Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d.

💡 Research Summary

The paper extends the powerful framework of bidimensionality, originally devised for H‑minor‑free graph families, to geometric graph classes that are not closed under taking minors, namely map graphs and unit‑disk graphs. The authors observe that these classes can contain arbitrarily large cliques, which prevents them from having the “truly sublinear treewidth” property required by the classic bidimensionality machinery. To overcome this obstacle they introduce a “clique‑cleaning” preprocessing step: large cliques are identified and either removed or handled separately, after which the remaining graph is K_t‑free. For K_t‑free unit‑disk and map graphs they prove that the maximum degree is bounded by O(t) and that the treewidth grows only as O(|X|^λ) for some λ<1 when a set X of vertices is added. This establishes truly sublinear treewidth for the cleaned graphs.

With this structural insight, the authors can apply the same pipeline used in earlier bidimensionality results: (1) obtain a treewidth‑bounded subgraph, (2) formulate the target problem as a CMSO (Counting Monadic Second‑Order) expression, (3) solve it by dynamic programming on the tree decomposition, and (4) lift the solution back to the original graph while incurring only a small additive error proportional to the size of the removed cliques. They show that the problems Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, Cycle Packing, and Minimum‑Vertex Feedback Edge Set are all minor‑bidimensional, satisfy a suitable separability property, and are reducible in the sense required by the framework.

Consequently they obtain, for both map graphs and unit‑disk graphs, (1+ε)‑approximation schemes whose running time is f(ε)·n^{O(1)} (EPTAS) and parameterized algorithms with running time 2^{O(√k)}·n^{O(1)} (sub‑exponential FPT) where k is the solution size. The algorithms are “robust”: they work directly on the combinatorial representation of the input graph and do not need an explicit geometric embedding, which is important because recognizing unit‑disk graphs is NP‑hard and recognizing map graphs has a prohibitive polynomial exponent.

The paper also investigates the limits of this approach. For unit‑ball graphs in ℝ³ (intersection graphs of equal‑radius balls in three dimensions) they prove that Feedback Vertex Set does not admit a PTAS unless P=NP and does not admit a sub‑exponential parameterized algorithm unless the Exponential Time Hypothesis fails. This negative result stems from the fact that even K_t‑free unit‑ball graphs can have linear treewidth, breaking the truly sublinear treewidth condition. Similarly, they show that disk graphs (intersection graphs of disks with arbitrary radii) do not enjoy truly sublinear treewidth even when K₄‑free, implying that extending the current technique to disk graphs would require new decomposition theorems. Nevertheless, they demonstrate that the EPTAS and sub‑exponential algorithms for Vertex Cover and Connected Vertex Cover can be lifted to disk graphs and to unit‑ball graphs in any fixed dimension d.

In summary, the work provides a unified, geometry‑agnostic method to obtain efficient approximation schemes and fast parameterized algorithms for a suite of classic combinatorial problems on map and unit‑disk graphs, while also delineating clear boundaries where the bidimensionality‑based approach breaks down. The results broaden the applicability of bidimensionality beyond minor‑closed families and open several avenues for future research, including more powerful clique‑cleaning techniques and new structural decompositions for richer geometric graph classes.