Large Induced Subgraphs via Triangulations and CMSO

  title={Large Induced Subgraphs via Triangulations and CMSO},
  author={F. Fomin and Ioan Todinca and Yngve Villanger},
  booktitle={SIAM J. Comput.},
We obtain an algorithmic metatheorem for the following optimization problem. Let $\varphi$ be a counting monadic second order logic (CMSO) formula and $t\geq 0$ be an integer. For a given graph $G=(V,E)$, the task is to maximize $|X|$ subject to the following: there is a set $ F\subseteq V$ such that $X\subseteq F $, the subgraph $G[F]$ induced by $F$ is of treewidth at most $t$, and the structure $(G[F],X)$ models $\varphi$, i.e., $(G[F],X)\models\varphi$. We give an algorithm solving this… 

Figures from this paper

Beyond Classes of Graphs with “Few” Minimal Separators: FPT Results Through Potential Maximal Cliques
It is proved that the generic optimization problem is fixed parameter tractable on Gpoly, with parameter k, if the modulator is also part of the input.
Finding large $H$-colorable subgraphs in hereditary graph classes
It is proved that for every fixed pattern graph H, the problem of finding the largest induced subgraph of G that admits a homomorphism into H can be solved, and that even a restricted variant of \textsc{Max Partial $H$-Coloring} is $\mathsf{NP}$-hard in the considered subclasses of $P_5$-free graphs, if the authors allow loops on $H$.
Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs
It is proved that the problem of finding a largest, in terms of the number of vertices, induced subgraph of a graph G that belongs to $\mathcal{C}$ can be solved in $2^{o(n)$ time.
Beyond Classes of Graphs with "Few" Minimal Separators: FPT Results Through Potential Maximal Cliques
It is proved that the generic optimization problem is fixed parameter tractable on G + kv, with parameter k, if the modulator is also part of the input.
Exact Algorithms via Multivariate Subroutines
This work derandomizes this algorithm at the cost of increasing the running time by a subexponential factor in $n, and adapts it to the enumeration setting where the authors need to enumerate all subsets of the universe with property $\Phi$.
On H-Topological Intersection Graphs
It is proved that it is NP-complete if $H$ contains the diamond graph as a minor and that the clique problem is APX-hard, and it is shown that both the k-clique and the list ofcoloring problems are solvable in FPT-time on $H-graphs.
Covering minimal separators and potential maximal cliques in Pt-free graphs
It is shown that the so-called Separator Covering Lemma, which asserts that every minimal separator in the graph can be covered by the union of neighborhoods of a constant number of vertices, generalizes to "$P_7$-free graphs and is false in $P_8$- free graphs".
Induced subgraphs of bounded treewidth and the container method
It is shown that, given an $n-vertex graph G, one can in time find a maximum-weight induced subgraph of $G$ of treewidth less than k, which implies both aforementioned results.
On Distance-d Independent Set and Other Problems in Graphs with "few" Minimal Separators
It is shown that the odd powers of a graph G have at most as many minimal separators as G, and Distance-d Independent Set, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on G, for any even d.
Efficient Enumerations for Minimal Multicuts and Multiway Cuts
An incremental polynomial delay enumeration algorithm for minimal node multicuts, which extends an enumeration algorithms due to Khachiyan et al. (Algorithmica, 2008) for minimal edge multicuts.


Maximum r-Regular Induced Subgraph Problem: Fast Exponential Algorithms and Combinatorial Bounds
We show that for a fixed $r$, the number of maximal $r$-regular induced subgraphs in any graph with $n$ vertices is upper bounded by $\mathcal{O}(c^n)$, where $c$ is a positive constant strictly less
Treewidth computation and extremal combinatorics
It is proved that every graph on nvertices contains at most $n\binom{b+f}{b}$ such vertex subsets, and this result from extremalcombinatorics appears to be very useful in the design of severalumeration and exact algorithms.
Finding Induced Subgraphs via Minimal Triangulations
It is shown 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 potentialmaximal cliques times O(n^{O(t)}) to find a maximum induced subgraph of treewidth t in G and for a given graph F to decide if G contains an induced sub graph isomorphic to F.
Exact Algorithms for Treewidth and Minimum Fill-In
It is shown that the treewidth and the minimum fill-in of an $n$-vertex graph can be computed in time $\mathcal{O}(1.8899^n)$ and the running time of the algorithms can be reduced to 1.4142 minutes.
An improved exact algorithm for undirected feedback vertex set
This paper improves the algorithm for finding a minimum feedback vertex set in an undirected graph by designing new reductions based on biconnectivity of instances and introducing a new measure scheme on the structure of reduced graphs.
Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms
A number of generic algorithmic results about F-DELETION, when F contains at least one planar graph are obtained, which unify, generalize, and improve a multitude of results in the literature.
Easy Problems for Tree-Decomposable Graphs
Finding Contractions and Induced Minors in Chordal Graphs via Disjoint Paths
The k-Disjoint Paths problem, which takes as input a graph G and k pairs of specified vertices (si,ti), is studied, and it is proved that the problem becomes polynomial-time solvable for any fixed integer k if the input graph is chordal.
Subexponential parameterized algorithm for minimum fill-in
This work gives the first subexponential parameterizedv algorithm solving Minimum Fill-in in time and substantially lowers the complexity of the problem.
Graph minors. V. Excluding a planar graph