Abstract We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Coveron temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at mostktime 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.

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.

Random forests and, more generally, (decision-)tree ensembles are widely used methods for classification and regression. Recent algorithmic advances allow to compute decision trees that are optimal for various measures such as their size or depth. We are not aware of such research for tree ensembles and aim to contribute to this area. Mainly, we provide two novel algorithms and corresponding lower bounds. First, we are able to carry over and substantially improve on tractability results for decision trees, obtaining a $(6\delta D S)^S \cdot \mathrm{poly}$-time algorithm, where $S$ is the number of cuts in the tree ensemble, $D$ the largest domain size, and $\delta$ is the largest number of features in which two examples differ. To achieve this, we introduce the witness-tree technique which also seems promising for practice. Second, we show that dynamic programming, which has been successful for decision trees, may also be viable for tree ensembles, providing an $\ell^n \cdot \mathrm{poly}$-time algorithm, where $\ell$ is the number of trees and $n$ the number of examples. Finally, we compare the number of cuts necessary to classify training data sets for decision trees and tree ensembles, showing that ensembles may need exponentially fewer cuts for increasing number of trees.

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.

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.

Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.