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ABSTRACT: We study the parameterized complexity of separating a small set of vertices
from a graph by a small vertex-separator. That is, given a graph $G$ and
integers $k$, $t$, the task is to find a vertex set $X$ with $|X| \le k$ and
$|N(X)| \le t$. We show that
- the problem is fixed-parameter tractable (FPT) when parameterized by $t$
but W[1]-hard when parameterized by $k$, and
- a terminal variant of the problem, where $X$ must contain a given vertex
$s$, is W[1]-hard when parameterized either by $k$ or by $t$ alone, but is FPT
when parameterized by $k + t$.
We also show that if we consider edge cuts instead of vertex cuts, the
terminal variant is NP-hard and FPT parameterized by $k+t$.
04/2013;
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ABSTRACT: We give an algorithm that for an input n-vertex graph G and integer k>0, in
time 2^[O(k)]n either outputs that the treewidth of G is larger than k, or
gives a tree decomposition of G of width at most 5k+4. This is the first
algorithm providing a constant factor approximation for treewidth which runs in
time single-exponential in k and linear in n. Treewidth based computations are
subroutines of numerous algorithms. Our algorithm can be used to speed up many
such algorithms to work in time which is single-exponential in the treewidth
and linear in the input size.
04/2013;
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ABSTRACT: The behavior of users in social networks is often observed to be affected by
the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp}
introduced a formal mathematical model for user engagement in social networks
where each individual derives a benefit proportional to the number of its
friends which are engaged. Given a threshold degree $k$ the equilibrium for
this model is a maximal subgraph whose minimum degree is $\geq k$. However the
dropping out of individuals with degrees less than $k$ might lead to a
cascading effect of iterated withdrawals such that the size of equilibrium
subgraph becomes very small. To overcome this some special vertices called
"anchors" are introduced: these vertices need not have large degree. Bhawalkar
et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored $k$-Core}
problem: Given a graph $G$ and integers $b, k$ and $p$ do there exist a set of
vertices $B\subseteq H\subseteq V(G)$ such that $|B|\leq b, |H|\geq p$ and
every vertex $v\in H\setminus B$ has degree at least $k$ is the induced
subgraph $G[H]$. They showed that the problem is NP-hard for $k\geq 2$ and gave
some inapproximability and fixed-parameter intractability results. In this
paper we give improved hardness results for this problem. In particular we show
that the \textsc{Anchored $k$-Core} problem is W[1]-hard parameterized by $p$,
even for $k=3$. This improves the result of Bhawalkar et al.
\cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by $b$) as our
parameter is always bigger since $p\geq b$. Then we answer a question of
Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored
$k$-Core} problem remains NP-hard on planar graphs for all $k\geq 3$, even if
the maximum degree of the graph is $k+2$. Finally we show that the problem is
FPT on planar graphs parameterized by $b$ for all $k\geq 7$.
04/2013;
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ABSTRACT: Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012] introduced
the Anchored k-Core problem, where the task is for a given graph G and integers
b, k, and p to find an induced subgraph H with at least p vertices (the core)
such that all but at most b vertices (called anchors) of H are of degree at
least k. In this paper, we extend the notion of k-core to directed graphs and
provide a number of new algorithmic and complexity results for the directed
version of the problem. We show that
- The decision version of the problem is NP-complete for every k>=1 even if
the input graph is restricted to be a planar directed acyclic graph of maximum
degree at most k+2.
- The problem is fixed parameter tractable (FPT) parameterized by the size of
the core p for k=1, and W[1]-hard for k>=2.
- When the maximum degree of the graph is at most \Delta, the problem is FPT
parameterized by p+\Delta if k>= \Delta/2.
04/2013;
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ABSTRACT: An undirected graph is Eulerian if it is connected and all its vertices are
of even degree. Similarly, a directed graph is Eulerian, if for each vertex its
in-degree is equal to its out-degree. It is well known that Eulerian graphs can
be recognized in polynomial time while the problems of finding a maximum
Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this
paper, we study the parameterized complexity of the following Euler subgraph
problems:
- Large Euler Subgraph: For a given graph G and integer parameter k, does G
contain an induced Eulerian subgraph with at least k vertices?
- Long Circuit: For a given graph G and integer parameter k, does G contain
an Eulerian subgraph with at least k edges?
