Catherine McCartin

On the Parameterized Complexity of Layered Graph Drawing

Abstract We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

Parameterized approximation problems

Up to now, most work in the area of parameterized complexity has focussed on exact algorithms for decision problems. The goal of this paper is to apply parameterized ideas to approximation. We begin exploration of parameterized approximation problems, where the problem in question is a parameterized decision problem, and the required approximation factor is treated as a second parameter for the problem.

Coordinatized Kernels and Catalytic Reductions: An Improved FPT Algorithm for Max Leaf Spanning Tree and Other Problems

We describe some new, simple and apparently general methods for designing FPT algorithms, and illustrate how these can be used to obtain a significantly improved FPT algorithm for the MAXIMUM LEAF SPANNING TREE problem. Furthermore, we sketch how the methods can be applied to a number of other well-known problems, including the parametric dual of DOMINATING SET (also known as NONBLOCKER), MATRIX DOMINATION, EDGE DOMINATING SET, and FEEDBACK VERTEX SET FOR UNDIRECTED GRAPHS. The main payoffs of these new methods are in improved functions f(k) in the FPT running times, and in general systematic approaches that seem to apply to a wide variety of problems.

A Fixed-Parameter Approach to Two-Layer Planarization

Abstract A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2-Layer Planarization problem: Can k edges be deleted from a given graph G so that the remaining graph is biplanar? This problem is NP-complete, and remains so if the permutation of the vertices in one layer is fixed (the 1-Layer Planarization problem). We prove that these problems are fixed-parameter tractable by giving linear-time algorithms for their solution (for fixed k). In particular, we solve the 2-Layer Planarization problem in O(k · 6k + |G|) time and the 1-Layer Planarization problem in O(3k · |G|) time. We also show that there are polynomial-time constant-approximation algorithms for both problems.

A 3-approximation algorithm for the subtree distance between phylogenies

In this paper, we give a (polynomial-time) 3-approximation algorithm for the rooted subtree prune and regraft distance between two phylogenetic trees. This problem is known to be NP-complete and the best previously known approximation algorithm is a 5-approximation. We also give a faster fixed-parameter algorithm for the rooted subtree prune and regraft distance than was previously known.

Parameterized approximation of dominating set problems

A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k-Dominating Set, or (2) produces a dominating set of size at most g(k), where g(k) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is no such FPT algorithm for g(k) of the form k+c (where c is a fixed constant, termed an additive FPT approximation), unless FPT=W[2]. We answer the analogous problem completely for the related Independent Dominating Set (IDS) problem, showing that IDS does not admit an FPT approximation algorithm, for any g(k), unless FPT=W[2].

Cluster analysis of Java dependency graphs

We present a novel approach to the analysis of dependency graphs of object-oriented programs. We propose to use the Girvan-Newman clustering algorithm to compute the modular structure of programs. This is useful in assisting software engineers to redraw component boundaries in software, in order to improve the level of reuse and maintainability. The results of this analysis can be used as a starting point for refactoring the software. We present BARRIO, an Eclipse plugin that can detect and visualise clusters in dependency graphs extracted from Java programs by means of source code and byte code analysis. These clusters are then compared with the modular structure of the analysed programs defined by package and container specifications. Two metrics are introduced to measure the degree of overlap between the defined and the computed modular structure. Some empirical results obtained from analysing non-trivial software packages are presented.

A Faster FPT Algorithm for the Maximum Agreement Forest Problem

Given two unrooted, binary trees, T1 and T2, leaf labelled bijectively by a set of species L, the Maximum Agreement Forest (MAF) problem asks to find a minimum cardinality collection F = {t1, ..., tk} of phylogenetic trees where each element of F is a subtree of both T1 and T2, the elements of F are pairwise disjoint, and the leaf labels for the elements of F partition the leaf label set L. We give an efficient fixed-parameter tractable (FPT) algorithm for the MAF problem, significantly improving on an FPT algorithm given in [2]. Whereas the algorithm from [2] has a running time of O(k3k) + p(|L|), our algorithm runs in time O(4k · k5) + p(|L|), where k bounds the size of the agreement forest and p(·) is a low order polynomial.

Barriers to modularity: an empirical study to assess the potential for modularisation of java programs

To deal with the challenges when building large and complex systems modularisation techniques such as component-based software engineering and aspect-oriented programming have been developed. In the Java space these include dependency injection frameworks and dynamic component models such as OSGi. The question arises as to how easy it will be to transform existing systems to take advantage of these new techniques. Anecdotal evidence from industry suggests that the presence of certain patterns presents barriers to refactoring of monolithic systems into a modular architecture. In this paper, we present such a set of patterns and analyse a large set of open-source systems for occurrences of these patterns. We use a novel, scalable static analyser that we have developed for this purpose. The key findings of this paper are that almost all programs investigated have a significant number of these patterns, implying that modularising will be therefore difficult and expensive.

Making Smart Moves to Untangle Programs

We present a novel algorithm to improve the design of programs by removing circular dependencies between packages without completely collapsing the package structure. This is achieved by moving classes between packages. The algorithm is based on a scoring function that is used to select the classes to be moved. The algorithm is validated against several open source case studies. The results show that our algorithm improves the program structure and removes inter-package cycles.