57 research outputs found
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
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
Backdoors to q-Horn
The class q-Horn, introduced by Boros, Crama and Hammer in 1990, is one of the largest known classes of propositional CNF formulas for which satisfiability can be decided in polynomial time. This class properly contains the fundamental classes of Horn and Krom formulas as well as the class of renamable (or disguised) Horn formulas.
In this paper we extend this class so that its favorable algorithmic properties can be made accessible to formulas that are outside but "close"\u27 to this class. We show that deciding satisfiability is fixed-parameter tractable parameterized by the distance of the given formula from q-Horn. The distance is measured by the smallest number of variables that we need to delete from the formula in order to get a q-Horn formula, i.e., the size of a smallest deletion backdoor set into the class q-Horn.
This result generalizes known fixed-parameter tractability results for satisfiability decision with respect to the parameters distance from Horn, Krom, and renamable Horn
Kernels for Feedback Arc Set In Tournaments
A tournament T=(V,A) is a directed graph in which there is exactly one arc
between every pair of distinct vertices. Given a digraph on n vertices and an
integer parameter k, the Feedback Arc Set problem asks whether the given
digraph has a set of k arcs whose removal results in an acyclic digraph. The
Feedback Arc Set problem restricted to tournaments is known as the k-Feedback
Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear
vertex kernel for k-FAST. That is, we give a polynomial time algorithm which
given an input instance T to k-FAST obtains an equivalent instance T' on O(k)
vertices. In fact, given any fixed e>0, the kernelized instance has at most
(2+e)k vertices. Our result improves the previous known bound of O(k^2) on the
kernel size for k-FAST. Our kernelization algorithm solves the problem on a
subclass of tournaments in polynomial time and uses a known polynomial time
approximation scheme for k-FAST
A Relaxation of the Directed Disjoint Paths Problem: A Global Congestion Metric Helps
In the Directed Disjoint Paths problem, we are given a digraph and a set
of requests , and the task is to find a
collection of pairwise vertex-disjoint paths such that
each is a path from to in . This problem is NP-complete
for fixed and W[1]-hard with parameter in DAGs. A few positive
results are known under restrictions on the input digraph, such as being planar
or having bounded directed tree-width, or under relaxations of the problem,
such as allowing for vertex congestion. Positive results are scarce, however,
for general digraphs. In this article we propose a novel global congestion
metric for the problem: we only require the paths to be "disjoint enough", in
the sense that they must behave properly not in the whole graph, but in an
unspecified part of size prescribed by a parameter. Namely, in the Disjoint
Enough Directed Paths problem, given an -vertex digraph , a set of
requests, and non-negative integers and , the task is to find a
collection of paths connecting the requests such that at least vertices of
occur in at most paths of the collection. We study the parameterized
complexity of this problem for a number of choices of the parameter, including
the directed tree-width of . Among other results, we show that the problem
is W[1]-hard in DAGs with parameter and, on the positive side, we give an
algorithm in time and a kernel of
size in general digraphs. This latter
result has consequences for the Steiner Network problem: we show that it is FPT
parameterized by the number of terminals and , where and
is the size of the solution.Comment: 25 pages, 9 figure
Testing first-order properties for subclasses of sparse graphs
We present a linear-time algorithm for deciding first-order (FO) properties
in classes of graphs with bounded expansion, a notion recently introduced by
Nesetril and Ossona de Mendez. This generalizes several results from the
literature, because many natural classes of graphs have bounded expansion:
graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs
of bounded degree, graphs with no subgraph isomorphic to a subdivision of a
fixed graph, and graphs that can be drawn in a fixed surface in such a way that
each edge crosses at most a constant number of other edges. We deduce that
there is an almost linear-time algorithm for deciding FO properties in classes
of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a
fixed class of graphs of bounded expansion. After a linear-time initialization
the data structure allows us to test an FO property in constant time, and the
data structure can be updated in constant time after addition/deletion of an
edge, provided the list of possible edges to be added is known in advance and
their simultaneous addition results in a graph in the class. All our results
also hold for relational structures and are based on the seminal result of
Nesetril and Ossona de Mendez on the existence of low tree-depth colorings
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