An Introduction to Clique Minimal Separator Decomposition

International audienceThis paper is a review which presents and explains the decomposition of graphs by clique minimal separators. The pace is leisurely, we give many examples and figures. Easy algorithms are provided to implement this decomposition. The historical and theoretical background is given, as well as sketches of proofs of the structural results involved

Computing hypergraph width measures exactly

Hypergraph width measures are a class of hypergraph invariants important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. A connection between these and tree decompositions is established. This enables us to almost seamlessly adapt the combinatorial and algorithmic results known for tree decompositions of graphs to the case of hypergraphs and obtain fast exact algorithms. As a consequence, we provide algorithms which, given a hypergraph H on n vertices and m hyperedges, compute the generalized hypertree-width of H in time O*(2^n) and compute the fractional hypertree-width of H in time O(m*1.734601^n).Comment: 12 pages, 1 figur

On the Enumeration of all Minimal Triangulations

We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where "proper" means that the tree decomposition cannot be improved by removing or splitting a bag

Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is at least $k$, or: * finds in an isomorphic-invariant way a graph $\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes $G$ together with a tree decomposition of $G$ of width $O(k^4)$. Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.Comment: Full version of a paper presented at FOCS 201

A structural Markov property for decomposable graph laws that allows control of clique intersections

We present a new kind of structural Markov property for probabilistic laws on decomposable graphs, which allows the explicit control of interactions between cliques, so is capable of encoding some interesting structure. We prove the equivalence of this property to an exponential family assumption, and discuss identifiability, modelling, inferential and computational implications.Comment: 10 pages, 3 figures; updated from V1 following journal review, new more explicit title and added section on inferenc

Canonizing Graphs of Bounded Tree Width in Logspace

Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.Comment: 26 page

Polynomial-time algorithm for Maximum Weight Independent Set on P6P_6-free graphs

In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves the polynomial-time algorithm on $P_5$-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$