404 research outputs found

### Editing to a Graph of Given Degrees

We consider the Editing to a Graph of Given Degrees problem that asks for a
graph G, non-negative integers d,k and a function \delta:V(G)->{1,...,d},
whether it is possible to obtain a graph G' from G such that the degree of v is
\delta(v) for any vertex v by at most k vertex or edge deletions or edge
additions. We construct an FPT-algorithm for Editing to a Graph of Given
Degrees parameterized by d+k. We complement this result by showing that the
problem has no polynomial kernel unless NP\subseteq coNP/poly

### 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

### 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

### Variants of Plane Diameter Completion

The {\sc Plane Diameter Completion} problem asks, given a plane graph $G$ and
a positive integer $d$, if it is a spanning subgraph of a plane graph $H$ that
has diameter at most $d$. We examine two variants of this problem where the
input comes with another parameter $k$. In the first variant, called BPDC, $k$
upper bounds the total number of edges to be added and in the second, called
BFPDC, $k$ upper bounds the number of additional edges per face. We prove that
both problems are {\sf NP}-complete, the first even for 3-connected graphs of
face-degree at most 4 and the second even when $k=1$ on 3-connected graphs of
face-degree at most 5. In this paper we give parameterized algorithms for both
problems that run in $O(n^{3})+2^{2^{O((kd)^2\log d)}}\cdot n$ steps.Comment: Accepted in IPEC 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

### Refined Complexity of PCA with Outliers

Principal component analysis (PCA) is one of the most fundamental procedures
in exploratory data analysis and is the basic step in applications ranging from
quantitative finance and bioinformatics to image analysis and neuroscience.
However, it is well-documented that the applicability of PCA in many real
scenarios could be constrained by an "immune deficiency" to outliers such as
corrupted observations. We consider the following algorithmic question about
the PCA with outliers. For a set of $n$ points in $\mathbb{R}^{d}$, how to
learn a subset of points, say 1% of the total number of points, such that the
remaining part of the points is best fit into some unknown $r$-dimensional
subspace? We provide a rigorous algorithmic analysis of the problem. We show
that the problem is solvable in time $n^{O(d^2)}$. In particular, for constant
dimension the problem is solvable in polynomial time. We complement the
algorithmic result by the lower bound, showing that unless Exponential Time
Hypothesis fails, in time $f(d)n^{o(d)}$, for any function $f$ of $d$, it is
impossible not only to solve the problem exactly but even to approximate it
within a constant factor.Comment: To be presented at ICML 201

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