Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in O(n1+ϵ) time

AbstractWe give constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs. For a planar graph G with n vertices, let bw(G) be the branchwidth of G and gm(G) the largest integer g such that G has a g×g grid as a minor. Let c≥1 be a fixed integer and α,β arbitrary constants satisfying α>c+1 and β>2c+1. We give an algorithm which constructs in O(n1+1clogn) time a branch-decomposition of G with width at most αbw(G). We also give an algorithm which constructs a g×g grid minor of G with g≥gm(G)β in O(n1+1clogn) time. The constants hidden in the Big-O notations are proportional to cα−(c+1) and cβ−(2c+1), respectively

A Unifying Framework for Characterizing and Computing Width Measures

Algorithms for computing or approximating optimal decompositions for decompositional parameters such as treewidth or clique-width have so far traditionally been tailored to specific width parameters. Moreover, for mim-width, no efficient algorithms for computing good decompositions were known, even under highly restrictive parameterizations. In this work we identify ?-branchwidth as a class of generic decompositional parameters that can capture mim-width, treewidth, clique-width as well as other measures. We show that while there is an infinite number of ?-branchwidth parameters, only a handful of these are asymptotically distinct. We then develop fixed-parameter and kernelization algorithms (under several structural parameterizations) that can approximate every possible ?-branchwidth, providing a unifying parameterized framework that can efficiently obtain near-optimal tree-decompositions, k-expressions, as well as optimal mim-width decompositions

Integer programming models for the branchwidth problem

We consider the problem of computing the branchwidth and an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced in 1991 by Robertson and Seymour and were used in the proof of Graph Minors Theorem (GMT), a well known conjecture (Wagner's conjecture) in graph theory. The notions of branchwidth and branch decompositions have been proved to be useful for solving many NP-hard problems that have applications in fields such as graph theory, network design, sensor networks and biology. Branch decompositions have been utilized for problems such as the traveling salesman problem by Cook and Seymour, general minor containment and the branchwidth problem by Hicks by means of the relevant branch decomposition-based algorithms. Branch decomposition-based algorithms are fixed parameter tractable algorithms obtained by combining dynamic programming techniques with branch decompositions. The running time and space of these algorithms strongly depend on the width of the utilized branch decomposition. Thus, finding optimal or close to optimal branch decompositions is very important for the efficiency of the branch decomposition-based algorithms. Motivated by the vastness of the fields of application, we aim to increase the efficiency of the branch decomposition-based algorithms by investigating effective techniques to find optimal branch decompositions. We present three integer programming models for the branchwidth problem. Two similar formulations are based on the relationship of branchwidth problem with a special case of the Steiner tree packing problem. The third formulation is based on the notion of laminar separations. We utilize upper and lower bounds obtained by heuristic algorithms, reduction techniques and cutting planes to increase the efficiency of our models. We use all three models for the branchwidth problem on hypergraphs as well. We compare the performance of three models both on graphs and hypergraphs. Furthermore we use the third model for rank-width problem and also offer a heuristic for finding good rank decompositions. We provide computational results for this problem, which can be a basis of comparison for future formulations

