Nonuniform Reductions and NP-Completeness (original) (raw)
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Theory of Computing Systems, 2010
Reductions and completeness notions form the heart of computational complexity theory. Recently non-uniform reductions have been naturally introduced in a variety of settings concerning completeness notions for NP and other classes. We follow up on these results by strengthening some of them. In particular, we show that under certain well studied hypotheses:
Reducing the complexity of reductions
Computational Complexity, 2001
We build on the recent progress regarding isomorphisms of complete sets that was reported in . In that paper, it was shown that all sets that are complete under (non-uniform) AC 0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC 0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a "Gap Theorem", showing that all sets complete under AC 0 reductions are in fact already complete under NC 0 reductions. The following questions were left open in that paper:
A comparison of polynomial time completeness notions
Theoretical Computer Science, 1987
showed the differences among the power of several types of polynomial time reductions for DEXT ( = lJcpo DTIME(2"')). But questions concerning complete degrees w.r.t. these reductions have remained open. Here we show, in DEXT, almost all the possible differences between the compl&ness notions w.r.t. any pair of the following reductions: 6:. SE, G;(~,)_~~, SF, ~1, G[,~, ~z_~~, G: and their combinations. In order to show differences between several types of truth-table completenesses, we introduce the notion 's-easy subset'. For each d p -reduction, we first prove the property that all 6 r-complete sets for DEXT have s-easy subsets where the class 9 depends on each reduction type r. Then the construction ofa<,P -complete set which is not up -complete becomes a simple diagonalization argument. These conr:ructions also work for all deterministic superpolynomial complexity classes which have a GL -complete set. In order to construct a set which is <F-complete for DEXT < not <L-complete for DEXT, we use strings which are not polynomially producible from small-sized inputs (these strings are said to have large generalized Kolmogorov complexity (cf. ).
Collapsing and Separating Completeness Notions Under Average-Case and Worst-Case Hypotheses
Theory of Computing Systems, 2012
This paper presents the following results on sets that are complete for NP. (i) If there is a problem in NP that requires 2 n Ω(1) time at almost all lengths, then every many-one NP-complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. (ii) If there is a problem in co-NP that cannot be solved by polynomial-size nondeterministic circuits, then every many-one complete set is complete under length-increasing reductions that are computed by polynomial-size circuits. (iii) If there exist a one-way permutation that is secure against subexponential-size circuits and there is a hard tally language in NP∩co-NP, then there is a Turing complete language for NP that is not many-one complete. Our first two results use worst-case hardness hypotheses whereas earlier work that showed similar results relied on average-case or almost-everywhere hardness assumptions. The use of average-case and worst-case hypotheses in the last result is unique as previous results obtaining the same consequence relied on almost-everywhere hardness results.
On the Power of Randomized Reductions and the Checkability of SAT
2010 IEEE 25th Annual Conference on Computational Complexity, 2010
The closure of complexity classes is a delicate question and the answer varies depending on the type of reduction considered. The closure of most classes under many-to-one (Karp) reductions is clear, but the question becomes complicated when oracle (Cook) reductions are allowed, and even more so when the oracle reductions are allowed to be randomized.
Abstract Some Results in Computational Complexity
2008
In this thesis, we present some results in computational complexity. We consider two approaches for showing that #P has polynomial-size circuits. These approaches use ideas from the interactive proof for #3-SAT. We show that these approaches fail. We discuss whether there are instance checkers for languages complete for the class of approximate counting problems. We provide evidence that such instance checkers do not exist. We discuss the extent to which proofs of hierarchy theorems are constructive. We examine the problems that arise when trying to make the proof of Fortnow and Santhanam’s nonuniform BPP hierarchy theorem more constructive. ii Acknowledgements First, I would like to thank my supervisor, Charles Rackoff. Working with Charlie has been an intellectually stimulating and enjoyable experience. I greatly appreciate the many hours that Charlie spent explaining new concepts to me and suggesting ideas for this thesis. I would like to thank my second reader, Stephen Cook, for...
Some connections between bounded query classes and non-uniform complexity
Information and Computation, 2003
Let A(x) be the characteristic function of A. Consider the function C A k (x 1 , . . . , x k ) = A(x 1 ) · · · A(x k ). We show that if C A k can be computed in polynomial time with fewer than k queries to some set X then A ∈ 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 / = p 3 , there are functions computable using f (n) + 1 queries to SAT that are not computable using f (n) queries to SAT, for f (n) = O(log n). (2) If C A k , restricted to length n inputs, can be computed by an unbounded fanin oracle circuit of size s(n) and depth d(n), with k − 1 queries to some set X, then A can be computed with an unbounded fanin (non-oracle) circuit of size n O(k) s(n) and depth d(n) + O(1).