Partitioning Perfect Graphs into Stars

The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor

Computing the chromatic number of t-(v,k,[lambda]) designs

Colouring t-designs has previously been shown to be an NP-complete problem; heuristics and a practical algorithm for this problem were developed for this thesis; the algorithm was then employed to find the chromatic numbers of the sixteen non- isomorphic 2-(25, 4, 1) designs and the four cyclic 2-(19, 3, 1) designs. This thesis additionally examines the existing literature on colouring and finding chromatic numbers of t-designs

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

New Steiner 2-designs from old ones by paramodifications

Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals

Parameterized Complexity Dichotomy for Steiner Multicut

The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt $\subseteq$ V(G) of size at most p, and an integer k, whether there is a set S of at most k edges or nodes s.t. of each set Ti at least one pair of terminals is in different connected components of G \ S. This problem generalizes several graph cut problems, in particular the Multicut problem (the case p = 2), which is fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. We provide a dichotomy of the parameterized complexity of Steiner Multicut. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). We highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+t on general graphs (but has no polynomial kernel, even on trees). We present two proofs: one using the randomized contractions technique of Chitnis et al, and one relying on new structural lemmas that decompose the Steiner cut into important separators and minimal s-t cuts. - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Hence, the results of Marx and Razgon, and Bousquet et al. do not generalize to Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).Comment: As submitted to journal. This version also adds a proof of fixed-parameter tractability for parameter k+t using the technique of randomized contraction

Order-Related Problems Parameterized by Width

In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints. While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory. Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory. In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin