164 research outputs found
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
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
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require 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 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
Graph Partitioning With Input Restrictions
In this thesis we study the computational complexity of a number of graph
partitioning problems under a variety of input restrictions. Predominantly,
we research problems related to Colouring in the case where the input
is limited to hereditary graph classes, graphs of bounded diameter or some
combination of the two.
In Chapter 2 we demonstrate the dramatic eect that restricting our
input to hereditary graph classes can have on the complexity of a decision
problem. To do this, we show extreme jumps in the complexity of three
problems related to graph colouring between the class of all graphs and every
other hereditary graph class.
We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be
H-free for some graph H if it contains no induced subgraph isomorphic to
H. Similarly, G is said to be H-free for some set of graphs H, if it does not
contain any graph in H as an induced subgraph. Here, the set H consists
usually of a single cycle or tree but may also contain a number of cycles, for
example we give results for graphs of bounded diameter and girth.
Chapter 4 is dedicated to three variants of the Colouring problem,
Acyclic Colouring, Star Colouring, and Injective Colouring.
We give complete or almost complete dichotomies for each of these decision
problems restricted to H-free graphs.
In Chapter 5 we study these problems, along with three further variants of
3-Colouring, Independent Odd Cycle Transversal, Independent
Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of
bounded diameter.
Finally, Chapter 6 deals with a dierent variety of problems. We study
the problems Disjoint Paths and Disjoint Connected Subgraphs for
H-free graphs
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM
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