Rodney G. Downey

Fixed-parameter tractability and completeness II: on completeness for W [1]

Abstract For many fixed-parameter problems that are trivially solvable in polynomial-time, such as k -DOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as FEEDBACK VERTEX SET, exhibit fixed-parameter tractability : for each fixed k the problem is solvable in time bounded by a polynomial of degree c , where c is a constant independent of k . In a previous paper, the W Hierarchy of parameterized problems was defined, and complete problems were identified for the classes W [ t ] for t ⩾ 2. Our main result shows that INDEPENDENT SET is complete for W [1].

On problems without polynomial kernels

Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.

Fixed-Parameter Tractability and Completeness I: Basic Results

For many fixed-parameter problems that are trivially soluable in polynomial-time, such as (

Parameterized Computational Feasibility

k

Fixed-parameter tractability and completeness IV: On completeness for W(P) and PSPACE analogues

-)DOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as (

Parameterized complexity analysis in computational biology

k

Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems.

-)FEEDBACK VERTEX SET, exhibit fixed-parameter tractability: for each fixed

Advice classes of parameterized tractability

k

Fixed-parameter intractability

the problem is soluable in time bounded by a polynomial of degree

On Problems without Polynomial Kernels (Extended Abstract)

c