63 research outputs found
Hardness Amplification of Optimization Problems
In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products.
We say that an optimization problem ? is direct product feasible if it is possible to efficiently aggregate any k instances of ? and form one large instance of ? such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem ?, our hardness amplification theorem may be informally stated as follows:
If there is a distribution D over instances of ? of size n such that every randomized algorithm running in time t(n) fails to solve ? on 1/?(n) fraction of inputs sampled from D, then, assuming some relationships on ?(n) and t(n), there is a distribution D\u27 over instances of ? of size O(n??(n)) such that every randomized algorithm running in time t(n)/poly(?(n)) fails to solve ? on 99/100 fraction of inputs sampled from D\u27.
As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium
Superlinear lower bounds based on ETH
Andras Z. Salamon acknowledges support from EPSRC grants EP/P015638/1 and EP/V027182/1.We introduce techniques for proving superlinear conditional lower bounds for polynomial time problems. In particular, we show that CircuitSAT for circuits with m gates and log(m) inputs (denoted by log-CircuitSAT) is not decidable in essentially-linear time unless the exponential time hypothesis (ETH) is false and k-Clique is decidable in essentially-linear time in terms of the graph's size for all fixed k. Such conditional lower bounds have previously only been demonstrated relative to the strong exponential time hypothesis (SETH). Our results therefore offer significant progress towards proving unconditional s uperlinear time complexity lower bounds for natural problems in polynomial time.Postprin
A PCP Characterization of AM
We introduce a 2-round stochastic constraint-satisfaction problem, and show
that its approximation version is complete for (the promise version of) the
complexity class AM. This gives a `PCP characterization' of AM analogous to the
PCP Theorem for NP. Similar characterizations have been given for higher levels
of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the
result for AM might be of particular significance for attempts to derandomize
this class.
To test this notion, we pose some `Randomized Optimization Hypotheses'
related to our stochastic CSPs that (in light of our result) would imply
collapse results for AM. Unfortunately, the hypotheses appear over-strong, and
we present evidence against them. In the process we show that, if some language
in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there
exist hard-on-average optimization problems of a particularly elegant form.
All our proofs use a powerful form of PCPs known as Probabilistically
Checkable Proofs of Proximity, and demonstrate their versatility. We also use
known results on randomness-efficient soundness- and hardness-amplification. In
particular, we make essential use of the Impagliazzo-Wigderson generator; our
analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page
Stretching demi-bits and nondeterministic-secure pseudorandomness
We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work [25], and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following: Demi-bit stretch: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are secure against nondeterministic statistical tests [25]. They were introduced to rule out certain approaches to proving strong complexity lower bounds beyond the limitations set out by the Natural Proofs barrier of Razborov and Rudich [23]. Whether demi-bits are stretchable at all had been an open problem since their introduction. We answer this question affirmatively by showing that: every demi-bit b : {0, 1}n → {0, 1}n+1 can be stretched into sublinear many demi-bits b′: {0, 1}n → {0, 1}n+nc , for every constant 0 < c < 1. Average-case hardness: Using work by Santhanam [26], we apply our results to obtain new averagecase Kolmogorov complexity results: we show that Kpoly[n-O(1)] is zero-error average-case hard against NP/poly machines iff Kpoly[n-o(n)] is, where for a function s(n) : N → N, Kpoly[s(n)] denotes the languages of all strings x ∈ {0, 1}n for which there are (fixed) polytime Turing machines of description-length at most s(n) that output x. Characterising super-bits by nondeterministic unpredictability: In the deterministic setting, Yao [31] proved that super-polynomial hardness of pseudorandom generators is equivalent to ("nextbit") unpredictability. Unpredictability roughly means that given any strict prefix of a random string, it is infeasible to predict the next bit. We initiate the study of unpredictability beyond the deterministic setting (in the cryptographic regime), and characterise the nondeterministic hardness of generators from an unpredictability perspective. Specifically, we propose four stronger notions of unpredictability: NP/poly-unpredictability, coNP/poly-unpredictability, ∩-unpredictability and ∪unpredictability, and show that super-polynomial nondeterministic hardness of generators lies between ∩-unpredictability and ∪unpredictability. Characterising super-bits by nondeterministic hard-core predicates: We introduce a nondeterministic variant of hard-core predicates, called super-core predicates. We show that the existence of a super-bit is equivalent to the existence of a super-core of some non-shrinking function. This serves as an analogue of the equivalence between the existence of a strong pseudorandom generator and the existence of a hard-core of some one-way function [8, 12], and provides a first alternative characterisation of super-bits. We also prove that a certain class of functions, which may have hard-cores, cannot possess any super-core
The Value of Help Bits in Randomized and Average-Case Complexity
"Help bits" are some limited trusted information about an instance or
instances of a computational problem that may reduce the computational
complexity of solving that instance or instances. In this paper, we study the
value of help bits in the settings of randomized and average-case complexity.
Amir, Beigel, and Gasarch (1990) show that for constant , if instances
of a decision problem can be efficiently solved using less than bits of
help, then the problem is in P/poly. We extend this result to the setting of
randomized computation: We show that the decision problem is in P/poly if using
help bits, instances of the problem can be efficiently solved with
probability greater than . The same result holds if using less than
help bits (where is the binary entropy function),
we can efficiently solve fraction of the instances correctly with
non-vanishing probability. We also extend these two results to non-constant but
logarithmic . In this case however, instead of showing that the problem is
in P/poly we show that it satisfies "-membership comparability," a notion
known to be related to solving instances using less than bits of help.
Next we consider the setting of average-case complexity: Assume that we can
solve instances of a decision problem using some help bits whose entropy is
less than when the instances are drawn independently from a particular
distribution. Then we can efficiently solve an instance drawn from that
distribution with probability better than .
Finally, we show that in the case where is super-logarithmic, assuming
-membership comparability of a decision problem, one cannot prove that the
problem is in P/poly by a "black-box proof.
Better Pseudorandom Generators from Milder Pseudorandom Restrictions
We present an iterative approach to constructing pseudorandom generators,
based on the repeated application of mild pseudorandom restrictions. We use
this template to construct pseudorandom generators for combinatorial rectangles
and read-once CNFs and a hitting set generator for width-3 branching programs,
all of which achieve near-optimal seed-length even in the low-error regime: We
get seed-length O(log (n/epsilon)) for error epsilon. Previously, only
constructions with seed-length O(\log^{3/2} n) or O(\log^2 n) were known for
these classes with polynomially small error.
The (pseudo)random restrictions we use are milder than those typically used
for proving circuit lower bounds in that we only set a constant fraction of the
bits at a time. While such restrictions do not simplify the functions
drastically, we show that they can be derandomized using small-bias spaces.Comment: To appear in FOCS 201
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