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

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.

Figures (18)

• Figure 1: (i) Two paths in a unit disk graphs that cannot be shortened: on the blue path from $a$ to $b$, the distance of the disk from $a$ is indicated with blue numbers, and on the red $c\rightarrow d$ path the red number is the distance from $c$. (ii) In a planar graph, if $C$ is a cycle where the distance of vertices on the cycle is less or equal to their graph distance, then we can shorten any path $P$ that intersects $C$ at least twice.
• Figure 2: (i) Adding the shortcut $P$ to the skeleton creates two faces inside $C$. (ii) The final skeleton after adding multiple shortcuts (red). The final orange face has no shortcuts. (iii) A face of the skeleton, with a North-North, a South-South, and a North-South path going through it.
• Figure 3: The brick $B$ inside a skeleton face, bounded by columns starting at $s_i$ and $s_{i+\kappa}$. The solution inside $B$ (purple forest) includes $\mathrm{East}(B)$ and $\mathrm{West}(B)$, and it is connected to the nearest portals (purple circled nodes) along $\mathrm{North}(B)$ and $\mathrm{South}(B)$.
• Figure 4: The graph modification that restricts interactions between bricks to the portals.
• Figure 5: (i) The intersection graph of the 15 red objects and the 3+3 objects on the boundary is a tree (shown by blue). However, the union of this tree and the green boundary is nonplanar, as it contains the subdivision of a $K_{3,3}$. (ii) Replacing the 15 red objects with an object frame of 18 subobjects. Union of the intersection graph of these objects and the boundary is now planar.

Theorems & Definitions (178)

• Theorem 1.1: Planar Subset TSP EPTAS via spanner Klein06
• Theorem 1.2: Planar Steiner Tree EPTAS via spanner BorradaileKM09
• Theorem 1.3: Faster EPTAS for planar Steiner Tree BorradaileKM09
• Theorem 1.4: Geometric Subset TSP EPTAS via spanner
• Theorem 1.5: Geometric Steiner Tree EPTAS via spanner
• Theorem 1.6: Faster EPTAS for geometric Steiner Tree
• Theorem 1.7: APX-hardness results
• Theorem 1.8: Reduction from planar graphs to unit disk graphs
• Lemma 2.0: Bipartite spanner, Klein Klein06
• Theorem 2.1: Contraction Decomposition for Planar Graphs, Demaine DemaineHM10