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

2025-11-10

AI Summary

This paper studies the Steiner Tree problem on intersection graphs of geometric objects (e.g., disks, squares, convex polygons): given $n$ objects in the plane and a subset $T$ of $t$ terminal objects, find a minimum-size set $S$ of non-terminal objects such that the intersection graph of $S cup T$ is connected. The authors prove, under the Exponential Time Hypothesis (ETH), that no subexponential fixed-parameter tractable (FPT) algorithm exists with respect to either $|S|$ or $t$. However, for several natural geometric families—including arbitrary disks, squares, and convex polygons—they present the first exact algorithms running in $n^{O(sqrt{t})}$ time, significantly generalizing prior results limited to unit-disk and planar graphs. Technically, the approach integrates geometric structural decomposition, divide-and-conquer strategies, and parameterized complexity analysis, complemented by matching lower-bound proofs.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $Scup T$ 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 $n^{o(k+t)/log(k+t)}$.

Problem

• Solving Steiner Tree problem on geometric intersection graphs
• Investigating subexponential algorithms for connected terminal sets
• Establishing complexity bounds for various geometric objects

Innovation

• Developed n^O(√t) algorithm for geometric objects
• Applied subexponential parameterized approach to Steiner Tree
• Extended technique to disks, squares and fat polygons