The parameterized computational complexity of a collection of well-known problems including: Bandwidth, Precedence constrained k-processor scheduling, Longest Common Subsequence, DNA physical mapping (or Intervalizing colored graphs), Perfect phylogeny (or Triangulating colored graphs), Colored cutwidth, and Feasible register assignment is explored. It is shown that these problems are hard for various levels of the W hierarchy. In the case of Precedence constrained k-processor scheduling the results can be interpreted as providing substantial new complexity lower bounds on the outcome of [OPEN 8] of the Garey and Johnson list. We also obtain the conjectured "third strike" against Perfect phylogeny.

In this paper, we consider a large class of vertex partitioning problems and apply to those the theory of algorithm design for problems restricted to partial k-trees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications.

In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a tree-decomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a tree-decomposition or (path-decomposition) of G of width at most k, and that use O(|V|) time. In contrast with previous solutions, our algorithms do not rely on non-constructive reasoning, and are single exponential in k and l. This result can be combined with a result of Reed [37], yielding explicit O(n log n) algorithms for the problem, given a graph G, to determine whether the treewidth (or pathwidth) of G is at most k, and if so, to find a tree- (or path-)decomposition of width at most k (k constant). Also, Bodlaender [13] has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constants k, there exists a polynomial time algorithm, that, when given a graph G = (V; E) with treewidth k, computes the pathwidth of G and a minimum path decomposition of G.

In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"-instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete. 1 Introduction This paper focuses on a number of graph decision problems which share the characteristic that all have a uniform upper bo...

The complexity of the simple maxcut problem is investigated for several special classes of graphs. It is shown that this problem is NP-complete when restricted to one of the following classes: chordal graphs, undirected path graphs, split graphs, tripartite graphs, and graphs that are the complement of a bipartite graph. The problem can be solved in polynomial time, when restricted to graphs with bounded treewidth, or cographs. We also give large classes of graphs that can be seen as generalizations of classes of graphs with bounded treewidth and of the class of the cographs, and allow polynomial time algorithms for the simple max cut problem. 1 Introduction One of the best known combinatorial graph problems is the max cut problem. In this problem, we have a weighted, undirected graph G = (V; E) and we look for a partition of the vertices of G into two disjoint sets, such that the total weight of the edges that go from one set to the other is as large as possible. In the simple max cu...

In this paper we propose a new method for deriving a practical linear-time algorithm from the specification of a maximum-weight sum problem: From the elements of a data structure x, find a subset which satisfies a certain property p and whose weightsum is maximum. Previously proposed methods for automatically generating linear-time algorithms are theoretically appealing, but the algorithms generated are hardly useful in practice due to a huge constant factor for space and time. The key points of our approach are to express the property p by a recursive boolean function over the structure x rather than a usual logical predicate and to apply program transformation techniques to reduce the constant factor. We present an optimization theorem, give a calculational strategy for applying the theorem, and demonstrate the effectiveness of our approach through several nontrivial examples which would be difficult to deal with when using the methods previously available.

We consider the framework of Parameterized Complexity and we investigate the issue of which problems do admit efficient fixed parameter parallel algorithms. In particular, we introduce two classes of efficiently parallelizable parameterized problems, PNC and FPP, according to the degree of efficiency we want to obtain. We sketch both some FPP-algorithms solving natural parameterized problems and a useful tool for proving membership to FPP based on the concept of treewidth. We then focus our attention on parameterized parallel intractability and prove that a necessary condition for a parameterized problem to be complete for the class of sequentially fixed parameter tractable problems (with respect to reductions preserving membership to PNC) is that it is not in NC for some fixed value of the parameter (unless P = NC). Finally, we give two alternative characterizations of both PNC and FPP and we use them to prove the PNC-completeness of two natural parameterized problems.

This thesis investigates the computational complexity of algorithmic problems defined on graphs. At the abstract level of the complexity spectrum we discriminate polynomial-time solvable problems from N P-complete problems, while at the concrete level we improve on polynomial-time algorithms for generally hard problems restricted to tree-decomposable graphs. One contribution of this thesis is a precise characterization of vertex partitioning problems which include variants of domination, coloring and packing. An elaboration of this characterization is given for problems defined over vertex subsets and over maximal/minimal vertex subsets. We introduce several new graph parameters as vertex partition generalizations of classical parameters. The given characterizations provide a basis for a taxonomy of vertex partitioning problems, facilitating their common algorithmic treatment and allowing for their uniform complexity classification. We explore the computational complexity of two important types of problems

We give linear time algorithms constructing canonical representations of partial 2-trees (series-parallel graphs) and partial 3-trees. 1 Introduction A canonical representation of a family of graphs assigns to each member of the family a label that is independent of any arbitrary vertex numbering: two graphs have the same canonical representation if and only if they are isomorphic. Thus, the graph isomorphism problem can be solved using canonical representations and solved efficiently if such representations can be efficiently computed and compared. Other uses of canonical representations are to investigate the structure of the automorphism group of a graph and to generate random graphs with some distribution over isomorphism classes. Most graph representations are not canonical since vertices are arbitrarily numbered. But if we consider all possible vertex permutations, compute the corresponding representations, and select the lexicographically smallest, then we get a canonical repre...

We derive an O(n2)-time algorithm for calculating the sequence of numbers g0(G), g1(G), g2(G),... of distinct ways to embed a 3-regular Halin graph G on the respective orientable surfaces S0, S1, S2,.... Key topological features are a quadrangular decomposition of plane Halin graphs and a new recombinant-strands reassembly process that fits pieces together three-at-a-vertex. Key algorithmic features are reassembly along a post-order traversal, with just-in-time dynamic assignment of roots for quadrangular pieces encountered along the tour. 1.