Liming Cai

Advice classes of parameterized tractability

Abstract Many natural computational problems have input consisting of two or more parts, one of which may be considered a parameter . For example, there are many problems for which the input consists of a graph and a positive integer. A number of results are presented concerning parameterized problems that can be solved (uniformly with respect to the parameter) in complexity classes below P , given a single word of advice for each parameter value. Different ways in which the word of advice can be employed are considered, and it is shown that the class FPT of tractable parameterized problems (the parameterized analog of P ) has interesting and natural internal structure.

On Fixed-Parameter Tractability and Approximability of NP Optimization Problems

Fixed-parameter tractability of NP optimization problems is studied by relating it to approximability of the problems. It is shown that an NP optimization problem is fixed-parameter tractable if it admits a fully polynomial-time approximation scheme, or if it belongs to the class MAX SNP or to the class MIN F+?1. This provides strong evidence that noW1-hard NP optimization problems belong to these optimization classes and includes a very large class of approximable optimization problems into the class of fixed-parameter tractable problems. Evidence is also demonstrated to support the current working hypothesis in the theory of fixed-parameter tractability.

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

On the structure of parameterized problems in NP

k

On fixed-parameter tractability and approximability of NP-hard optimization problems

steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word

On the amount of nondeterminism and the power of verifying

x

On input read-modes of alternating Turing machines

can be derived in a grammar

Circuit Bottom Fan-in and Computational Power

G

Circuit bottom fan-in and computational power

in

On log-time alternating Turing machines of alternation depth k

k