We present a new optical method for solving bounded (input-length-restricted) NP-complete combinatorial problems. We have chosen to demonstrate the method with an NP-complete problem called the traveling salesman problem (TSP). The power of optics in this method is realized by using a fast matrix–vector multiplication between a binary matrix, representing all feasible TSP tours, and a grayscale vector, representing the weights among the TSP cities. The multiplication is performed optically by using an optical correlator. To synthesize the initial binary matrix representing all feasible tours, an efficient algorithm is provided. Simulations and experimental results prove the validity of the new

Abstract. A fundamental goal of computational complexity (and foundations of cryptography) is to find a polynomial-time samplable distribution (e.g., the uniform distribution) and a language in NTIME(f(n)) for some polynomial function f, such that the language is hard on the average with respect to this distribution, given that NP is worst-case hard (i.e. NP ̸ = P, or NP ̸ ⊆ BPP). Currently, no such result is known even if we relax the language to be in nondeterministic sub-exponential time. There has been a long line of research trying to explain our failure in proving such worst-case/average-case connections [FF93,Vio03,BT03,AGGM06]. The bottom line of this research is essentially that (under plausible assumptions) non-adaptive Turing reductions cannot prove such results. In this paper we revisit the problem. Our first observation is that the above mentioned negative arguments extend to a non-standard notion of average-case complexity, in which the distribution on the inputs with respect to which we measure the average-case complexity of the language, is only samplable in super-polynomial time. The significance of this result stems from the fact that in this non-standard setting, [GSTS05] did show a worst-case/average-case connection. In other words, their techniques give a way to bypass the impossibility arguments. By taking a closer look at the proof of [GSTS05], we discover that the worst-case/averagecase connection is proven by a reduction that ”almost ” falls under the category ruled out by the negative result. This gives rise to an intriguing new notion of (almost black-box) reductions. After extending the negative results to the non-standard average-case setting of [GSTS05], we ask whether their positive result can be extended to the standard setting, to prove some new worst-case/average-case connections. While we can not do that unconditionally, we are able to show that under a mild derandomization assumption, the worst-case hardness of NP implies the average-case hardness of NTIME(f(n)) (under the uniform distribution) where f is computable in quasi-polynomial time. 1

We may believe SAT does not have small Boolean circuits. But is it possible that some language with small circuits looks indistiguishable from SAT to every polynomialtime bounded adversary? We rule out this possibility. More precisely, assuming SAT does not have small circuits, we show that for every language ¡ with small circuits, there exists a probabilistic polynomial-time algorithm that makes black-box queries to ¡, and produces, for a given input length, a Boolean formula on which ¡

We may believe SAT does not have small Boolean circuits. But is it possible that some language with small circuits looks indistiguishable from SAT to every polynomial-time bounded adversary? We rule out this possibility. More precisely, assuming SAT does not have small circuits, we show that for every language with small circuits, there exists a probabilistic polynomial-time algorithm that makes black-box queries to, and produces, for a given input length, a Boolean formula on which differs from SAT. A key step for obtaining this result is a new proof of the main result by Gutfreund, Shaltiel, and Ta-Shma reducing average-case hardness to worst-case hardness via uniform adversaries that know the algorithm they fool. The new adversary we construct has the feature of being black-box on the algorithm it fools, so it makes sense in the non-uniform setting as well. Our proof makes use of a refined analysis of the learning algorithm of Bshouty et al..

This paper introduces an optical solution to (bounded-length input instances of) an NP-complete problem called the traveling salesman problem using a pure optical system. The solution is based on the multiplication of a binary-matrix, representing all feasible routes, by a weight-vector, representing the weights of the problem. The multiplication of the binary-matrix by the weight-vector can be implemented by any optical vector-matrix multiplier. In this paper, we show that this multiplication can be obtained by an optical correlator. In order to synthesize the binary-matrix, a unique iterative algorithm is presented. This algorithm synthesizes an N-node binary-matrix using rather small number of vector duplications from the (N−1)-node binary-matrix. We also show that the algorithm itself can be implemented optically and thus we ensure the entire optical solution to the problem. Simulation and experimental results prove the validity of the optical method.