Sphere-cut decompositions and dominating sets in planar graphs

Ένα σημαντικό αποτέλεσμα στη Θεωρία Γραφημάτων αποτελεί η απόδειξη της εικασίας του Wagner απο τους Neil Robertson και Paul. D. Seymour στη σειρά εργασιών "Ελλάσονα Γραφήματα" απο το 1983 εώς το 2011. Η εικασία αυτή λέει ότι στην κλάση των γραφημάτων δεν υπάρχει άπειρη αντιαλυσίδα ως προς τη σχέση των ελλασόνων γραφημάτων. Η Θεωρία που αναπτύχθηκε για την απόδειξη αυτής της εικασίας είχε και έχει ακόμα σημαντικό αντίκτυπο τόσο στην δομική όσο και στην αλγοριθμική Θεωρία Γραφημάτων, άλλα και σε άλλα πεδία όπως η Παραμετρική Πολυπλοκότητα. Στα πλαίσια της απόδειξης οι συγγραφείς εισήγαγαν και νέες παραμέτρους πλάτους. Σε αυτές ήταν η κλαδοαποσύνθεση και το κλαδοπλάτος ενός γραφήματος. Η παράμετρος αυτή χρησιμοποιήθηκε ιδιαίτερα στο σχεδιασμό αλγορίθμων και στην χρήση της τεχνικής "διαίρει και βασίλευε". Επιπλέον εισήχθησαν νέες παρεμφερείς έννοιες όπως οι αποσυνθέσεις σφαιρικών τομών που είναι κλαδοαποσυνθέσεις στην κλάση των επίπεδων γραφημάτων που έχουν κάποιες επιπλέον ιδιότητες. Στην εξέλιξη της έρευνας υπήρξαν σημαντικά αποτέλεσμα σχετικά με το κλαδοπλάτος στην κλάση των επίπεδων γραφημάτων. Οι Fedor. V. Fomin και Δημήτριος Μ. Θηλυκός απέδειξαν ότι το κλαδοπλάτος ενός επίπεδου γραφήματος με n κορυφές είναι το πολύ \sqrt{4.5\cdot n}. Βασιζόμενος σε αυτό το αποτέλεσμα, ο Δημήτριος Μ. Θηλυκός συσχέτισε το κλαδοπλάτος με μια άλλη παράμετρο σε επίπεδα γραφήματα, το r-ακτινικό σύνολο κυριαρχίας. Απέδειξε ότι αν εμβαπτισμένο επίπεδο γράφημα έχει r- ακτινικό σύνολο κυριαρχίας το πολύ k, τότε το κλαδοπλάτος του γραφήματος θα είναι το πολύ r\cdot\sqrt{4.5\cdot k}. H παρούσα διπλωματική εργασία κάνει μια ποιοτική επέκταση του αυτού του αποτελέσματος. Αποδεικνύουμε ότι το παραπάνω όριο μπορεί να αναζητηθεί σε ένα δάσος που είναι υπογράφημα του δέντρου μιας αποσύνθεσης σφαιρικών τομών του γραφήματος όπου το μέγεθος του είναι γραμμικό ως προς το k.An important result in Graph Theory is the proof of Wagner's Conjecture by Neil Robertson and Paul D. Seymour in Graph Minor Series from 1983 until 2011. This conjecture state that there is no innite anti-chain in the class of graphs under the minor relation. The theory that was built for the proof of this conjecture had, and continues to have, an important impact not only in structural and algorithmic Graph Theory, but also in other elds such as Parameterized Complexity. In the context of this proof, the authors have introduced some new width parameters. Within these were branchwidth and branch decompositions. This parameter was used for algorithm design via the \divide and conquer" technique. Moreover, the authors have introduced, similar to branch decompositions, concepts such as sphere-cut decompositions which are a special type of branch decompositions in planar graphs that have some additional properties. In the course of the research there was a lot of important results about branchwidth in the class of planar graphs. Fedor V. Fomin and Dimitrios M. Thilikos proved that the branchwidth of a n-vertex planar graph is at most p4:5 n. Based on this result Dimitrios M. Thilikos connected the branchwidth with r-radial dominating set which is another parameter in plane graphs. He proved that if a plane graph has an r-radial dominating set of size at most k, then the branchwidth of the graph is at most r p4:5 k. The purpose of this thesis is to provide a qualitative extension of this result. What we show is that this upper bound is attained by a number of edges of a sphere-cut decomposition, that is a linear function of k

Computer Science Research Institute 2004 annual report of activities.

This report summarizes the activities of the Computer Science Research Institute (CSRI) at Sandia National Laboratories during the period January 1, 2004 to December 31, 2004. During this period the CSRI hosted 166 visitors representing 81 universities, companies and laboratories. Of these 65 were summer students or faculty. The CSRI partially sponsored 2 workshops and also organized and was the primary host for 4 workshops. These 4 CSRI sponsored workshops had 140 participants--74 from universities, companies and laboratories, and 66 from Sandia. Finally, the CSRI sponsored 14 long-term collaborative research projects and 5 Sabbaticals

Planar branch decompositions II: The cycle method

informs ® doi 10.1287/ijoc.1040.0074 © 2005 INFORMS This is the second of two papers dealing with the relationship of branchwidth and planar graphs. Branchwidth and branch decompositions, introduced by Robertson and Seymour, have been shown to be beneficial for both proving theoretical results on graphs and solving NP-hard problems modeled on graphs. The first practical implementation of an algorithm of Seymour and Thomas for computing optimal branch decompositions of planar hypergraphs is presented. This algorithm encompasses another algorithm of Seymour and Thomas for computing the branchwidth of any planar hypergraph, whose implementation is discussed in the first paper. The implementation also includes the addition of a heuristic to decrease the run times of the algorithm. This method, called the cycle method, is an improvement on the algorithm by using a “divide-and-conquer” approach. Key words: planar graph; branchwidth; branch decomposition; carvingwidt

Branch decompositions and their applications

Many real-life problems can be modeled as optimization or decision problems on graphs. Also, many of those real-life problems are NP-hard. One traditional method to solve these problems is by branch and bound while another method is by graph decompositions. In the 1980's, Robertson and Seymour conceived of two new ways to decompose the graph in order to solve these problems. These ingenious ideas were only by-products of their work proving Wagner's Conjecture. A branch decomposition is one of these ideas. A paper by Arnborg, Lagergren and Seeseshowed that many NP-complete problems can be solved in polynomial time using divide and conquer techniques on input graphs with bounded branchwidth, but a paper by Seymour and Thomas proved that computing an optimal branch decomposition is also NP-complete. Although computing optimal branch decompositions is NP-complete, there is a plethora of theory about branchwidth and branch decompositions. For example, a paper by Seymour and Thomas offered a polynomial time algorithm to compute the branchwidth and optimal branch decomposition for planar graphs. This doctoral research is concentrated on constructing branch decompositions for graphs and using branch decompositions to solve NP-complete problems modeled on graphs. In particular, a heuristic to compute near-optimal branch decompositions is presented and the heuristic is compared to previous heuristics in the subject. Furthermore, a practical implementation of an algorithm given in a paper by Seymour and Thomas for computing optimal branch decompositions of planar graphs is implemented with the addition of heuristics to give the algorithm a "divide and conquer" design. In addition, this work includes a theoretical result relating the branchwidth of planar graphs to their duals, characterizations of branchwidth for Halin and chordal graphs. Also, this work presents an algorithm for minor containment using a branch decomposition and a parallel implementation of the heuristic for general graphs using p-threads