303 research outputs found
Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts
A skew-symmetric graph is a directed graph with an
involution on the set of vertices and arcs. In this paper, we
introduce a separation problem, -Skew-Symmetric Multicut, where we are given
a skew-symmetric graph , a family of of -sized subsets of
vertices and an integer . The objective is to decide if there is a set
of arcs such that every set in the family has a vertex
such that and are in different connected components of
. In this paper, we give an algorithm for
this problem which runs in time , where is the
number of arcs in the graph, the number of vertices and the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time and
respectively. This resolves an open problem posed by Reed,
Smith and Vetta [Operations Research Letters, 2003] and improves upon the
earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].
We also show that Deletion q-Horn Backdoor Set Detection is a special case of
3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor
Set Detection which runs in time . This gives the first
fixed-parameter tractable algorithm for this problem answering a question posed
in a paper by a superset of the authors [STACS, 2013]. Using this result, we
get an algorithm for Satisfiability which runs in time where
is the size of the smallest q-Horn deletion backdoor set, with being
the length of the input formula
NodeTrix Planarity Testing with Small Clusters
We study the NodeTrix planarity testing problem for flat clustered graphs
when the maximum size of each cluster is bounded by a constant . We consider
both the case when the sides of the matrices to which the edges are incident
are fixed and the case when they can be chosen arbitrarily. We show that
NodeTrix planarity testing with fixed sides can be solved in
time for every flat clustered graph that can be
reduced to a partial 2-tree by collapsing its clusters into single vertices. In
the general case, NodeTrix planarity testing with fixed sides can be solved in
time for , but it is NP-complete for any . NodeTrix
planarity testing remains NP-complete also in the free sides model when .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Colorings of oriented planar graphs avoiding a monochromatic subgraph
For a fixed simple digraph and a given simple digraph , an -free
-coloring of is a vertex-coloring in which no induced copy of in
is monochromatic. We study the complexity of deciding for fixed and
whether a given simple digraph admits an -free -coloring. Our main focus
is on the restriction of the problem to planar input digraphs, where it is only
interesting to study the cases . From known results it follows
that for every fixed digraph whose underlying graph is not a forest, every
planar digraph admits an -free -coloring, and that for every fixed
digraph with , every oriented planar graph admits an
-free -coloring.
We show in contrast, that
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring.
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring
Machine Learning Techniques as Applied to Discrete and Combinatorial Structures
Machine Learning Techniques have been used on a wide array of input types: images, sound waves, text, and so forth. In articulating these input types to the almighty machine, there have been all sorts of amazing problems that have been solved for many practical purposes.
Nevertheless, there are some input types which don’t lend themselves nicely to the standard set of machine learning tools we have. Moreover, there are some provably difficult problems which are abysmally hard to solve within a reasonable time frame.
This thesis addresses several of these difficult problems. It frames these problems such that we can then attempt to marry the allegedly powerful utility of existing machine learning techniques to the practical solvability of said problems
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