Leon Kellerhals, Mitja Krebs, André Nichterlein et al.
We study the following problem that is motivated by demand-aware network design: Given a tree~$G$, the task is to find a binary tree~$H$ on the same vertex set. The objective is to minimize the sum of distances in~$H$ between vertex pairs that are adjacent in~$G$. We present a linear-time factor-4 approximation for this problem.
Tom-Lukas Breitkopf, Vincent Froese, Anton Herrmann et al.
We consider the well-studied problem of finding a spanning tree with minimum average distance between vertex pairs (called a MAD tree). This is a classic network design problem which is known to be NP-hard. While approximation algorithms and polynomial-time algorithms for some graph classes are known, the parameterized complexity of the problem has not been investigated so far. We start a parameterized complexity analysis with the goal of determining the border of algorithmic tractability for the MAD tree problem. To this end, we provide a linear-time algorithm for graphs of constant modular width and a polynomial-time algorithm for graphs of bounded treewidth; the degree of the polynomial depends on the treewidth. That is, the problem is in FPT with respect to modular width and in XP with respect to treewidth. Moreover, we show it is in FPT when parameterized by vertex integrity or by an above-guarantee parameter. We complement these algorithms with NP-hardness on split graphs.
Jakob Raupach, Tom-Lukas Breitkopf, Anton Herrmann et al.
We explore the parameterized complexity of Bounded Density Vertex Deletion (BDVD): given a graph $G$, an integer budget $k$, and a target density $τ_ρ$, the task is to determine whether the density (i.e. number of edges divided by number of vertices) of the densest subgraph of $G$ can be reduced to at most $τ_ρ$ by deleting at most $k$ vertices. Our primary focus is on structural graph parameters related to treewidth, as the parameterized complexity of BDVD with respect to treewidth was left as open question by Bazgan et al. [JCSS, 2025]. We resolve this question by showing W[1]-hardness with respect to various parameters, including treedepth and feedback vertex number. These results imply W[1]-hardness with respect to treewidth. We obtain positive results for parameters larger than treedepth and feedback vertex number, namely we show BDVD is in FPT parameterized by the max leaf number or vertex integrity. Under the assumption that the target density $τ_ρ$ is a fixed constant the parameterized complexity landscape of BDVD changes drastically, allowing a fixed-parameter tractable algorithm even for parameters smaller than treewidth, namely cliquewidth. Altogether, our results provide a refined complexity landscape for Bounded Density Vertex Deletion, sharply distinguishing between tractable and intractable parameter regimes under structural parameterizations.
Niclas Boehmer, Robert Bredereck, André Nichterlein
A recent report of Littmann [Commun. ACM '21] outlines the existence and the fatal impact of collusion rings in academic peer reviewing. We introduce and analyze the problem Cycle-Free Reviewing that aims at finding a review assignment without the following kind of collusion ring: A sequence of reviewers each reviewing a paper authored by the next reviewer in the sequence (with the last reviewer reviewing a paper of the first), thus creating a review cycle where each reviewer gives favorable reviews. As a result, all papers in that cycle have a high chance of acceptance independent of their respective scientific merit. We observe that review assignments computed using a standard Linear Programming approach typically admit many short review cycles. On the negative side, we show that Cycle-Free Reviewing is NP-hard in various restricted cases (i.e., when every author is qualified to review all papers and one wants to prevent that authors review each other's or their own papers or when every author has only one paper and is only qualified to review few papers). On the positive side, among others, we show that, in some realistic settings, an assignment without any review cycles of small length always exists. This result also gives rise to an efficient heuristic for computing (weighted) cycle-free review assignments, which we show to be of excellent quality in practice.