Omer Giménez

Asymptotic enumeration and limit laws of planar graphs

A graph is planar if it can be embedded in the plane, or in the sphere, so that no two edges cross at an interior point. A planar graph together with a particular embedding is called a map. There is a rich theory of counting maps, started by Tutte in the 1960s. However, in this paper we are interested in counting graphs as combinatorial objects, regardless of how many nonequivalent topological embeddings they may have. As we are going to see, this makes the counting considerably more difficult.

The complexity of planning problems with simple causal graphs

We present three new complexity results for classes of planning problems with simple causal graphs. First, we describe a polynomial-time algorithm that uses macros to generate plans for the class 3S of planning problems with binary state variables and acyclic causal graphs. This implies that plan generation may be tractable even when a planning problem has an exponentially long minimal solution. We also prove that the problem of plan existence for planning problems with multi-valued variables and chain causal graphs is NP-hard. Finally, we show that plan existence for planning problems with binary state variables and polytree causal graphs is NP-complete.

Computing the Tutte Polynomial on Graphs of Bounded Clique-Width

The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree-width. The notion of clique-width extends the definition of cograhs (graphs without induced P4), and it is a more general notion than that of tree-width. We show a subexponential algorithm (running in time expO(n2/3)) for computing the Tutte polynomial on cographs. The algorithm can be extended to a subexponential algorithm computing the Tutte polynomial on on all graphs of bounded clique-width. In fact, our algorithm computes the more general U-polynomial. 2000 Math Subjects Classification: 05C85, 68R10.

Jutge.org: an educational programming judge

Jutge.org is an open access educational online programming judge where students can try to solve more than 800 problems using 22 programming languages. The verdict of their solutions is computed using exhaustive test sets run under time, memory and security restrictions. By contrast to many popular online judges, Jutge.org is designed for students and instructors: On one hand, the problem repository is mainly aimed to beginners, with a clear organization and gradding. On the other hand, the system is designed as a virtual learning environment where instructors can administer their own courses, manage their roster of students and tutors, add problems, attach documents, create lists of problems, assignments, contests and exams. This paper presents Jutge.org and offers some case studies of courses using it.

Graph classes with given 3-connected components: Asymptotic enumeration and random graphs†

Consider a family of 3-connected graphs of moderate growth, and let be the class of graphs whose 3-connected components are graphs in . We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied cases such as planar graphs and series-parallel graphs. We provide a general result for the asymptotic number of graphs in , based on the singularities of the exponential generating function associated to . We derive limit laws, which are either normal or Poisson, for several basic parameters, including the number of edges, number of blocks and number of components. For the size of the largest block we find a fundamental dichotomy: classes similar to planar graphs have almost surely a unique block of linear size, while classes similar to series-parallel graphs have only sublinear blocks. This dichotomy was already observed by Panagiotou and Steger [25], and we provide a finer description. For some classes under study both regimes occur, because of a critical phenomenon as the edge density in the class varies. Finally, we analyze the size of the largest 3-connected component in random planar graphs.

Causal graphs and structurally restricted planning

The causal graph is a directed graph that describes the variable dependencies present in a planning instance. A number of papers have studied the causal graph in both practical and theoretical settings. In this work, we systematically study the complexity of planning restricted by the causal graph. In particular, any set of causal graphs gives rise to a subcase of the planning problem. We give a complete classification theorem on causal graphs, showing that a set of graphs is either polynomial-time tractable, or is not polynomial-time tractable unless an established complexity-theoretic assumption fails; our theorem describes which graph sets correspond to each of the two cases. We also give a classification theorem for the case of reversible planning, and discuss the general direction of structurally restricted planning.

Surveys in Combinatorics 2009: Counting planar graphs and related families of graphs

In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree distribution or the size of the largest k-connected component. Most of the results we present use generating functions and analytic tools.

