366 research outputs found

### Finding Induced Subgraphs via Minimal Triangulations

Potential maximal cliques and minimal separators are combinatorial objects
which were introduced and studied in the realm of minimal triangulations
problems including Minimum Fill-in and Treewidth. We discover unexpected
applications of these notions to the field of moderate exponential algorithms.
In particular, we show that given an n-vertex graph G together with its set of
potential maximal cliques Pi_G, and an integer t, it is possible in time |Pi_G|
* n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a
given graph F of treewidth t, to decide if G contains an induced subgraph
isomorphic to F. Combined with an improved algorithm enumerating all potential
maximal cliques in time O(1.734601^n), this yields that both problems are
solvable in time 1.734601^n * n^(O(t)).Comment: 14 page

### Long Circuits and Large Euler Subgraphs

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

### 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

### Preventing Unraveling in Social Networks Gets Harder

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 $k\geq 7$.Comment: To appear in AAAI 201

### Approximating acyclicity parameters of sparse hypergraphs

The notions of hypertree width and generalized hypertree width were
introduced by Gottlob, Leone, and Scarcello in order to extend the concept of
hypergraph acyclicity. These notions were further generalized by Grohe and
Marx, who introduced the fractional hypertree width of a hypergraph. All these
width parameters on hypergraphs are useful for extending tractability of many
problems in database theory and artificial intelligence. In this paper, we
study the approximability of (generalized, fractional) hyper treewidth of
sparse hypergraphs where the criterion of sparsity reflects the sparsity of
their incidence graphs. Our first step is to prove that the (generalized,
fractional) hypertree width of a hypergraph H is constant-factor sandwiched by
the treewidth of its incidence graph, when the incidence graph belongs to some
apex-minor-free graph class. This determines the combinatorial borderline above
which the notion of (generalized, fractional) hypertree width becomes
essentially more general than treewidth, justifying that way its functionality
as a hypergraph acyclicity measure. While for more general sparse families of
hypergraphs treewidth of incidence graphs and all hypertree width parameters
may differ arbitrarily, there are sparse families where a constant factor
approximation algorithm is possible. In particular, we give a constant factor
approximation polynomial time algorithm for (generalized, fractional) hypertree
width on hypergraphs whose incidence graphs belong to some H-minor-free graph
class

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