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The problem of deletion of vertices to a hereditary graph class is a well-studied problem in parameterized complexity. Recently, a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes and we determine whether k vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be fixed parameter tractable (FPT) when the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. This paper focuses on pairs of specific graph classes (Π 1 , Π 2) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms. [ABSTRACT FROM AUTHOR]

Trees with many leaves have applications for broadcasting a message to all recipients simultaneously. Internal nodes of a broadcasting tree require more expensive technology to forward the messages received. We address a problem that captures the main goal: finding spanning trees with few internal nodes in a given network. The Maximum Leaf Spanning Arborescence problem consists of, given a directed graph D , finding a spanning arborescence of D , if one exists, with the maximum number of leaves. This problem is NP-hard in general and MaxSNP-hard in the rooted directed acyclic graphs class. This paper explores a relationship between Maximum Leaf Spanning Arborescence in rooted directed acyclic graphs and maximum weight set packing. The latter problem is related to independent sets on particular classes of intersection graphs. Exploiting this relation, we derive a 7/5-approximation for Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs, improving on the previous 3/2-approximation. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the shortest link scheduling (SLS) in wireless networks under the Signal-to-Interference-plus-Noise-Ratio (SINR) constraints. We introduce a consistent graph G ¯ = (V ¯ , E ¯) where V ¯ is the links set and each edge in E ¯ indicates that its two ends can be active simultaneously. We interpret SLS as the finding of a clique partition in G ¯ and propose a fast approximation algorithm, MCBS (Maximum Clique Based Scheduling), for SLS with a uniform power assignment and with an approximation ratio to the optimal solution of O (1). Theoretical analysis demonstrates the correctness and effectiveness of the proposed algorithm. Our simulation results show that the MCBS algorithm overcomes the best known algorithms in terms of execution time and the number of time slots. [ABSTRACT FROM AUTHOR]

We study clustering problems such as k -Median, k -Means, and Facility Location in graphs of low highway dimension, which is a graph parameter modeling transportation networks. It was previously shown that approximation schemes for these problems exist, which either run in quasi-polynomial time (assuming constant highway dimension) (Feldmann et al., 2018) [8] or run in FPT time (parameterized by the number of clusters k , the highway dimension, and the approximation factor) (Becker et al., 2018; Braverman et al., 2021) [9,10]. In this paper we show that a polynomial-time approximation scheme (PTAS) exists (assuming constant highway dimension). We also show that the considered problems are NP-hard on graphs of highway dimension 1. [ABSTRACT FROM AUTHOR]