On the parameterized complexity of short computation and factorization

Abstract

SummaryA 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,DEF,FH,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. These include a natural parameterized version of the famous Post Correspondence Systems. 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. The result could be viewed as one analogue of Cook's theorem and we believe provides strong evidence for the parameterized intractability of W [1].

On the Parameterised Complexity of String Morphism Problems

“…, V k of V , such that every V i is an independent set, whether or not G has a clique of size k. It is a well-known fact that both k-Clique and k-Multicoloured-Clique are complete for W [1] (with respect to parameterised reductions) [12]. Another W [1]-complete problem that we use is Short-NTM-Comp, i. e., to decide for a nondeterministic Turing machine M , a string w over the input alphabet of M and a parameter k ∈ N whether or not M has an accepting computation for w with at most k steps (see Cai et al [5], Downey et al [10], Cesati [7]). The classes of polynomial and nondeterministically polynomial time solvable problems are denoted by P and N P, respectively.…”

The Parameterized Complexity of Local Consistency

“…Observing that c is the variable to which the partial instantiation α cannot be extended in both directions of the proof of Claim 2, the co-W Proof. Cesati and Di Ianni [6] showed that the following parameterized problem is in W [1] (see also [3] where W[1]-completeness is established for the single-tape version of the problem).…”

“…Problems that can be solved in uniform polynomial time are called fixed-parameter tractable (FPT), problems that can be solved in nonuniform polynomial time are further classified within a hierarchy of parameterized complexity classes forming the chain FPT ⊆ W[1] ⊆ W[2] ⊆ W [3] ⊆ · · · , where all inclusions are believed to be strict.…”

W-Hierarchies Defined by Symmetric Gates

“…The reader familiar with the characterization of W [1] in terms of nondeterministic random access machines [4] will easily see that the algorithm A can be simulated by a program for such a machine. The second option: It is not hard, using the algorithm A, to construct an fpt-reduction of p-WSAT(c ≤ , f ) to the parameterized short halting problem for nondeterministic single-tape Turing machines, a problem in W[1] (see [2]). …”