Advice Classes of Parameterized Tractability

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. ∗The authors thank Ashish Karkare for pointing out an error in an earlier version of the paper. †Cai is supported by an Engineering Excellence Award from Texas A & M University, and Chen by NSF Grant CCR-9110824. ‡Research supported in part by a grant from the Victoria University IGC, by the United States / New Zealand Cooperative Science Foundation under grant INT 90-20558, and by the Mathematical Sciences Institute at Cornell and Cornell University. Email address: [email protected]. §Research supported in part by the National Science and Engineering Research Council of Canada, and by the United States National Science Foundation under grant MIP-8919312. Email address: [email protected].