82 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

### Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators

### 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 triangulation problems in- cluding 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, and an integer t, it is possible in time the number of potential maximal cliques times O(nO(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(1.734601n ), this yields that both the problems are solvable in time 1.734601n * nO(t) .publishedVersio

### TREEWIDTH and PATHWIDTH parameterized by vertex cover

After the number of vertices, Vertex Cover is the largest of the classical
graph parameters and has more and more frequently been used as a separate
parameter in parameterized problems, including problems that are not directly
related to the Vertex Cover. Here we consider the TREEWIDTH and PATHWIDTH
problems parameterized by k, the size of a minimum vertex cover of the input
graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3^k)
time. This complements recent polynomial kernel results for TREEWIDTH and
PATHWIDTH parameterized by the Vertex Cover

### Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k^{3/2}) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios O(log{k}) on planar graphs and O(sqrt{k} log{k}) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs.publishedVersio

### Exploring Subexponential Parameterized Complexity of Completion Problems

Let ${\cal F}$ be a family of graphs. In the ${\cal F}$-Completion problem,
we are given a graph $G$ and an integer $k$ as input, and asked whether at most
$k$ edges can be added to $G$ so that the resulting graph does not contain a
graph from ${\cal F}$ as an induced subgraph. It appeared recently that special
cases of ${\cal F}$-Completion, the problem of completing into a chordal graph
known as Minimum Fill-in, corresponding to the case of ${\cal
F}=\{C_4,C_5,C_6,\ldots\}$, and the problem of completing into a split graph,
i.e., the case of ${\cal F}=\{C_4, 2K_2, C_5\}$, are solvable in parameterized
subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$. The exploration of this
phenomenon is the main motivation for our research on ${\cal F}$-Completion.
In this paper we prove that completions into several well studied classes of
graphs without long induced cycles also admit parameterized subexponential time
algorithms by showing that:
- The problem Trivially Perfect Completion is solvable in parameterized
subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$, that is ${\cal
F}$-Completion for ${\cal F} =\{C_4, P_4\}$, a cycle and a path on four
vertices.
- The problems known in the literature as Pseudosplit Completion, the case
where ${\cal F} = \{2K_2, C_4\}$, and Threshold Completion, where ${\cal F} =
\{2K_2, P_4, C_4\}$, are also solvable in time $2^{O(\sqrt{k}\log{k})}
n^{O(1)}$.
We complement our algorithms for ${\cal F}$-Completion with the following
lower bounds:
- For ${\cal F} = \{2K_2\}$, ${\cal F} = \{C_4\}$, ${\cal F} = \{P_4\}$, and
${\cal F} = \{2K_2, P_4\}$, ${\cal F}$-Completion cannot be solved in time
$2^{o(k)} n^{O(1)}$ unless the Exponential Time Hypothesis (ETH) fails.
Our upper and lower bounds provide a complete picture of the subexponential
parameterized complexity of ${\cal F}$-Completion problems for ${\cal
F}\subseteq\{2K_2, C_4, P_4\}$.Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in
the proceedings of STACS'1

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