363 research outputs found

### Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

A skew-symmetric graph $(D=(V,A),\sigma)$ is a directed graph $D$ with an
involution $\sigma$ on the set of vertices and arcs. In this paper, we
introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given
a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of
vertices and an integer $k$. The objective is to decide if there is a set
$X\subseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex
$v$ such that $v$ and $\sigma(v)$ are in different connected components of
$D'=(V,A\setminus (X\cup \sigma(X))$. In this paper, we give an algorithm for
this problem which runs in time $O((4d)^{k}(m+n+\ell))$, where $m$ is the
number of arcs in the graph, $n$ the number of vertices and $\ell$ the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time $O(4^kk^4\ell)$ and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and
$O(4^kk^5(m+n))$ 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 $O(12^kk^5\ell)$. 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 $O(12^kk^5\ell)$ where
$k$ is the size of the smallest q-Horn deletion backdoor set, with $\ell$ being
the length of the input formula

### A Linear Time Parameterized Algorithm for Node Unique Label Cover

The optimization version of the Unique Label Cover problem is at the heart of
the Unique Games Conjecture which has played an important role in the proof of
several tight inapproximability results. In recent years, this problem has been
also studied extensively from the point of view of parameterized complexity.
Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable
(FPT) and Wahlstr\"om [SODA 2014] gave an FPT algorithm with an improved
parameter dependence. Subsequently, Iwata, Wahlstr\"om and Yoshida [2014]
proved that the edge version of Unique Label Cover can be solved in linear
FPT-time. That is, there is an FPT algorithm whose dependence on the input-size
is linear. However, such an algorithm for the node version of the problem was
left as an open problem. In this paper, we resolve this question by presenting
the first linear-time FPT algorithm for Node Unique Label Cover

### 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

### Subexponential Parameterized Odd Cycle Transversal on Planar Graphs

In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, the objective is to determine whether there exists a vertex set O in G of size at most k such that G - O is bipartite. Reed, Smith and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with running time 3^kn^{O(1)}. Assuming the exponential time hypothesis of Impagliazzo, Paturi and Zane, the running time can not be improved to 2^{o(k)}n^{O(1)}. We show that OCT admits a randomized algorithm running in O(n^{O(1)} + 2^{O(sqrt{k} log k)}n) time when the input graph is planar. As a byproduct we also obtain a linear time algorithm for OCT on planar graphs with running time O(n^O(1) + 2O( sqrt(k) log k) n) time. This improves over an algorithm of Fiorini et al. [Disc. Appl. Math., 2008]

### Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and
two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth
at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running
time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a
given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is
at least $k$, or: * finds in an isomorphic-invariant way a graph
$\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes $G$ together with a
tree decomposition of $G$ of width $O(k^4)$.
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.Comment: Full version of a paper presented at FOCS 201

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