Obstructions to within a few vertices or edges of acyclic

Finite obstruction sets for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. It has been known for several years that, in principle, obstruction sets can be mechanically computed for most natural lower ideals. In this paper, we describe a general-purpose method for finding obstructions by using a bounded treewidth (or pathwidth) search. We illustrate this approach by characterizing certain families of cycle-cover graphs based on the two well-known problems: $k$-{\sc Feedback Vertex Set} and $k$-{\sc Feedback Edge Set}. Our search is based on a number of algorithmic strategies by which large constants can be mitigated, including a randomized strategy for obtaining proofs of minimality.Comment: 16 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

Dynamic Programming on Nominal Graphs

Many optimization problems can be naturally represented as (hyper) graphs, where vertices correspond to variables and edges to tasks, whose cost depends on the values of the adjacent variables. Capitalizing on the structure of the graph, suitable dynamic programming strategies can select certain orders of evaluation of the variables which guarantee to reach both an optimal solution and a minimal size of the tables computed in the optimization process. In this paper we introduce a simple algebraic specification with parallel composition and restriction whose terms up to structural axioms are the graphs mentioned above. In addition, free (unrestricted) vertices are labelled with variables, and the specification includes operations of name permutation with finite support. We show a correspondence between the well-known tree decompositions of graphs and our terms. If an axiom of scope extension is dropped, several (hierarchical) terms actually correspond to the same graph. A suitable graphical structure can be found, corresponding to every hierarchical term. Evaluating such a graphical structure in some target algebra yields a dynamic programming strategy. If the target algebra satisfies the scope extension axiom, then the result does not depend on the particular structure, but only on the original graph. We apply our approach to the parking optimization problem developed in the ASCENS e-mobility case study, in collaboration with Volkswagen. Dynamic programming evaluations are particularly interesting for autonomic systems, where actual behavior often consists of propagating local knowledge to obtain global knowledge and getting it back for local decisions.Comment: In Proceedings GaM 2015, arXiv:1504.0244

Establishing a Connection Between Graph Structure, Logic, and Language Theory

The field of graph structure theory was given life by the Graph Minors Project of Robertson and Seymour, which developed many tools for understanding the way graphs relate to each other and culminated in the proof of the Graph Minors Theorem. One area of ongoing research in the field is attempting to strengthen the Graph Minors Theorem to sets of graphs, and sets of sets of graphs, and so on. At the same time, there is growing interest in the applications of logic and formal languages to graph theory, and a significant amount of work in this field has recently been consolidated in the publication of a book by Courcelle and Engelfriet. We investigate the potential applications of logic and formal languages to the field of graph structure theory, suggesting a new area of research which may provide fruitful

Graph Parameters, Universal Obstructions, and WQO

We introduce the notion of universal obstruction of a graph parameter, with respect to some quasi-ordering relation. Universal obstructions may serve as compact characterizations of the asymptotic behavior of graph parameters. We provide order-theoretic conditions which imply that such a characterization is finite and, when this is the case, we present some algorithmic implications on the existence of fixed-parameter algorithms

On the computability of obstruction sets for well-quasi-ordered graph classes

Στην παρούσα διπλωματική εργασία θα μελετήσουμε αλγόριθμους για τον υπολογισμό συνόλων παρεμπόδησης καλώς μερικώς διατεταγμένων κλάσεων γραφημάτων. Το Θεώρημα Ελασσόνων Γραφημάτων (ΘΕΓ), των Neil Robertson και Paul Seymour (Graph Minor Theorem) εγγυάται πως κάθε κλάση κλειστή ως προς τη σχέση των ελασσόνων έχει πεπερασμένο σύνολο παρεμόδησης. Αν η C είναι μια τέτοια κλάση, τότε το σύνολο παρεμπόδησης της C είναι το ελαχιστικό σύνολο γραφημάτων H έτσι ώστε, ένα γράφημα G ανήκει στην κλάση C αν και μόνο αν κανένα από τα γραφήματα στο σύνολο H δεν περιέχεται ως ελάσσον στο G. Το αντίστοιχο αποτέλεσμα για μια άλλη καλή μερική διάταξη, την σχέση της εμβύθισης, αποδείχθηκε στην ίδια σειρά εργασιών (Graph Minors). Όμως αυτά τα αποτελέσματα είναι μη-κατασκευαστικά: ξέρουμε πως κάθε κλάση κλειστή ως προς ελάσσονα ή εμβυθίσεις έχει πεπερασμένο σύνολο παρεμπόδησης αλλά από αυτά τα αποτελέσματα δεν υποδεικνύουν κάποιο αλγόριθμο για να το υπολογίσουμε. Οι K. Cattell, M. J. Dinneen, R. Downey, M. R. Fellows and M. Langston στην εργασία "On computing graph minor obstruction sets" και οι I. Adler, M. Grohe and S. Kreutzer στην εργασία "Computing Excluded Minors" παρουσιάζουν αλγόριθμους για να ξεπεράσουμε αυτό το πρόβλημα στις κλάσεις γραφημάτων κλειστές ως προς ελάσσονα, καθώς και εφαρμογές των μεθόδων τους, όπως το πρόβλημα της ένωσης. Προσαρμόζοντας τις μεθόδους της δεύτερης από τις προηγουμένες εργασίες σε εμβυθήσεις οι Α. Γιαννοπούλου, Δ. Ζώρος και ο συγγραφέας, ύπο την επίβλεψη του Δ. Μ. Θηλυκού αποδεικνύουν το αντίστοιχο αποτέλεσμα για κλάσεις γραφημάτων κλειστές ως προς εμβύθιση, καθώς και έναν αλγόριθμο για το πρόβλημα της ένωσης για εμβυθίσεις.In this MSc thesis we are going to present algorithms for computing obstruction sets of well--quasi--ordered graph classes. Neil Robertson and Paul Seymour's Graph Minor Theorem (GMT) guarantees that any minor-closed graph class has a finite obstruction set. If C is such a class, the obstruction set of C is the minimal set of graphs H such that G belongs to C if and only if none of the graphs in H is contained as a minor in G. The analogous result for another well-quasi-ordering, the immersion ordering, was shown in the same series of papers (Graph Minors). But these results are non-constructive; we know that a minor or immersion-closed graph class has a finite obstruction set but the GMT does not imply any algorithm for computing it. K. Cattell, M. J. Dinneen, R. Downey, M. R. Fellows and M. Langston in "On computing graph minor obstruction sets" and I. Adler, M. Grohe and S. Kreutzer in "Computing Excluded Minors" present algorithms to overcome this problem for minor-closed graph classes, as well as, applications of their methods proving that the obstruction sets of various graph classes are computable, such as the union problem. By adapting some of the methods of Adler, Grohe and Kreutzer to immersions, the analogue result for immersion obstruction sets and an algorithm for the union problem on immersion-closed graph classes are proven by A. Giannopoulou, D. Zoros and the author, under the supervision of D. M. Thilikos