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

Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure

Hyperbolic intersection graphs and (quasi)-polynomial time

We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in $d$-dimensional hyperbolic space, which we denote by $\mathbb{H}^d$. Using a new separator theorem, we show that unit ball graphs in $\mathbb{H}^d$ enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in $2^{O(n^{1-1/(d-1)})}$ time for any fixed $d\geq 3$, while the same problems need $2^{O(n^{1-1/d})}$ time in $\mathbb{R}^d$. We also show that these algorithms in $\mathbb{H}^d$ are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in $\mathbb{H}^2$, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial ($n^{O(\log n)}$) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and $3$-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in $\mathbb{H}^2$ have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require $2^{\Omega(\sqrt{n})}$ time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching $n^{\Omega(\log n)}$ lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams

We study a general family of facility location problems defined on planar graphs and on the 2-dimensional plane. In these problems, a subset of $k$ objects has to be selected, satisfying certain packing (disjointness) and covering constraints. Our main result is showing that, for each of these problems, the $n^{O(k)}$ time brute force algorithm of selecting $k$ objects can be improved to $n^{O(\sqrt{k})}$ time. The algorithm is based on an idea that was introduced recently in the design of geometric QPTASs, but was not yet used for exact algorithms and for planar graphs. We focus on the Voronoi diagram of a hypothetical solution of $k$ objects, guess a balanced separator cycle of this Voronoi diagram to obtain a set that separates the solution in a balanced way, and then recurse on the resulting subproblems. We complement our study by giving evidence that packing problems have $n^{O(\sqrt{k})}$ time algorithms for a much more general class of objects than covering problems have.Comment: 64 pages, 16 figure

QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails