The 1-Versus-2 Queries Problem Revisited

The 1-versus-2 queries problem, which has been extensively studied in computational complexity theory, asks in its generality whether every efficient algorithm that makes at most 2 queries to a Σ _k ^p -complete language L _k has an efficient simulation that makes at most 1 query to L _k . We obtain solutions to this problem for hypotheses weaker than previously considered. We prove that:

Lower Bounds for Agnostic Learning via Approximate Rank

We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose that there exist functions f₁, ?, f_r\phi_{1}, \ldots , \phi_{r} : {− 1, 1}ⁿ → \mathbbR{\mathbb{R}} with the property that every disjunction f on n variables has $\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3$\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3 for some reals a₁, ?, a_r\alpha_{1}, \ldots , \alpha_{r}. We prove that then $r \geq exp \{\Omega(\sqrt{n})\}$r \geq exp \{\Omega(\sqrt{n})\}, which is tight. We prove an incomparable lower bound for the concept class of decision lists. For the concept class of majority functions, we obtain a lower bound of W(2ⁿ/n)\Omega(2^{n}/n) , which almost meets the trivial upper bound of 2ⁿ for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi (1992) and show that the regression-based agnostic learning algorithm of Kalai et al. (2005) is optimal.     

When Does Eddy Viscosity Damp Subfilter Scales Sufficiently?

Large eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence is that the LES-solution contains only scales of size ≥Δ, where Δ denotes some user-chosen length scale. This property enables us to perform a LES when it is not feasible to compute the full, turbulent solution of the Navier-Stokes equations. Therefore, in case the large eddy simulation is based on an eddy viscosity model we determine the eddy viscosity such that any scales of size <Δ are dynamically insignificant. In this paper, we address the following two questions: how much eddy diffusion is needed to (a) balance the production of scales of size smaller than Δ; and (b) damp any disturbances having a scale of size smaller than Δ initially. From this we deduce that the eddy viscosity ν _e has to depend on the invariants q = \frac12tr(S²)q = \frac{1}{2}\mathrm{tr}(S^{2}) and r = -\frac13tr(S³)r= -\frac{1}{3}\mathrm{tr}(S^{3}) of the (filtered) strain rate tensor S. The simplest model is then given by n_e = \frac32(D/p)² |r|/q\nu_{e} = \frac{3}{2}(\Delta/\pi)^{2} |r|/q. This model is successfully tested for a turbulent channel flow (Re  _τ =590).     

An improved algorithm on the content of realizable fuzzy matrices

An n×nn{\times}n fuzzy matrix A is called realizable if there exists an n×tn{\times}t fuzzy matrix B such that A=B\odot B^T,A=B\odot B^{T}, where \odot\odot is the max–min composition. Let r(A)=min{p:A=B\odot B^T, B ? L^n×p}.r(A)={min}\{p:A=B\odot B^{T}, B\in L^{n\times p}\}. Then r(A)r(A) is called the content of A. Since 1982, how to calculate r(A) for a given n×nn{\times}n realizable fuzzy matrix A was a focus problem, many researchers have made a lot of research work. X. P. Wang in 1999 gave an algorithm to find the fuzzy matrix B and calculate r(A) within [r(A)]ⁿ²[r(A)]^{n^{2}} steps. Therefore, to find a simpler algorithm is a problem what we have to consider. This paper makes use of the symmetry of the realizable fuzzy matrix A to simplify the algorithm of content r(A)r(A) based on the work of Wang (Chin Ann Math A 6: 701–706, 1999).     

Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues

The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA ₁, A₂, …, A_p, is the least integert for wich there existt diadsu ^(r) v ^(r)τ, τ=1,2,...,t, such that . IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log² k) steps with max(n ²+p (n+k), k² n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log² k) parallel steps and by a sequential algorithm inn(1+k ²)+p (n+k)+0 (k ³) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown. Dedicated to Professor S. Faedo on his 70th birthday Part of the results of this paper has been presented at the Oberwolfach Conference on Komplexitatstheorie, November 1983     

Small Space Representations for Metric Min-sum k-Clustering and Their Applications

The min-sum k -clustering problem is to partition a metric space (P,d) into k clusters C ₁,…,C _k ?P such that $\sum_{i=1}^{k}\sum_{p,q\in C_{i}}d(p,q)The min-sum k -clustering problem is to partition a metric space (P,d) into k clusters C ₁,…,C _k ⊆P such that ?_i=1^k?_{p,q ? Ci}d(p,q)\sum_{i=1}^{k}\sum_{p,q\in C_{i}}d(p,q) is minimized. We show the first efficient construction of a coreset for this problem. Our coreset construction is based on a new adaptive sampling algorithm. With our construction of coresets we obtain two main algorithmic results.     

Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

Large deviations for distributions of sums of random variables: Markov chain method

Let {ξ _k } _k=0^∞ be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} , n → ∞, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x| ^p , p > 0, and exponential random variables with g(x) = x for x ≥ 0.     

Prediction of phosphorus content of electroless nickel–phosphorous coatings using artificial neural network modeling

A multilayer feedforward neural network with two hidden layers was designed and developed for prediction of the phosphorus content of electroless Ni–P coatings. The input parameters of the network were the pH, metal turnover, and loading of an electroless bath. The output parameter was the phosphorus content of the electroless Ni–P coatings. The temperature and molar rate of the bath were constant ( 91^° \textC, 0.4 \textNi^{\text + +} /\textH₂ \textPO₂ ^{- -} 91^\circ {\text{C}},\:0.4\,{\text{Ni}}^{{{\text{ + + }}}} /{\text{H}}_{2} {\text{PO}}_{2}^{{ - - }} ). The network was trained and tested using the data gathered from our own experiments. The goal of the study was to estimate the accuracy of this type of neural network in prediction of the phosphorus content. The study result shows that this type of network has high accuracy even when the number of hidden neurons is very low. Some comparison between the network’s predictions and own experimental data are given.     

Rounds in Combinatorial Search

A set system $\mathcal{H} \subseteq2^{[m]}$ is said to be separating if for every pair of distinct elements x,y∈[m] there exists a set $H\in\mathcal{H}$ such that H contains exactly one of them. The search complexity of a separating system $\mathcal{H} \subseteq 2^{[m]}$ is the minimum number of questions of type “x∈H?” (where $H \in\mathcal{H}$ ) needed in the worst case to determine a hidden element x∈[m]. If we receive the answer before asking a new question then we speak of the adaptive complexity, denoted by $\mathrm{c} (\mathcal{H})$ ; if the questions are all fixed beforehand then we speak of the non-adaptive complexity, denoted by $\mathrm{c}_{na} (\mathcal{H})$ . If we are allowed to ask the questions in at most k rounds then we speak of the k-round complexity of $\mathcal{H}$ , denoted by $\mathrm{c}_{k} (\mathcal{H})$ . It is clear that $|\mathcal{H}| \geq\mathrm{c}_{na} (\mathcal{H}) = \mathrm{c}_{1} (\mathcal{H}) \geq\mathrm{c}_{2} (\mathcal{H}) \geq\cdots\geq\mathrm{c}_{m} (\mathcal{H}) = \mathrm{c} (\mathcal{H})$ . A group of problems raised by G.O.H. Katona is to characterize those separating systems for which some of these inequalities are tight. In this paper we are discussing set systems $\mathcal{H}$ with the property $|\mathcal{H}| = \mathrm{c}_{k} (\mathcal{H}) $ for any k≥3. We give a necessary condition for this property by proving a theorem about traces of hypergraphs which also has its own interest.     

Reoptimization of constraint satisfaction problems with approximation resistant predicates

If k = O(log n) and a predicate P is approximation resistant for the reoptimization of the Max-EkCSP-P problem, then, after inserting a truth-value into the predicate and imposing some constraint, there exists a polynomial algorithm with the approximation ratio q(P) = \frac12 - d(P) q(P) = \frac{1}{{2 - d(P)}} , where d(P) = 2 ^{- k}| P ^{- 1}(1) | d(P) = {2^{ - k}}\left| {{P^{ - 1}}(1)} \right| is a “random” threshold approximation ratio of the predicate P. The ratio q(P) is a threshold approximation ratio.     

Online variance minimization

We consider the following type of online variance minimization problem: In every trial t our algorithms get a covariance matrix C ^t and try to select a parameter vector w ^t−1 such that the total variance over a sequence of trials ?_t=1^T (w^t-1)^T C^tw^t-1\sum_{t=1}^{T} (\boldsymbol {w}^{t-1})^{\top} \boldsymbol {C}^{t}\boldsymbol {w}^{t-1} is not much larger than the total variance of the best parameter vector u chosen in hindsight. Two parameter spaces in ℝ ⁿ are considered—the probability simplex and the unit sphere. The first space is associated with the problem of minimizing risk in stock portfolios and the second space leads to an online calculation of the eigenvector with minimum eigenvalue of the total covariance matrix ?_t=1^T C^t\sum_{t=1}^{T} \boldsymbol {C}^{t}. For the first parameter space we apply the Exponentiated Gradient algorithm which is motivated with a relative entropy regularization. In the second case, the algorithm has to maintain uncertainty information over all unit directions u. For this purpose, directions are represented as dyads uu ^⊤ and the uncertainty over all directions as a mixture of dyads which is a density matrix. The motivating divergence for density matrices is the quantum version of the relative entropy and the resulting algorithm is a special case of the Matrix Exponentiated Gradient algorithm. In each of the two cases we prove bounds on the additional total variance incurred by the online algorithm over the best offline parameter.     

On the solution of the fuzzy Sylvester matrix equation

In this paper, we consider the fuzzy Sylvester matrix equation AX+XB=C,AX+XB=C, where A ? \mathbbR^{n ×n}A\in {\mathbb{R}}^{n \times n} and B ? \mathbbR^{m ×m}B\in {\mathbb{R}}^{m \times m} are crisp M-matrices, C is an n×mn\times m fuzzy matrix and X is unknown. We first transform this system to an (mn)×(mn)(mn)\times (mn) fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.     