Our main algorithmic result is that Large Euler Subgraph is fixed parameter
tractable (FPT) on undirected graphs. We find this a bit surprising because the
problem of finding an induced Eulerian subgraph with exactly k vertices is
known to be W[1]-hard. The complexity of the problem changes drastically on
directed graphs. On directed graphs we obtained the following complexity
dichotomy: Large Euler Subgraph is NP-hard for every fixed k>3 and is solvable
in polynomial time for k<=3. For Long Circuit, we prove that the problem is FPT
on directed and undirected graphs.
04/2013;
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ABSTRACT: We give two algorithms computing representative families of linear and
uniform matroids and demonstrate how to use representative families for
designing single-exponential parameterized and exact exponential time
algorithms. The applications of our approach include
- LONGEST DIRECTED CYCLE
- MINIMUM EQUIVALENT GRAPH (MEG)
- Algorithms on graphs of bounded treewidth
-k-PATH, k-TREE, and more generally, k-SUBGRAPH ISOMORPHISM, where the
k-vertex pattern graph is of constant treewidth.
04/2013;
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ABSTRACT: We give the first linear kernels for {\sc Dominating Set} and {\sc Connected
Dominating Set} problems on graphs excluding a fixed graph $H$ as a topological
minor. In other words, we give polynomial time algorithms that, for a given
$H$-topological-minor free graph $G$ and a positive integer $k$, output an
$H$-topological-minor free graph $G'$ on $\cO(k)$ vertices such that $G$ has a
(connected) dominating set of size $k$ if and only if $G'$ has.
Our results extend the known classes of graphs on which {\sc Dominating Set}
and {\sc Connected Dominating Set} problems admit linear kernels. The
kernelization algorithm is based on a non-trivial combination of the following
ingredients \bullet The structural theorem of Grohe and Marx [STOC 2012] for
graphs excluding a fixed graph $H$ as a topological subgraph; \bullet A novel
notion of protrusions, different that the one defined in [FOCS 2009]; \bullet
Reinterpretations of reduction techniques developed for kernelization
algorithms for {\sc Dominating Set} and {\sc Connected Dominating Set} from
[SODA 2012].
A protrusion is a subgraph of constant treewidth separated from the remaining
vertices by a constant number of vertices.
Roughly speaking, in the new notion of protrusion instead of demanding the
subgraph of being of constant treewidth, we ask it to contain a {\sl constant}
number of vertices from a solution. We believe that the new notion of
protrusion will be useful in many other algorithmic settings.
09/2012;
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ABSTRACT: We prove a number of results around kernelization of problems parameterized
by a vertex cover of the input graph. We provide two simple general conditions
characterizing problems admitting kernels of polynomial size. Our
characterizations not only give generic explanations for the existence of many
known polynomial kernels for problems like Odd Cycle Transversal, Chordal
Deletion, Eta Transversal, Long Path, Long Cycle, or H-Packing, parameterized
by the size of a vertex cover, they also imply new polynomial kernels for
problems like F-Minor-Free Deletion, which is to delete at most k vertices to
obtain a graph with no minor from a fixed finite set F.
While our characterization captures many interesting problems, the
kernelization complexity landscape of problems parameterized by vertex cover is
much more involved. We demonstrate this by several results about induced
subgraph and minor containment, which we find surprising. While it was known
that testing for an induced complete subgraph has no polynomial kernel unless
NP is contained in coNP/poly, we show that the problem of testing if a graph
contains a given complete graph on t vertices as a minor admits a polynomial
kernel. On the other hand, it was known that testing for a path on t vertices
as a minor admits a polynomial kernel, but we show that testing for containment
of an induced path on t vertices is unlikely to admit a polynomial kernel.
06/2012;
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ABSTRACT: In this paper we present branching algorithms for infinite classes of problems.
The novelty in the design and analysis of our branching algorithms lies in the fact that the weights are redistributed over
the graph by the algorithms. Our particular setting to make this idea work is a combination of a branching approach with a
recharging mechanism. We call it Branch & Recharge. To demonstrate this approach we consider a generalized domination problem.
Let σ and ϱ be two nonempty sets of nonnegative integers. A vertex subset S⊆V of an undirected graph G=(V(G),E(G)) is called a (σ,ϱ)-dominating set of G if |N(v)∩S|∈σ for all v∈S and |N(v)∩S|∈ϱ for all v∈V∖S. This notion generalizes many domination-type graph invariants.
We present Branch & Recharge algorithms enumerating all (σ,ϱ)-dominating sets of an input graph G in time O
*(c
n
) for some c<2, if σ is successor-free, i.e., it does not contain two consecutive integers, and either both σ and ϱ are finite, or one of them is finite and σ∩ϱ=∅. Our algorithm implies a non trivial upper bound of O
*(c
n
) on the number of (σ,ϱ)-dominating sets in an n-vertex graph under the above conditions on σ and ϱ.
