Fixed-parameter Tractable Problem Research Articles

A fixed-parameter algorithm for scheduling unit dependent tasks on parallel machines with time windows

This paper proves that the existence of a feasible schedule for a set of dependent tasks of unit execution times with release dates and deadlines on a limited number of processors is a fixed-parameter tractable problem. The parameter considered is the pathwidth of the interval graph associated with the time windows of the tasks. A fixed-parameter algorithm based on a dynamic programming approach is developed and proved to solve this decision problem.Fixed-parameter algorithms for the two classical problems P|prec,pi=1,ri|Cmax and P|prec,pi=1,ri|Lmax are then derived using a binary search. They are, as far as we know, the first fixed-parameter algorithms for these scheduling problems.

Parameterized Testability

This article studies property testing for NP optimization problems with parameter k under the general graph model with an augmentation of random edge sampling capability. It is shown that a variety of such problems, including k -Vertex Cover, k -Feedback Vertex Set, k -Multicut, k -Path-Free, and k -Dominating Set, are constant-query testable if k is constant. It should be noted that the first four problems are fixed parameter tractable (FPT) and it turns out that algorithmic techniques for their FPT algorithms (branch-and-bound search, color coding, etc.) are also useful for our testers. k -Dominating Set is W [2]-hard, but we can still test the property with a constant number of queries, since the definition of ε-farness makes the problem trivial for non-sparse graphs that are the source of hardness for the original optimization problem. We also consider k -Odd Cycle Transversal, which is another well-known FPT problem, but we only give a sublinear-query tester when k is a constant.

Inferring local transition functions of discrete dynamical systems from observations of system behavior

We consider the problem of inferring the local transition functions of discrete dynamical systems from observed behavior. Our focus is on synchronous systems whose local transition functions are threshold functions. We assume that the topology of the system is known and that the goal is to infer a threshold value for each node so that the system produces the observed behavior. We show that some of these inference problems are efficiently solvable while others are NP-complete, even when the underlying graph of the dynamical system is a simple path. We identify a fixed parameter tractable problem in this context. We also consider constrained versions of threshold inference problems where the input includes a set of equality or inequality constraints (which specify pairs of nodes which must have the same threshold value or different threshold values). We present algorithmic and complexity results for several constrained threshold inference problems.

Quantifying Hybridization in Realistic Time

Recently, numerous practical and theoretical studies in evolutionary biology aim at calculating the extent to which reticulation-for example, horizontal gene transfer, hybridization, or recombination-has influenced the evolution for a set of present-day species. It has been shown that inferring the minimum number of hybridization events that is needed to simultaneously explain the evolutionary history for a set of trees is an NP-hard and also fixed-parameter tractable problem. In this article, we give a new fixed-parameter algorithm for computing the minimum number of hybridization events for when two rooted binary phylogenetic trees are given. This newly developed algorithm is based on interleaving-a technique using repeated kernelization steps that are applied throughout the exhaustive search part of a fixed-parameter algorithm. To show that our algorithm runs efficiently to be applicable to a wide range of practical problem instances, we apply it to a grass data set and highlight the significant improvements in terms of running times in comparison to an algorithm that has previously been implemented.

A Parametrized Algorithm for Matroid Branch-Width

Branch-width is a structural parameter very closely related to tree-width, but branch-width has an immediate generalization from graphs to matroids. We present an algorithm that, for a given matroid M of bounded branch-width t which is represented over a finite field, finds a branch decomposition of M of width at most 3t in cubic time. Then we show that the branch-width of M is a uniformly fixed-parameter tractable problem. Other applications include recognition of matroid properties definable in the monadic second-order logic for bounded branch-width, and [S.-I. Oum, Approximating rank-width and clique-width quickly, in Proceedings of the 31st International Workshop on Graph-Theoretic Concepts in Computer Science, Springer-Verlag, Heidelberg, to appear] a cubic time approximation algorithm for graph rank-width and clique-width.

Linear-time recognition of bipartite graphs plus two edges

Cai and Schieber (1997) proved that bipartite graphs plus one edge can be recognized in linear time. We extend their result to bipartite graphs plus two edges. Our algorithm works on a depth-first-search spanning tree. This gives, as a byproduct, also a simplified solution to the one-edge case. It is a notoriously open question whether recognizing bipartite graphs plus k edges is a fixed-parameter tractable problem. The present result might support the affirmative conjecture.