The <Emphasis Type="Italic">k</Emphasis>-Homotopic Thinning and a Torus-Like Digital Image in <Emphasis Type="Bold">Z</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In order to discuss digital topological properties of a digital image (X,k), many recent papers have used the digital fundamental group and several digital topological invariants such as the k-linking number, the k-topological number, and so forth. Owing to some difficulties of an establishment of the multiplicative property of the digital fundamental group, a k-homotopic thinning method can be essentially used in calculating the digital fundamental group of a digital product with k-adjacency. More precisely, let be a simple closed k _i -curve with l _i elements in . For some k-adjacency of the digital product which is a torus-like set, proceeding with the k-homotopic thinning of , we obtain its k-homotopic thinning set denoted by DT _k . Writing an algorithm for calculating the digital fundamental group of , we investigate the k-fundamental group of by the use of various properties of a digital covering (Z×Z,p ₁×p ₂,DT _k ), a strong k-deformation retract, and algebraic topological tools. Finally, we find the pseudo-multiplicative property (contrary to the multiplicative property) of the digital fundamental group. This property can be used in classifying digital images from the view points of both digital k-homotopy theory and mathematical morphology.

Redundant modeling in permutation weighted constraint satisfaction problems

In classical constraint satisfaction, redundant modeling has been shown effective in increasing constraint propagation and reducing search space for many problem instances. In this paper, we investigate, for the first time, how to benefit the same from redundant modeling in weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and over-constrained problems. Our work focuses on a popular and special class of problems, namely, permutation problems. First, we show how to automatically generate a redundant permutation WCSP model from an existing permutation WCSP using generalized model induction. We then uncover why naively combining mutually redundant permutation WCSPs by posting channeling constraints as hard constraints and relying on the standard node consistency (NC*) and arc consistency (AC*) algorithms would miss pruning opportunities, which are available even in a single model. Based on these observations, we suggest two approaches to handle the combined WCSP models. In our first approach, we propose m\text -NC_{\text c}^*m\text {-NC}_{\text c}^* and m\text -AC_{\text c}^*m\text {-AC}_{\text c}^* and their associated algorithms for effectively enforcing node and arc consistencies in a combined model with m sub-models. The two notions are strictly stronger than NC* and AC* respectively. While the first approach specifically refines NC* and AC* so as to apply to combined models, in our second approach, we propose a parameterized local consistency LB(m,Φ). The consistency can be instantiated with any local consistency Φ for single models and applied to a combined model with m sub-models. We also provide a simple algorithm to enforce LB(m,Φ). With the two suggested approaches, we demonstrate their applicabilities on several permutation problems in the experiments. Prototype implementations of our proposed algorithms confirm that applying 2\text -NC_{\text c}^*,  2\text -AC_{\text c}^*2\text {-NC}_{\text c}^*,\;2\text {-AC}_{\text c}^*, and LB(2,Φ) on combined models allow far more constraint propagation than applying the state-of-the-art AC*, FDAC*, and EDAC* algorithms on single models of hard benchmark problems.     

Homogeneous Formulas and Symmetric Polynomials

We investigate the arithmetic formula complexity of the elementary symmetric polynomials S^k_n{S^k_n} . We show that every multilinear homogeneous formula computing S^k_n{S^k_n} has size at least k^W(logk)n{k^{\Omega(\log k)}n} , and that product-depth d multilinear homogeneous formulas for S^k_n{S^k_n} have size at least 2^W(k1/d)n{2^{\Omega(k^{1/d})}n} . Since Sⁿ_2n{S^{n}_{2n}} has a multilinear formula of size O(n ²), we obtain a superpolynomial separation between multilinear and multilinear homogeneous formulas. We also show that S^k_n{S^k_n} can be computed by homogeneous formulas of size k^O(logk)n{k^{O(\log k)}n} , answering a question of Nisan and Wigderson. Finally, we present a superpolynomial separation between monotone and non-monotone formulas in the noncommutative setting, answering a question of Nisan.     

On the Exact Complexity of Evaluating Quantified k -CNF

We relate the exponential complexities 2 ^s(k)n of $\textsc {$k$-sat}$ and the exponential complexity $2^{s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))n}$ of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ (the problem of evaluating quantified formulas of the form $\forall\vec{x} \exists\vec{y} \textsc {F}(\vec {x},\vec{y})$ where F is a 3-cnf in $\vec{x}$ variables and $\vec{y}$ variables) and show that s(∞) (the limit of s(k) as k→∞) is at most $s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))$ . Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ running in time 2 ^cn with c<1. On the other hand, a nontrivial exponential-time algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ would provide a $\textsc {$k$-sat}$ solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of the evaluation problem $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ have nontrivial algorithms, and provide strong evidence that the hardest cases of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ must have a mixture of clauses of two types: one universally quantified literal and two existentially quantified literals, or only existentially quantified literals. Moreover, the hardest cases must have at least n?o(n) universally quantified variables, and hence only o(n) existentially quantified variables. Our proofs involve the construction of efficient minimally unsatisfiable $\textsc {$k$-cnf}$ s and the application of the Sparsification lemma.