For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction which we call Sub-Exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that CircuitSAT is SERF-complete for all NP-search problems, and that for any fixed k, k-SAT, k-Colorability, k-Set Cover, Independent Set, Clique, Vertex Cover, are SERF--complete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, sub-exponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC 0 ; that is, bounds of the form 2 \Omega\Gamma n) . This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even l...

In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixed-parameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that k-Step Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph k-Coloring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.

The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.

The purposes of this paper are two: (1) to give an exposition of the main ideas of parameterized complexity, and (2) to discuss the connections of parameterized complexity to the systematic design of heuristics and approximation algorithms.

Abstract. It is well known that for a fixed relational database query φ in m free variables, it can be determined in time polynomial in the size n of the database whether there exists an m-tuple x that belongs to the relation defined by the query. For the best known algorithms, however, the exponent of the polynomial is proportional to the size of the query. We study the data complexity of this problem parameterized by the size k = |φ | of the query, and answer a question recently raised by Yannakakis [Yan95]. Our main results show: (1) the general problem is complete for the parametric complexity class AW[∗], and (2) when restricted to monotone queries, the problem is complete for the fundamental parametric complexity class W[1]. The practical significance of these results is that unless the parameterized complexity hierarchy collapses, there are unlikely to be algorithms that solve this problem (even under the restriction to monotone queries) in time f(k)n c where f is an arbitrary function of k and c is a constant independent of k. An important consequence of the proof of (2) is a significantly improved characterization of the parameterized complexity class W[1]. Previous results by Downey and Fellows characterize W[1] in terms of the k-Weighted Circuit Satisfiability problem, for families of circuits that satisfy: (1) the depth of the circuits is bounded by a constant c, (2) on any input-output path there is at most one gate having unbounded fanin (termed a large gate), with all other gates having fan-in bounded by c (that is, small gates). We show that the definition can be broadened by allowing circuits of depth bounded by an arbitrary function f(k). If we denote this parameterized complexity class W ∗ [1], then our corollary

We describe a point of view about the parameterized computational complexity framework in the broad context of one of the central issues of theoretical computer science as a field: the problem of systematically coping with computational intractability. Those already familiar with the basic ideas of parameterized complexity will nevertheless find here something new: the emerging systematic connections between fixed-parameter tractability techniques and the design of useful heuristic algorithms, and also perhaps the philosophical maturation of the parameterized complexity program.

We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilinear-time computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m -complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on first-order definability, as well as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

A parameterized problem hL; ki belongs to W [t] if there exists k computed from k such that hL; ki reduces to the weight-k satisfiability problem for weft-t circuits. We relate the fundamental question of whether the W [t] hierarchy is proper to parameterized problems for constant-depth circuits. We define classes G[t] as the analogues of AC depth-t for parameterized problems, and N [t] by weight-k existential quantification on G[t], by analogy with NP = 9 \Delta P. We prove that for each t, W [t] equals the closure under fixed-parameter reductions of N [t]. Then we prove, using Sipser's results on the AC depth-t hierarchy, that both the G[t] and the N [t] hierarchies are proper. If this separation holds up under parameterized reductions, then the W [t] hierarchy is proper.

We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilinear-time computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy hasseveral alternative characterizations. We de ne both SBH (QL) and its corresponding hierarchy of function classes, ql and present a variety of problems in these classes, including m-complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that determining its precise relationship to deterministic time classes can imply P 6 = PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on rst-order de nability, aswell as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

We give a review of the development and some of the achievements of the theory of parameterized complexity in the last (nearly) 10 years. We highlight what we see as some of the major open questions and programmes for future development.