A simple linear-time algorithm for finding path-decompositions of small width

We described a simple algorithm running in linear time for each fixed constant $k$, that either establishes that the pathwidth of a graph $G$ is greater than $k$, or finds a path-decomposition of $G$ of width at most $O(2^{k})$. This provides a simple proof of the result by Bodlaender that many families of graphs of bounded pathwidth can be recognized in linear time.Comment: 9 page

Obstructions to within a few vertices or edges of acyclic

Finite obstruction sets for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. It has been known for several years that, in principle, obstruction sets can be mechanically computed for most natural lower ideals. In this paper, we describe a general-purpose method for finding obstructions by using a bounded treewidth (or pathwidth) search. We illustrate this approach by characterizing certain families of cycle-cover graphs based on the two well-known problems: $k$-{\sc Feedback Vertex Set} and $k$-{\sc Feedback Edge Set}. Our search is based on a number of algorithmic strategies by which large constants can be mitigated, including a randomized strategy for obtaining proofs of minimality.Comment: 16 page

FPT is Characterized by Useful Obstruction Sets

Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-Pathwidth, complementing the trivial AND-composition that is known for this problem. In the other direction, we show that OR-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the NO-instances by polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201

A Generalization of Nemhauser and Trotter\u27s Local Optimization Theorem

The Nemhauser-Trotter local optimization theorem applies to the NP-hard textsc{Vertex Cover} problem and has applications in approximation as well as parameterized algorithmics. We present a framework that generalizes Nemhauser and Trotter\u27s result to vertex deletion and graph packing problems, introducing novel algorithmic strategies based on purely combinatorial arguments (not referring to linear programming as the Nemhauser-Trotter result originally did). We exhibit our framework using a generalization of textsc{Vertex Cover}, called textrm{sc Bounded-Degree Deletion}, that has promise to become an important tool in the analysis of gene and other biological networks. For some fixed~$dgeq 0$, textrm{sc Bounded-Degree Deletion} asks to delete as few vertices as possible from a graph in order to transform it into a graph with maximum vertex degree at most~$d$. textsc{Vertex Cover} is the special case of $d=0$. Our generalization of the Nemhauser-Trotter theorem implies that textrm{sc Bounded-Degree Deletion} has a problem kernel with a linear number of vertices for every constant~$d$. We also outline an application of our extremal combinatorial approach to the problem of packing stars with a bounded number of leaves. Finally, charting the border between (parameterized) tractability and intractability for textrm{sc Bounded-Degree Deletion}, we provide a W[2]-hardness result for textrm{sc Bounded-Degree Deletion} in case of unbounded $d$-values