Julian Christoph Brinkmann

arXiv2026

Faster Algorithms for Deciding the Unbiased Maker-Breaker Triangle Game on General Graphs

In this paper, we present new polynomial-time algorithms for determining the winner of the unbiased triangle game played on the edge set of general graphs. To that end, we propose to view the game through the edge-triangle incidence graph instead of the standard hypergraph model. We identify a necessary and sufficient winning condition for Maker in terms of the edge-triangle incidence graph and show that winning strategies achieving this condition as fast as possible play monotonically in the sense that they only consider monotonically decreasing connected subgraphs of the edge-triangle incidence graph. We give three different algorithms for different classes of graphs. For general graphs $G$, the outcome of the unbiased triangle game can be decided in time $\mathcal{O}(n+m^{3.5})=\mathcal{O}(n^7)$. This significantly improves on the $\mathcal{O}(n^{16})$ algorithm implied by the work of Galliot, Gravier and Sivignon (arXiv 2022). For graphs $G$ which contain $K_4$, the complete graph on four vertices, as a subgraph and whose edge-triangle incidence graph is connected, the winner can be decided in time $\mathcal{O}(n+\min\{n^{ω+1},m^2\})=\mathcal{O}(n^{ω+1})$, where $ω<2.372$ is the exponent of matrix multiplication (Alman et. al., SODA 2025). For graphs $G$ whose edge-triangle incidence graph is a cactus graph, i.e. all its cycles are edge-disjoint, the winner can be decided in time $\mathcal{O}(n+m^{1.5})=\mathcal{O}(n^3)$. Such $G$ are $K_4$-free. The algorithms for the special cases are based on novel structural characterizations of Maker's win for each graph class. We also give a linear time reduction from triangle detection to deciding the unbiased triangle game.

cs.CC↗

arXiv2026

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

We consider the Minimum-$(k,ρ)$-$\mathrm{Shortcut}$ problem ($\min(k,ρ)\text{-}\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $ρ$ 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 $k\geqρ$. While recent work by Leonhardt, Meyer, and Penschuck (ESA 2024) showed that in undirected graphs $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq 3$ if $ρ=Θ(n^ε)$, 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 $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq2$ and $ρ\geq k+2$ in both directed and undirected graphs. Complementing this, we use the symmetry of the undirected case to show that $ρ=k+1$ 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 $\min(k,ρ)\text{-}\mathrm{Shortcut}$, with the sole remaining open case being $ρ= k+1$ in the directed setting.

cs.CC↗