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Consider the function F A k (x 1 ; : : :; x k) = A(x 1) A(x k). We show that if F A k can be computed with less than k queries to some set X then A 2 P=poly. A generalization of this result has applications to bounded query classes, circuits, and enumerability. In particular we obtain the following. (1) Assuming p 3 6 = p 3 the hierarchy of functions(More)
‡e will ™onsider — l—ter formul—tion ˜y ghor et —lF ‘PQ“F ‡e model — d—t—˜—se —s @IA —n nE ˜it string x = x 1 x 2 · · · x n D together with @PA — ™omput—tion—l —gent th—t ™—n do ™omput—tions ˜—sed on ˜oth x —nd queries m—de to itF eli™e w—nts to o˜t—in x i su™h th—t the d—t—˜—se does not le—rn iF e™tu—lly eli™e w—nts more th—n th—t! she w—nts the d—t—˜—se(More)
This paper investigates the computational complexity of approximating several NP-optimization problems using the number of queries to an NP oracle as a complexity measure. The results show a trade-off between the closeness of the approximation and the number of queries required. For an approximation factor k(n), log log k(n) n queries to an NP oracle can be(More)
A two-dimensional grid is a set G n,m = [n] × [m]. A grid G n,m is c-colorable if there is a function χ n,m : G n,m → [c] such that there are no rectangles with all four corners the same color. We address the following question: for which values of n and m is G n,m c-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to the(More)
We classify functions in recursive graph theory in terms of how many queries to K (or ∅ ′′ or ∅ ′′′) are required to compute them. We show that (1) binary search is optimal (in terms of the number of queries to K) for finding the chromatic number of a recursive graph and that no set of Turing degree less than 0 ′ will suffice, (2) determining if a recursive(More)
1) Introduction We examine the problem of coloring part of a k-colorable graph, while not knowing the rest of it. To illustrate some of our concepts, we describe a somewhat whimsical scenario, which we call the Mapmaker's Dilemma. Picture a 12th-century mapmaker who is given a map of Europe and the countries adjacent to Europe, and is told to 4-color the(More)