Abstract. We prove several results about the average-case complexity of problems in the Polynomial Hierarchy (PH). We give a connection among average-case, worst-case, and non-uniform complexity of optimization problems. Specifically, we show that if P NP is hard in the worst-case then it is either hard on the average (in the sense of Levin) or it is non-uniformly hard (i.e. it does not have small circuits). Recently, Gutfreund, Shaltiel and Ta-Shma (IEEE Conference on Computational Complexity, 2005) showed an interesting worst-case to averagecase connection for languages in NP, under a notion of average-case hardness defined using uniform adversaries. We show that extending their connection to hardness against quasi-polynomial time would imply that NEXP doesn’t have polynomial-size circuits. Finally we prove an unconditional average-case hardness result. We show that for each k, there is an explicit language in P Σ2 which is hard on average for circuits of size n k. 1

We show that for every integer k> 1, if Σk, the k’th level of the polynomial-time hierarchy, is worst-case hard for probabilistic polynomial-time algorithms, then there is a language L ∈ Σk such that for every probabilistic polynomial-time algorithm that attempts to decide it, there is a samplable distribution over the instances of L, on which the algorithm errs with probability at least 1/2−1/poly(n) (where the probability is over the choice of instances and the randomness of the algorithm). In other words, on this distribution the algorithm essentially does not perform any better than the algorithm that simply decides according to the outcome of an unbiased coin toss.

Abstract. A new state space representation of a class of combinatorial optimization problems is introduced. The representation enables efficient implementation of exhaustive search for an optimal solution in bounded NP complete problems such as the traveling salesman problem (TSP) with a relatively small number of cities. Furthermore, it facilitates effective heuristic search for sub optimal solutions for problems with large number of cities. This paper surveys structures for representing solutions to the TSP and the use of these structures in iterative hill climbing (ITHC) and genetic algorithms (GA). The mapping of these structures along with respective operators to a newly proposed electro-optical vector by matrix multiplication (VMM) architecture is detailed. In addition, time space tradeoffs related to using a record keeping mechanism for storing intermediate solutions are presented and the effect of record keeping on the performance of these heuristics in the new architecture is evaluated. Results of running these algorithms on sequential architecture as well as a simulation-based estimation of the speedup obtained are supplied. The results show that the VMM architecture can speedup various variants of the TSP algorithm by a factor of 30x to 50x.

We prove that relative to an oracle, there is no worst-case to average-case reduction for NP. We also handle classes that are somewhat larger than NP, as well as worst-case to errorlessaverage-case reductions. In fact, we prove that relative to an oracle, there is no worst-case. We also handle reductions from NP to the polynomial-time hierarchy and beyond, under restrictions on the number of queries the reductions can make. to errorless-average-case reduction from NP to BPP NP 1

Hardness amplification results show that for every Boolean function f there exists a Boolean function Amp(f) such that the following holds: if every circuit of size s computes f correctly on at most a 1 − δ fraction of inputs, then every circuit of size s ′ computes Amp(f) correctly on at most a 1/2+ϵ fraction of inputs. All hardness amplification results in the literature suffer from “size loss ” meaning that s ′ ≤ ϵ · s. In this paper we show that proofs using “non-uniform reductions ” must suffer from such size loss. To the best of our knowledge, all proofs in the literature are by non-uniform reductions. Our result is the first lower bound that applies to non-uniform reductions that are adaptive. A reduction is an oracle circuit R (·) such that when given oracle access to any function D that computes Amp(f) correctly on a 1/2 + ϵ fraction of inputs, R D computes f correctly on a 1 − δ fraction of inputs. A non-uniform reduction is allowed to also receive a short advice string that may depend on both f and D in an arbitrary way. The well known connection between hardness amplification and list-decodable error-correcting codes implies that reductions showing hardness amplification cannot be uniform for δ, ϵ < 1/4. A reduction is non-adaptive if it makes non-adaptive queries to its oracle. Shaltiel and Viola (SICOMP 2010) showed lower bounds on the number of queries made by nonuniform