Comparing Nontriviality for E and EXP

A set A is nontrivial for the linear-exponential-time class E=DTIME(2 ^lin ) if for any k≥1 there is a set B _k ∈E such that B _k is (p-m-)reducible to A and B_k ? DTIME(2^k·n)B_{k} \not\in \mathrm{DTIME}(2^{k\cdot n}). I.e., intuitively, A is nontrivial for E if there are arbitrarily complex sets in E which can be reduced to A. Similarly, a set A is nontrivial for the polynomial-exponential-time class EXP=DTIME(2 ^poly ) if for any k≥1 there is a set [^(B)]_k ? EXP\hat{B}_{k} \in \mathrm {EXP} such that [^(B)]_k\hat{B}_{k} is reducible to A and [^(B)]_k ? DTIME(2^nk)\hat{B}_{k} \not\in \mathrm{DTIME}(2^{n^{k}}). We show that these notions are independent, namely, there are sets A ₁ and A ₂ in E such that A ₁ is nontrivial for E but trivial for EXP and A ₂ is nontrivial for EXP but trivial for E. In fact, the latter can be strengthened to show that there is a set A∈E which is weakly EXP-hard in the sense of Lutz (SIAM J. Comput. 24:1170–1189, 11) but E-trivial.     

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).     

Minimum Augmentation of Edge-Connectivity between Vertices and Sets of Vertices in Undirected Graphs

Given an undirected multigraph G=(V,E), a family $\mathcal{W}Given an undirected multigraph G=(V,E), a family W\mathcal{W} of areas W⊆V, and a target connectivity k≥1, we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least k edge-disjoint paths between v and W for every pair of a vertex v∈V and an area W ? WW\in \mathcal{W} . So far this problem was shown to be NP-complete in the case of k=1 and polynomially solvable in the case of k=2. In this paper, we show that the problem for k≥3 can be solved in O(m+n(k ³+n ²)(p+kn+nlog n)log k+pkn ³log (n/k)) time, where n=|V|, m=|{{u,v}|(u,v)∈E}|, and p=|W|p=|\mathcal{W}| .     

Approximate Pattern Matching with the <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> Metrics

Given an alphabet Σ={1,2,…,|Σ|} text string T∈Σ ⁿ and a pattern string P∈Σ ^m , for each i=1,2,…,n−m+1 define L _p (i) as the p-norm distance when the pattern is aligned below the text and starts at position i of the text. The problem of pattern matching with L _p distance is to compute L _p (i) for every i=1,2,…,n−m+1. We discuss the problem for d=1,2,∞. First, in the case of L ₁ matching (pattern matching with an L ₁ distance) we show a reduction of the string matching with mismatches problem to the L ₁ matching problem and we present an algorithm that approximates the L ₁ matching up to a factor of 1+ε, which has an O(\frac1e²nlogmlog|S|)O(\frac{1}{\varepsilon^{2}}n\log m\log|\Sigma|) run time. Then, the L ₂ matching problem (pattern matching with an L ₂ distance) is solved with a simple O(nlog m) time algorithm. Finally, we provide an algorithm that approximates the L _∞ matching up to a factor of 1+ε with a run time of O(\frac1enlogmlog|S|)O(\frac{1}{\varepsilon}n\log m\log|\Sigma|) . We also generalize the problem of String Matching with mismatches to have weighted mismatches and present an O(nlog ⁴ m) algorithm that approximates the results of this problem up to a factor of O(log m) in the case that the weight function is a metric.     

Small deviations for two classes of Gaussian stationary processes and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-functionals, 0 < <Emphasis Type="Italic">p</Emphasis> ≤ ∞

Let w(t) be a standard Wiener process, w(0) = 0, and let η _a (t) = w(t + a) − w(t), t ≥ 0, be increments of the Wiener process, a > 0. Let Z _a (t), t ∈ [0, 2a], be a zeromean Gaussian stationary a.s. continuous process with a covariance function of the form E Z _a (t)Z _a (s) = 1/2[a − |t − s|], t, s ∈ [0, 2a]. For 0 < p < ∞, we prove results on sharp asymptotics as ɛ → 0 of the probabilities

Approximating Minimum-Power Degree and Connectivity Problems

Power optimization is a central issue in wireless network design. Given a graph with costs on the edges, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected network design problems under the power minimization criteria. Given a graph G=(V,E)\mathcal{G}=(V,\mathcal{E}) with edge costs {c(e):e∈ℰ} and degree requirements {r(v):v∈V}, the Minimum-Power Edge-Multi-Cover\textsf{Minimum-Power Edge-Multi-Cover} (MPEMC\textsf{MPEMC} ) problem is to find a minimum-power subgraph G of G\mathcal{G} so that the degree of every node v in G is at least r(v). We give an O(log n)-approximation algorithms for MPEMC\textsf{MPEMC} , improving the previous ratio O(log ⁴ n). This is used to derive an O(log n+α)-approximation algorithm for the undirected $\textsf{Minimum-Power $\textsf{Minimum-Power ($\textsf{MP$\textsf{MP ) problem, where α is the best known ratio for the min-cost variant of the problem. Currently, _boxclosen-k)\alpha=O(\log k\cdot \log\frac{n}{n-k}) which is O(log k) unless k=n−o(n), and is O(log ² k)=O(log ² n) for k=n−o(n). Our result shows that the min-power and the min-cost versions of the $\textsf{$\textsf{ problem are equivalent with respect to approximation, unless the min-cost variant admits an o(log n)-approximation, which seems to be out of reach at the moment.     

