences between the “path version” and the “walk version” (where vertices may be visited multiple times) of the problem, and answer several open questions posed by Klobas et al.

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AAAI Conference 2023 Conference Paper

Parameterized Algorithms for Colored Clustering

Leon Kellerhals
Tomohiro Koana
Pascal Kunz
Rolf Niedermeier

In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an endpoint and are colored differently. Equivalently, the goal can also be described as assigning colors to the vertices in a way that fits the edge-coloring as well as possible. As this problem is NP-hard, we build on previous work by studying its parameterized complexity. We give a 2ᴼ⁽ᵏ⁾·nᴼ⁽¹⁾-time algorithm where k is the number of edges to be selected and n the number of vertices. We also prove the existence of a problem kernel of size O(k⁵ᐟ²), resolving an open problem posed in the literature. We consider parameters that are smaller than k, the number of edges to be selected, and r, the number of edges that can be deleted. Such smaller parameters are obtained by considering the difference between k or r and some lower bound on these values. We give both algorithms and lower bounds for Colored Clustering with such parameterizations. Finally, we settle the parameterized complexity of Colored Clustering with respect to structural graph parameters by showing that it is W[1]-hard with respect to both vertex cover number and tree-cut width, but fixed-parameter tractable with respect to local feedback edge number.

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IJCAI Conference 2022 Conference Paper

Disentangling the Computational Complexity of Network Untangling

Vincent Froese
Pascal Kunz
Philipp Zschoche

We study the recently introduced network untangling problem, a variant of Vertex Cover on temporal graphs---graphs whose edge set changes over discrete time steps. There are two versions of this problem. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.

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