Approximation Schemes for Subset TSP and Steiner Tree on Geometric Intersection Graphs

2026-03-27

AI Summary

This work addresses the Subset TSP and Steiner Tree problems on unit disk graphs and, more generally, intersection graphs of similarly-sized, connected, fat polygons in the plane. By constructing quality-preserving sparse spanners, the authors reduce these problems to smaller instances and combine dynamic programming with geometric graph techniques to design efficient approximation algorithms. Both problems admit $(1+\varepsilon)$-approximation algorithms running in $2^{\mathrm{poly}(1/\varepsilon)} \cdot n^{O(1)}$ time. This is the first extension of planar graph approximation schemes to such geometric intersection graphs. Moreover, the authors establish the tightness of their results by proving that dropping any one of the assumptions—similar size, connectivity, or fatness—renders the problems APX-hard.

Abstract

We give approximation schemes for Subset TSP and Steiner Tree on unit disk graphs, and more generally, on intersection graphs of similarly sized connected fat (not necessarily convex) polygons in the plane. As a first step towards this goal, we prove spanner-type results: finding an induced subgraph of bounded size that is $(1+\varepsilon)$-equivalent to the original instance in the sense that the optimum value increases only by a factor of at most $(1+\varepsilon)$ when the solution can use only the edges in this subgraph. - For Subset TSP, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $\mathrm{poly}(1/\varepsilon)\cdot\mathrm{OPT}$ in polynomial time, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. - For Steiner Tree, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $2^{\mathrm{poly}(1/\varepsilon)}\cdot\mathrm{OPT}$ in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{2^{\mathrm{poly}(1/\varepsilon)}}\cdot n^{O(1)}$. - An improved algorithm finds a $(1+\varepsilon)$-approximate solution for Steiner Tree in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. An easy reduction shows that approximation schemes for unit disks imply approximation schemes for planar graphs. Thus our results are far-reaching generalizations of analogous results of Klein [STOC'06] and Borradaile, Klein, and Mathieu [ACM TALG'09] for Subset TSP and Steiner Tree in planar graphs. We show that our results are best possible in the sense that dropping any of (i) similarly sized, (ii) connected, or (iii) fat makes both problems APX-hard.

Problem

• Subset TSP
• Steiner Tree
• geometric intersection graphs
• approximation schemes
• unit disk graphs

Innovation

• approximation schemes
• geometric intersection graphs
• spanner
• Subset TSP
• Steiner Tree