Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs

TL;DR

This work has shown that many classic graph problems—Independent Set, Dominating Set, Hamiltonian Cycle, and more—can be solved in subexponential time on unit-ball graphs in \(\mathbb {R}^d\), which is tight under ETH.

Abstract

Recently it was shown that many classic graph problems -- Independent Set, Dominating Set, Hamiltonian Cycle, and more -- can be solved in subexponential time on unit-ball graphs. More precisely, these problems can be solved in $2^{O(n^{1-1/d})}$ time on unit-ball graphs in $\mathbb R^d$, which is tight under ETH. The result can be generalized to intersection graphs of similarly-sized fat objects. For Independent Set the same running time can be achieved for non-similarly-sized fat objects, and for the weighted version of the problem. We show that such generalizations most likely are not possible for Dominating Set: assuming ETH, we prove that - there is no algorithm with running time $2^{o(n)}$ for Dominating Set on (non-unit) ball graphs in $\mathbb R^3$; - there is no algorithm with running time $2^{o(n)}$ for Weighted Dominating Set on unit-ball graphs in $\mathbb R^3$; - there is no algorithm with running time $2^{o(n)}$ for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.

Figures (6)

• Figure 1: An intersection graph of a set of disks in the plane.
• Figure 2: Realizing a split graph as an intersection graph of fat objects. The circles $\gamma_{\mathrm{in}}$ and $\gamma_{\mathrm{out}}$, and the points $p_i$, are depicted in grey. For clarity, the objects $\hbox{obj}(a_i)$ for $i\neq 6$ have been omitted.
• Figure 3: Top: A variable gadget and a literal gadget. Bottom: The graph $\mathcal{G}_\phi$ for the formula $\phi := (x_1\vee \neg x_2\vee x_3)\wedge (x_4\vee \neg x_1\vee x_2) \wedge (\neg x_2 \vee \neg x_4) \wedge (\neg x_3\vee \neg x_2\vee x_4)$; edges in the clique on the vertices $l_i^1$ are omitted for clarity.
• Figure 4: Cross-section of the construction in the plane $x=0$ (left) and $y=0$ (right). Note that the large balls $l_i^1$ have their centers outside these planes. On the left, we have a literal $\neg x_j$ that occurs twice.
• Figure 5: The weighted graph $\mathcal{G}_{\phi}$. Vertices in each group $C_i$ form a clique. Large nodes have infinite weight, the dummies $d_4$ and $d_6$ have weight $0$, and all other nodes have weight $1$.

Theorems & Definitions (9)

• theorem 1
• proposition 1
• lemma 1
• lemma 2
• lemma 3
• theorem 2
• lemma 4
• lemma 5
• theorem 3