KeywordsExact exponential algorithms–NP-hard problems–Generalized domination–Branch and recharge
Algorithmica 05/2012; 61(2):252-273. · 0.60 Impact Factor
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ABSTRACT: It is known that the problem of deleting at most k vertices to obtain a
proper interval graph (Proper Interval Vertex Deletion) is fixed parameter
tractable. However, whether the problem admits a polynomial kernel or not was
open. Here, we answers this question in affirmative by obtaining a polynomial
kernel for Proper Interval Vertex Deletion. This resolves an open question of
van Bevern, Komusiewicz, Moser, and Niedermeier.
04/2012;
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ABSTRACT: We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be
used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between
this game-theoretic approach and graph decompositions called q
-branched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard)
tree decomposition are two extreme cases of q-branched tree decompositions. The equivalence between nondeterministic graph searching and q-branched tree decomposition enables us to design an exact (exponential time) algorithm computing q-branched treewidth for all q≥0, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm
for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required
to search a graph with the minimum number of searchers.
Algorithmica 04/2012; 53(3):358-373. · 0.60 Impact Factor
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ABSTRACT: The Hamiltonian Cycle problem is the problem of deciding whether an n-vertex graph G has a cycle passing through all vertices of G. This problem is a classic NP-complete problem. Finding an exact algorithm that solves it in O*(an){\mathcal {O}}^{*}(\alpha^{n}) time for some constant α<2 was a notorious open problem until very recently, when Björklund presented a randomized algorithm that uses O*(1.657n){\mathcal {O}}^{*}(1.657^{n}) time and polynomial space. The Longest Cycle problem, in which the task is to find a cycle of maximum length, is a natural generalization of the Hamiltonian Cycle problem. For a claw-free graph G, finding a longest cycle is equivalent to finding a closed trail (i.e., a connected even subgraph, possibly consisting of
a single vertex) that dominates the largest number of edges of some associated graph H. Using this translation we obtain two deterministic algorithms that solve the Longest Cycle problem, and consequently the Hamiltonian Cycle problem, for claw-free graphs: one algorithm that uses O*(1.6818n){\mathcal {O}}^{*}(1.6818^{n}) time and exponential space, and one algorithm that uses O*(1.8878n){\mathcal {O}}^{*}(1.8878^{n}) time and polynomial space.
Algorithmica 04/2012; · 0.60 Impact Factor
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ABSTRACT: We study the problem of determining the spanning tree congestion of agraph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free
graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the
problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed
k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe
that for k≤3 the problem becomes polynomially time solvable.
KeywordsSpanning tree congestion–Graph minor–Parameterized algorithms–Apex graph
Algorithmica 04/2012; · 0.60 Impact Factor
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ABSTRACT: We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our
approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined
techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach
we show how to obtain an
O(26.903Ön)O(2^{6.903\sqrt{n}})
time algorithm solving weighted Hamiltonian Cycle on an n-vertex planar graph. Similar technique solves Planar Graph Travelling Salesman Problem with n cities in time
O(29.8594Ön)O(2^{9.8594\sqrt{n}})
. Our approach can be used to design parameterized algorithms as well. For example, we give an algorithm that for a given
k decides if a planar graph on n vertices has a cycle of length at least k in time
O(213.6Ökn+n3)O(2^{13.6\sqrt{k}}n+n^{3})
.
KeywordsExact and parameterized algorithms-Planar graphs-Treewidth-Branchwidth-Traveling salesman problem-Hamiltonian cycle
Algorithmica 04/2012; 58(3):790-810. · 0.60 Impact Factor
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ABSTRACT: Cops & Robber is a classical pursuit-evasion game on undirected graphs, where the task is to identify the minimum number of cops sufficient to catch the robber. In this paper, we investigate the changes in problem's complexity and combinatorial properties with constraining the following natural game parameters • Fuel: The number of steps each cop can make; • Cost: The total sum of steps along edges all cops can make; • Time: The number of rounds of the game.
SIAM Journal on Discrete Mathematics 01/2012; 26:571-590. · 0.65 Impact Factor
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Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011; 01/2012
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29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France; 01/2012
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Theory Comput. Syst. 01/2012; 50:611-620.
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Theory Comput. Syst. 01/2012; 50:420-432.
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Algorithmica. 01/2012; 63:692-706.