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.     

Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function

Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t ? 1), where p, q > 0, q = 1 ? p, p ≠ q. The variables M _n = ∫₀¹t ⁿ dL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), where the function τ(x) is periodic in log₂x with the period 1 and is given as \(\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}\), \({z_k} = \frac{{2\pi ik}}{{1n2}}\), k ≠ 0. The proof is based on poissonization and the Mellin transform.     

Disuguaglianze relative all'approssimazione polinomiale in norma uniforme

Let Π _n denote the space of algebraic polynomials of degreen or less. In this paper we establish the inequality for everyf _∈ C ⁽ⁿ⁻¹⁾ ([−1, 1]) andf ⁽ⁿ⁻¹⁾ absolutely continuous. A way for obtaining similar inequalities forf _∈ C ^(t−1) ([−1, 1]) andf ^(l−1) absolutely continuous is given.

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Ricerca effettuata mentre l'autore fruiva di una Borsa di Studio del C.N.R.     

On the computability power and the robustness of set agreement-oriented failure detector classes

Solving agreement problems deterministically, such as consensus and k-set agreement, in asynchronous distributed systems prone to an unbounded number of process failures has been shown to be impossible. To circumvent this impossibility, unreliable failure detectors for the crash failure model have been widely studied. These are oracles that provide information on failures. The exact nature of such information is defined by a set of abstract properties that a particular class of failure detectors satisfy. The weakest class of such failure detectors that allow to solve consensus is Ω. This paper considers failure detector classes from the literature that solve k-set agreement in the crash failure model, and studies their relative power. It shows that the family of failure detector classes (1 ≤ x ≤ n), and (0 ≤ y ≤ n), can be “added” to provide a failure detector of the class Ω ^z (1 ≤ z ≤ n, a generalization of Ω). It also characterizes the power of such an “addition”, namely, , can construct Ω ^z iff y + z > t, and can construct Ω ^z iff x + z > t + 1, where t is the maximum number of processes that can crash in a run. As an example, the paper shows that, while allows solving 2-set agreement (but not consensus) and allows solving t-set agreement (but not (t − 1)-set agreement), a system with failure detectors of both classes can solve consensus for any value of t. More generally, the paper studies the failure detector classes , and Ω ^z , and shows which reductions among these classes are possible and which are not. The paper also presents a message-passing Ω ^k -based k-set agreement protocol and shows that Ω ^k is not enough to solve (k − 1)-set agreement. In that sense, it can be seen as a step toward the characterization of the weakest failure detector class that allows solving the k-set agreement problem. An extended abstract of this paper has appeared in the proceedings of PODC 2006 [20]. This work has been supported partially by a grant from LAFMI (Franco-Mexican Lab in Computer Science), the European Network of Excellence ReSIST and PAPIIT-UNAM.     

Block Toeplitz matrices and preconditioning

We study the asymptotic behaviour of the eigenvalues of Hermitiann×n block Topelitz matricesT _n , withk×k blocks, asn tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices{T _n } are generated by the Fourier coefficients of a Hermitian matrix valued functionf∈L ², and we study the distribution of their eigenvalues for largen, relating their behaviour to some properties of the functionf. We also study the eigenvalues of the preconditioned matrices{P _n ⁻¹ T_n}, where the sequence{P _n } is generated by a positive definite matrix valued functionp. We show that the spectrum of anyP _n ⁻¹ T _n is contained in the interval [r, R], wherer is the smallest andR the largest eigenvalue ofp ⁻¹ f. We also prove that the firstm eigenvalues ofP _n ⁻¹ T_n tend tor and the lastm tend toR, for anym fixed. Finally, exact limit values for both the condition number and the conjugate gradient convergence factor for the preconditioned matricesP _n ⁻¹ T_n are computed.     

The Directed Orienteering Problem

This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed k-TSP problem: given an asymmetric metric (V,d), a root r∈V and a target k≤|V|, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time O(\fraclog² nloglogn·logk)O(\frac{\log^{2} n}{\log\log n}\cdot\log k)-approximation algorithm for this problem. We use this algorithm for directed k-TSP to obtain an O(\fraclog² nloglogn)O(\frac{\log^{2} n}{\log\log n})-approximation algorithm for the directed orienteering problem. This answers positively, the question of poly-logarithmic approximability of directed orienteering, an open problem from Blum et al. (SIAM J. Comput. 37(2):653–670, 2007). The previously best known results were quasi-polynomial time algorithms with approximation guarantees of O(log ² k) for directed k-TSP, and O(log n) for directed orienteering (Chekuri and Pal in IEEE Symposium on Foundations in Computer Science, pp. 245–253, 2005). Using the algorithm for directed orienteering within the framework of Blum et al. (SIAM J. Comput. 37(2):653–670, 2007) and Bansal et al. (ACM Symposium on Theory of Computing, pp. 166–174, 2004), we also obtain poly-logarithmic approximation algorithms for the directed versions of discounted-reward TSP and vehicle routing problem with time-windows.     

