Sándor Kisfaludi-Bak

On the Exact Complexity of Hamiltonian Cycle and q-Colouring in Disk Graphs

We study the exact complexity of the Hamiltonian Cycle and the q-Colouring problem in disk graphs. We show that the Hamiltonian Cycle problem can be solved in \(2^{O(\sqrt{n})}\) on n-vertex disk graphs where the ratio of the largest and smallest disk radius is O(1). We also show that this is optimal: assuming the Exponential Time Hypothesis, there is no \(2^{o(\sqrt{n})}\)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Exponential Time Hypothesis, for any fixed q, q-Colouring does not admit a \(2^{o(\sqrt{n})}\)-time algorithm, even when restricted to unit disk graphs, and it is solvable in \(2^{O(\sqrt{n})}\)-time on disk graphs.

A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs

We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time 2O(n1−1/d) for any fixed dimension d≥ 2 for many well known graph problems, including Independent Set, r-Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower-bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching 2Ω(n1−1/d) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.

On the number of touching pairs in a set of planar curves

Given a set of planar curves (Jordan arcs), each pair of which meets -- either crosses or touches -- exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has

The Homogeneous Broadcast Problem in Narrow and Wide Strips

O(t^2n)

The Dominating Set Problem in Geometric Intersection Graphs

touchings, where

An ETH-Tight Exact Algorithm for Euclidean TSP

t

The complexity of dominating set in geometric intersection graphs

is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.

The homogeneous broadcast problem in narrow and wide strips

Let P be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let s ∈ P be a given source node. Each node p can transmit information to all other nodes within unit distance, provided p is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source s can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem (in the latter, s must be able to reach every node within a specified number of hops), with the restriction that all points lie inside a strip of width w. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width w.