R. G. Downey, M. R. Fellows and K. W. Regan, "Parameterized Circuit Complexity and the W Hierarchy, " Theoretical Computer Science A 191 (1998), 91--115,  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... quali ers) such that, when the variables in t 1 [ t r are made true and all other variables are made false, formula X is true Parameter: r, k 1 ; k r AW[ complete (kernel problem for the class AW[t] result follows from Trade o Theorems AW[1] AW[2] AW[ [1, 2, 57, 58, 53]) Note: See also Unitary Quantified Boolean t Normalized Formula Satisfiability. Quantified Circuit Satisfiability Instance: An integer r; a sequence s 1 ; s r of pairwise disjoint sets of boolean variables; a decision circuit X with the variables s 1 [ s r as inputs; integers k 1 ....

R. G. Downey, M. R. Fellows, and K. W. Regan. Parameterized circuit complexity and the W hierarchy. Theoretical Computer Science, 191(1{ 2):97-115, 1998.

.... quali ers) such that, when the variables in t 1 [ t r are made true and all other variables are made false, formula X is true Parameter: r, k 1 ; k r AW[ complete (kernel problem for the class AW[t] result follows from Trade o Theorems AW[1] AW[2] AW[ [1, 2, 57, 58, 53]) Note: See also Unitary Quantified Boolean t Normalized Formula Satisfiability. Quantified Circuit Satisfiability Instance: An integer r; a sequence s 1 ; s r of pairwise disjoint sets of boolean variables; a decision circuit X with the variables s 1 [ s r as inputs; integers k 1 ....

R. G. Downey, M. R. Fellows, and K. W. Regan. Parameterized circuit complexity and the W hierarchy. In P. Beame and S. Buss, editors, AMS-DIMACS Proceedings Series for the

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R. G. Downey, M. R. Fellows and K. W. Regan, "Parameterized Circuit Complexity and the W Hierarchy, " Theoretical Computer Science A 191 (1998), 91--115,

....known. Some would have very interesting concrete consequences. For example, if there is a parameterized analog of Toda s theorem, then the problem UNIQUE CLIQUE of determining whether a graph has a unique k clique (for parameter k) is as hard as any problem at any level of the W [t] hierarchy [DFR98a]. Currently, however, we lack even the basic probability amplification results that such an analog would presumably require. What about Oracle results The ones from [DF93] are somehow unsatisfying. Perhaps there are better models. What about highness lowness This might be relevant to GRAPH ....

R. G. Downey, M. R. Fellows and K. W. Regan, "Parameterized Circuit Complexity and the W Hierarchy," Theoretical Computer Science A 191 (1998), 91--115,

....similar discussion see [Gur89] on sparse hard sets [CF96] There is a partial analog of Ladner s Density Theorem [DF93, DF98] Notably lacking are parameterized analogs of probability amplification. An analog of Toda s Theorem [To91] would be very interesting, as together with the results of [DFR98a] it would show that Unique k Clique is hard for W [t] for all t via randomized parametric reductions. FPT or W Hard The following concrete problems seem important to classify because of their significant applications. 1. The naturally parameterized Directed Feedback Vertex Set problem. This can ....

R. G. Downey, M. R. Fellows and K. W. Regan, "Parameterized Circuit Complexity and the W Hierarchy," Theoretical Computer Science A 191 (1998), 91--115.

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R.G.Downey, M.R. Fellows, and K.R. Regan, Parameterized circuit complexity and the W hierarchy, Theoretical Comp. Science, 189, (1997), to appear.

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