Evaluations asymptotiques du nombre des cliques des hypergraphes uniformes

In this paper it is shown that, whenn→∞, for almost all (h+1)-hypergraphs withn nodes every clique contains at most nodes. By using the method developed by V. V. Glagolev for some asymptotical estimations of disjunctive normal forms of Boolean functions, which implies the use of Tchebychev's inequality, method applied to the study of graphs (h=1) by Phan Dinh Diêu, it is proved that whenn→∞ for almost all (h+1)-hypergraphs withn nodes (h≥2), the number of cliques (or of maximal cliques) is of degreen ^(h/(h+1))(h!logn)1/h. This fact implies that every enumerative algorithm for determining the maximum cardinality of a clique of a hypergraph cannot work in polynomial time.      

On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction

Given a directed graph G=(V,A) with a non-negative weight (length) function on its arcs w:A→ℝ₊ and two terminals s,t∈V, our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v∈V a fixed number k(v) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra’s algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c<2 the maximum s–t distance d(s,t) obtainable by removing (at most) k arcs from G. Furthermore, given d, it is NP-hard to approximate within a factor the minimum number of arcs which has to be removed to guarantee d(s,t)≥d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination. This research was supported in part by NSF grant IIS-0118635 and by DIMACS, the NSF Center for Discrete Mathematics & Theoretical Computer Science. Preprints DTR-2005-04 and DTR-2006-13 are available at and . Our co-author Leonid Khachiyan passed away with tragic suddenness on April 29th, 2005.     

Faster Approximate String Matching for Short Patterns

We study the classical approximate string matching problem, that is, given strings P and Q and an error threshold k, find all ending positions of substrings of Q whose edit distance to P is at most k. Let P and Q have lengths m and n, respectively. On a standard unit-cost word RAM with word size w≥log n we present an algorithm using time

Communication Complexity Under Product and Nonproduct Distributions

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R_∈(f) and $D^\mu_\in (f)$D^\mu_\in (f) denote the randomized and μ-distributional communication complexities of f, respectively (∈ a small constant). Yao’s well-known minimax principle states that $R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}$R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}. Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function f : {0, 1}ⁿ ×{0, 1}ⁿ ? {0, 1}f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\} for which max_{μ product} {D^m _? (f)} = Q(1)  but R _? (f) = Q(n)\{D^\mu_\in (f)\} = \Theta(1) \,{\textrm but}\, R_{\in} (f) = \Theta(n). We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).     

Spreading of Messages in Random Graphs

Consider the following model on the spreading of messages. A message initially convinces a set of vertices, called the seeds, of the Erdős-Rényi random graph G(n,p). Whenever more than a ρ∈(0,1) fraction of a vertex v’s neighbors are convinced of the message, v will be convinced. The spreading proceeds asynchronously until no more vertices can be convinced. This paper derives lower bounds on the minimum number of initial seeds, min-seed(n,p,d,r)\mathrm{min\hbox{-}seed}(n,p,\delta,\rho), needed to convince a δ∈{1/n,…,n/n} fraction of vertices at the end. In particular, we show that (1) there is a constant β>0 such that min-seed(n,p,d,r)=W(min{d,r}n)\mathrm{min\hbox{-}seed}(n,p,\delta,\rho)=\Omega(\min\{\delta,\rho\}n) with probability 1−n ^−Ω(1) for p≥β (ln (e/min {δ,ρ}))/(ρ n) and (2) min-seed(n,p,d,1/2)=W(dn/ln(e/d))\mathrm{min\hbox{-}seed}(n,p,\delta,1/2)=\Omega(\delta n/\ln(e/\delta)) with probability 1−exp (−Ω(δ n))−n ^−Ω(1) for all p∈[ 0,1 ]. The hidden constants in the Ω notations are independent of p.     

Generalized fuzzy interior ideals of semigroups

The concept of $(\overline{\in},\overline{\in} \vee \overline{q})The concept of ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups is introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy interior ideals, (∈, ∈ ∨ q)-fuzzy interior ideals and ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups. Finally, we give some characterization of [F] _t by means of (∈, ∈ ∨ q)-fuzzy interior ideals.     

Zur Konvergenz der Ableitungen von Interpolationspolynomen

Letp _n denote the polynomial of degreen or less that interpolates a given smooth functionf at the ?eby?ev nodest _j ⁿ =cos(jπ/n), 0≤j≤n, and let ‖·‖ be the maximum norm inC[?1, 1]. It is proved that fork-th derivatives (2≤k≤n) estimates of the following type hold $$\parallel f^{(k)} - p_n^{(k)} \parallel \leqslant c_k n^{k - 1} \inf \{ \parallel f^{(k)} - q\parallel :q \in \Pi _{n - k} \} .$$ In this relationc _k only depends onk andΠ _n?k denotes the space of polynomials up to degreen?k.