Vertices of given degree in series-parallel graphs: Vertices of Given Degree in Series-Parallel Graphs

We show that the number of vertices of a given degree k in several kinds of series-parallel labelled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labelled planar graphs. The proof method is based on generating functions and singularity analysis. In particular we need systems of equations for multivariate generating functions and transfer results for singular representations of analytic functions. 1. Statement of main results A graph is series-parallel if it does not contain the complete graph K4 as a minor; equivalently, if it does not contain a subdivision of K4. Since both K5 and K3,3 contain a subdivision of K4, by Kuratowski’s theorem a series-parallel graph is planar. An outerplanar graph is a planar graph that can be embedded in the plane so that all vertices are incident to the outer face. They are characterized as those graphs not containing a minor isomorphic to (or a subdivision of) either K4 or K2,3. These are important subfamilies of planar graphs, as they are much simpler but often they already capture the essential structural properties of planar graphs. In particular, they are used as a natural first benchmark for many algorithmic problems and conjectures related to planar graphs The purpose of this paper is to study the number of vertices of given degree in certain classes of labelled planar graphs. In particular, we study labelled outerplanar graphs and series-parallel graphs; in what follows, all graphs are labelled. In order to state our results we introduce the notion of the degree distribution of a random outerplanar graph (the definition for series-parallel graphs is exactly the same). For every n we consider the class of all vertex labelled outerplanar graphs with n vertices. Let Dn denote the degree of a randomly chosen vertex in this class of graphs. Then we say that this class of graphs has a degree distribution if there exist non-negative numbers dk with ∑ k≥0 dk = 1 such that for all k lim n→∞ Pr{Dn = k} = dk. In a companion paper [8], we have established that the classes of 2-connected, connected or all outerplanar graphs, as well as the corresponding classes of seriesparallel graphs have a degree distribution. We describe briefly the degree distribution in the outerplanar case, which is the simplest one, and refer to [8] for the other 1Alternatively we can define Dn as the degree of the vertex with label 1. 1 2 MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY cases. Let dk be defined as before for outerplanar graphs, let D(x) = 1 + x−√1− 6x + x2 4 , and let g(x,w) = xw + xw 2 2D(x)− x 1− (2D(x)− x)w The function D(x) is a minor modification of function A(x) in (2.1), and g(x,w) is the generating function of rooted of 2-connected outerplanar graphs, where w marks the degree of the root. Then we have ∑ k≥0 dkw k = ρ · ∂ ∂x e |x=τ , where ρ and τ are constants defined analytically and having approximate values ρ ≈ 0.1366 and τ ≈ 0.1708. A plot of the distribution is given in Figure 1. For comparison, we have added also a plot of the distribution for 2-connected outerplanar graphs, whose probability generating function can be computed directly from g(x,w). 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Planning over chain causal graphs for variables with domains of size 5 Is NP-hard

Recently, considerable focus has been given to the problem of determining the boundary between tractable and intractable planning problems. In this paper, we study the complexity of planning in the class Cn of planning problems, characterized by unary operators and directed path causal graphs. Although this is one of the simplest forms of causal graphs a planning problem can have, we show that planning is intractable for Cn (unless P = NP), even if the domains of state variables have bounded size. In particular, we show that plan existence for Ckn is NP-hard for k ≥ 5 by reduction from CNF-SAT. Here, k denotes the upper bound on the size of the state variable domains. Our result reduces the complexity gap for the class Ckn to cases k = 3 and k = 4 only, since C2n is known to be tractable.

The HOM problem is decidable

We provide an algorithm that, given a tree homomorphism H and a regular tree language L represented by a tree automaton, determines whether H(L) is regular. This settles a question that has been open for a long time. Along the way, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular tree languages through tree homomorphisms. Our algorithms are based on the following constructions. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with inequality constraints recognizing the complementary language. We also define a new class of automata with arbitrary inequality constraints and a particular kind of equality constraints. An automaton of this new class essentially recognizes the intersection of a tree automaton with inequality constraints and the image of a regular tree language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism.