On the parameterized complexity of short computation and factorization

Abstract.

A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graph-theoretic problems, VLSI layout, games, computational biology, cryptography, and computational learning [ADF,BDHW,BFH, DEF,DF1-7,FHW,FK]. We here study the parameterized complexity of two kinds of problems: (1) problems concerning parameterized computations of Turing machines, such as determining whether a nondeterministic machine can reach an accept state in $k$ steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word $x$ can be derived in a grammar $G$ in $k$ steps, or whether a permutation has a factorization of length $k$ over a given set of generators. We show hardness and completeness for these problems for various levels of the $W$ hierarchy. In particular, we show that Short TM Computation is complete for $W[1]$. This gives a new and useful characterization of the most important of the apparently intractable parameterized complexity classes.