arXiv subjects

Alexander Leonhardt

Publications and source records attributed to Alexander Leonhardt.

4 recordsLinked to original sources

arXiv2026

On the Complexity of the Minimum-($k,\rho$)-Shortcut Problem

We consider the Minimum-$(k,\rho)$-$\mathrm{Shortcut}$ problem ($\min(k,\rho)\text{-}\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $\rho$ closest vertices using paths of at most $k$ edges. This is a fundamental graph optimization problem used to accelerate parallel shortest path algorithms. It is well-known that the problem is trivially solvable for the cases $k=1$ and $k\geq\rho$. While recent work by Leonhardt, Meyer, and Penschuck (ESA 2024) showed that in undirected graphs $\min(k,\rho)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq 3$ if $\rho=\Theta(n^\epsilon)$, the boundary where the problem transitions from polynomial-time solvable to NP-hard remained open. In this paper, we narrow this gap significantly. We present a simpler and more direct reduction from the Hitting Set problem which establishes that $\min(k,\rho)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq2$ and $\rho\geq k+2$ in both directed and undirected graphs. Complementing this, we use the symmetry of the undirected case to show that $\rho=k+1$ is solvable in polynomial time, a regime where the directed version remains a candidate for NP-hardness. Therefore, we obtain an almost complete characterization of the complexity of $\min(k,\rho)\text{-}\mathrm{Shortcut}$, with the sole remaining open case being $\rho = k+1$ in the directed setting.

cs.CC↗

arXiv2025

Revisiting a Successful Reduction Rule for Dominating Set

Given a graph $G = (V, E)$ with $n$ vertices and $m$ edges, the DominatingSet problem asks for a set $D \subseteq V$ of minimal cardinality such that every vertex either is in $D$ or adjacent to a member of $D$. Although there is little hope for a kernelization algorithm on general graphs due to the W[2]-hardness of DominatingSet, data reduction rules are extensively used in practice. In this context, Rule1 due to Alber, Fellows, and Niedermeier [JACM 2004] has been shown to be very powerful, yet its best-known running time is $\mathcal{O}(n^3)$ ($= \mathcal{O}(nm)$) for general graphs. In this work, we propose, to the best of our knowledge, the first $\mathcal{O}(n + m)$-time algorithm for Rule1 on general graphs. We additionally propose simple, but practically significant, extensions to our algorithmic framework to further prune the input instances. We complement our theoretical claims with experiments that confirm the practicality of our approach. On average, we see significant speedups of over one order of magnitude while removing $59.8\times$ more nodes and $410.9\times$ more edges than the original formulation across a large dataset comprised of real-world and synthetic networks.

cs.DS↗

arXiv2024

Efficient Uniform Negative Edge Weights

We consider a maximum entropy edge weight model that allows for negative weights. Given a graph $G$ and possible weights $\mathcal{W}$ typically consisting of positive and negative values, the model selects edge weights $w \in \mathcal{W}^m$ uniformly at random from all weights that do not introduce a negative cycle. We propose an MCMC process and show that it converges to the required distribution. We then engineer an implementation of the process using a dynamic version of Johnson's algorithm in connection with a bidirectional Dijkstra search as well as an innovative resampling method. We empirically study the performance characteristics of these novel sampling algorithms as well as the output produced by the model.

cs.DS↗

arXiv2024

Insights into $(k,\rho)$-shortcutting algorithms

A graph is called a $(k,\rho)$-graph iff every node can reach $\rho$ of its nearest neighbors in at most k hops. This property proved useful in the analysis and design of parallel shortest-path algorithms. Any graph can be transformed into a $(k,\rho)$-graph by adding shortcuts. Formally, the $(k,\rho)$-Minimum-Shortcut problem asks to find an appropriate shortcut set of minimal cardinality. We show that the $(k,\rho)$-Minimum-Shortcut problem is NP-complete in the practical regime of $k \ge 3$ and $\rho = \Theta(n^\epsilon)$ for $\epsilon > 0$. With a related construction, we bound the approximation factor of known $(k,\rho)$-Minimum-Shortcut problem heuristics from below and propose algorithmic countermeasures improving the approximation quality. Further, we describe an integer linear problem (ILP) solving the $(k,\rho)$-Minimum-Shortcut problem optimally. Finally, we compare the practical performance and quality of all algorithms in an empirical campaign.

cs.DS↗