On Subexponential Parameterized Algorithms for Steiner Tree on Intersection Graphs of Geometric Objects

TL;DR

The paper analyzes Steiner Tree on intersection graphs of geometric objects, proving ETH-based lower bounds that rule out subexponential in k+t time for unit disks and unit squares, while delivering n^{O(√t)}-time algorithms for broad classes including disks of arbitrary size, axis-aligned squares of arbitrary size, and similarly-sized fat polygons. It develops a unified framework combining a planarity-reduction, a disjoint representation of possibly overlapping solutions, and Voronoi-separator techniques to drive subexponential-time algorithms parameterized by the number of terminals. The work also establishes sharp lower bounds that limit extending the object classes (e.g., to almost-squares or near-rectangles) and shows these bounds are essentially tight under ETH. Overall, it generalizes and unifies subexponential parameterized results for geometric Steiner Tree beyond planar graphs, while clearly delineating the boundaries imposed by ETH on possible extensions.

Abstract

We study the Steiner Tree problem on the intersection graph of most natural families of geometric objects, e.g., disks, squares, polygons, etc. Given a set of $n$ objects in the plane and a subset $T$ of $t$ terminal objects, the task is to find a subset $S$ of $k$ objects such that the intersection graph of $S\cup T$ is connected. Given how typical parameterized problems behave on planar graphs and geometric intersection graphs, we would expect that exact algorithms with some form of subexponential dependence on the solution size or the number of terminals exist. Contrary to this expectation, we show that, assuming the Exponential-Time Hypothesis (ETH), there is no $2^{o(k+t)}\cdot n^{O(1)}$ time algorithm even for unit disks or unit squares, that is, there is no FPT algorithm subexponential in the size of the Steiner tree. However, subexponential dependence can appear in a different form: we show that Steiner Tree can be solved in time $n^{O(\sqrt{t})}$ for many natural classes of objects, including: Disks of arbitrary size. Axis-parallel squares of arbitrary size. Similarly-sized fat polygons. This in particular significantly improves and generalizes two recent results: (1) Steiner Tree on unit disks can be solved in time $n^{\Oh(\sqrt{k + t})}$ (Bhore, Carmi, Kolay, and Zehavi, Algorithmica 2023) and (2) Steiner Tree on planar graphs can be solved in time $n^{O(\sqrt{t})}$ (Marx, Pilipczuk, and Pilipczuk, FOCS 2018). We complement our algorithms with lower bounds that demonstrate that the class of objects cannot be significantly extended, even if we allow the running time to be $n^{o(k+t)/\log(k+t)}$.

Figures (11)

• Figure 1: A minimal solution with 10 terminals and 15 objects with degree at least 3.
• Figure 2: A solution with a separator $Q$ that breaks the solution into two disconnected parts.
• Figure 3: Replacing two parts of the solution by alternative solutions that are compatible with the original other part may not result in a solution connecting all three red terminals.
• Figure 4: Representing a solution (red terminals and black non-terminals) with disjoint objects (green).
• Figure 5: An example of a graph $G$ with the leaves $T$ depicted in red, such that $|T| = 3$ and $V_{\geqslant 3}(G) = 3 = 3 \cdot \left( |T| - 2\right)$. It is easy to verify that $T$ is a critically connected set.

Theorems & Definitions (117)

• Theorem 1.1: DBLP:conf/focs/MarxPP18
• Theorem 1.2: DBLP:journals/siamcomp/FominLMPPS22
• Theorem 1.3: DBLP:conf/focs/MarxPP18
• Theorem 1.4
• Theorem 1.5: DBLP:journals/algorithmica/BhoreCKZ23
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Corollary 1.10