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

The vertex leafage of chordal graphs

Every chordal graph $G$ can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called {\em tree model} of $G$. The leafage $\ell(G)$ of a connected chordal graph $G$ is the minimum number of leaves of the host tree of a tree model of $G$. The vertex leafage \vl(G) is the smallest number $k$ such that there exists a tree model of $G$ in which every subtree has at most $k$ leaves. The leafage is a polynomially computable parameter by the result of \cite{esa}. In this contribution, we study the vertex leafage. We prove for every fixed $k\geq 3$ that deciding whether the vertex leafage of a given chordal graph is at most $k$ is NP-complete by proving a stronger result, namely that the problem is NP-complete on split graphs with vertex leafage of at most $k+1$. On the other hand, for chordal graphs of leafage at most $\ell$, we show that the vertex leafage can be calculated in time $n^{O(\ell)}$. Finally, we prove that there exists a tree model that realizes both the leafage and the vertex leafage of $G$. Notably, for every path graph $G$, there exists a path model with $\ell(G)$ leaves in the host tree and it can be computed in $O(n^3)$ time

On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi

Qualitative calculi play a central role in representing and reasoning about qualitative spatial and temporal knowledge. This paper studies distributive subalgebras of qualitative calculi, which are subalgebras in which (weak) composition distributives over nonempty intersections. It has been proven for RCC5 and RCC8 that path consistent constraint network over a distributive subalgebra is always minimal and globally consistent (in the sense of strong $n$-consistency) in a qualitative sense. The well-known subclass of convex interval relations provides one such an example of distributive subalgebras. This paper first gives a characterisation of distributive subalgebras, which states that the intersection of a set of $n\geq 3$ relations in the subalgebra is nonempty if and only if the intersection of every two of these relations is nonempty. We further compute and generate all maximal distributive subalgebras for Point Algebra, Interval Algebra, RCC5 and RCC8, Cardinal Relation Algebra, and Rectangle Algebra. Lastly, we establish two nice properties which will play an important role in efficient reasoning with constraint networks involving a large number of variables.Comment: Adding proof of Theorem 2 to appendi

On the Computational Complexity of Vertex Integrity and Component Order Connectivity

The Weighted Vertex Integrity (wVI) problem takes as input an $n$-vertex graph $G$, a weight function $w:V(G)\to\mathbb{N}$, and an integer $p$. The task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ plus the weight of a heaviest component of $G-X$ is at most $p$. Among other results, we prove that: (1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight $1$; (2) wVI can be solved in $O(p^{p+1}n)$ time; (3) wVI admits a kernel with at most $p^3$ vertices. Result (1) refutes a conjecture by Ray and Deogun and answers an open question by Ray et al. It also complements a result by Kratsch et al., stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an $n$-vertex graph $G$, a weight function $w:V(G)\to \mathbb{N}$, and two integers $k$ and $l$, and the task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ is at most $k$ and the weight of a heaviest component of $G-X$ is at most $l$. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We prove, among other results, that: (4) wCOC can be solved in $O(\min\{k,l\}\cdot n^3)$ time on interval graphs, while the unweighted version can be solved in $O(n^2)$ time on this graph class; (5) wCOC is W[1]-hard on split graphs when parameterized by $k$ or by $l$; (6) wCOC can be solved in $2^{O(k\log l)} n$ time; (7) wCOC admits a kernel with at most $kl(k+l)+k$ vertices. We also show that result (6) is essentially tight by proving that wCOC cannot be solved in $2^{o(k \log l)}n^{O(1)}$ time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the conference proceedings of ISAAC 201

Limits of conformal images and conformal images of limits for planar random curves

Consider a chordal random curve model on a planar graph, in the scaling limit when a fine-mesh graph approximates a simply-connected planar domain. The well-known precompactness conditions of Kemppainen and Smirnov show that certain "crossing estimates" guarantee the subsequential weak convergence of the random curves in the topology of unparametrized curves, as well as in a topology inherited from curves on the unit disc via conformal maps. We complement this result by proving that proceeding to weak limit commutes with changing topology, i.e., limits of conformal images are conformal images of limits, without imposing any boundary regularity assumptions on the domains where the random curves lie. Treating such rough boundaries becomes necessary, e.g., in convergence proofs to multiple SLEs. The result in this generality has not been explicated before and is not trivial, which we demonstrate by giving warning examples and deducing strong consequences.Comment: 34 pages, 11 figures. v2: minor improvement

Conformally invariant scaling limits in planar critical percolation

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE(6). No prior knowledge is assumed beyond some general complex analysis and probability theory.Comment: 55 pages, 10 figure

Sequences of regressions and their independences

Ordered sequences of univariate or multivariate regressions provide statistical models for analysing data from randomized, possibly sequential interventions, from cohort or multi-wave panel studies, but also from cross-sectional or retrospective studies. Conditional independences are captured by what we name regression graphs, provided the generated distribution shares some properties with a joint Gaussian distribution. Regression graphs extend purely directed, acyclic graphs by two types of undirected graph, one type for components of joint responses and the other for components of the context vector variable. We review the special features and the history of regression graphs, derive criteria to read all implied independences of a regression graph and prove criteria for Markov equivalence that is to judge whether two different graphs imply the same set of independence statements. Knowledge of Markov equivalence provides alternative interpretations of a given sequence of regressions, is essential for machine learning strategies and permits to use the simple graphical criteria of regression graphs on graphs for which the corresponding criteria are in general more complex. Under the known conditions that a Markov equivalent directed acyclic graph exists for any given regression graph, we give a polynomial time algorithm to find one such graph.Comment: 43 pages with 17 figures The manuscript is to appear as an invited discussion paper in the journal TES