Improved Approximation Results for the Minimum Energy Broadcasting Problem

In this paper we present new results on the performance of the Minimum Spanning Tree heuristic for the Minimum Energy Broadcast Routing (MEBR) problem. We first prove that, for any number of dimensions d≥2, the approximation ratio of the heuristic does not increase when the power attenuation coefficient α, that is the exponent to which the coverage distance must be raised to give the emission power, grows. Moreover, we show that, for any fixed instance, as a limit for α going to infinity, the ratio tends to the lower bound of Clementi et al. (Proceedings of the 18th annual symposium on theoretical aspects of computer science (STACS), pp. 121–131, 2001), Wan et al. (Wirel. Netw. 8(6):607–617, 2002) given by the d-dimensional kissing number, thus closing the existing gap between the upper and the lower bound. We then introduce a new analysis allowing to establish a 7.45-approximation ratio for the 2-dimensional case, thus significantly decreasing the previously known 12 upper bound (Wan et al. in Wirel. Netw. 8(6):607–617, 2002) (actually corrected to 12.15 in Klasing et al. (Proceedings of the 3rd IFIP-TC6 international networking conference, pp. 866–877, 2004)). Finally, we extend our analysis to any number of dimensions d≥2 and any α≥d, obtaining a general approximation ratio of 3 ^d −1, again independent of α. The improvements of the approximation ratios are specifically significant in comparison with the lower bounds given by the kissing numbers, as these grow at least exponentially with respect to d. The research was partially funded by the European project COST Action 293, “Graphs and Algorithms in Communication Networks” (GRAAL). Preliminary version of this paper appeared in Flammini et al. (Proceedings of ACM joint workshop on foundations of mobile computing (DIALM-POMC), pp. 85–91, 2004).     

Approximation of functions by perceptrons: a new approach

We provide a constructive proof of the theorem of function approximation by perceptrons (cf Leshno et al. [1], Hornik [2]) when the activation function ψ isC∞ with all its derivatives non 0 at 0. We deal with uniform approximation on compact sets of continuous functions on ℜ ^d ,d≥1. This approach is elementary and provides some approximation results for the derivatives along with some bounds for the hidden layer.     

Kolmogorov-Loveland Stochasticity and Kolmogorov Complexity

Merkle et al. (Ann. Pure Appl. Logic 138(1–3):183–210, 2006) showed that all Kolmogorov-Loveland stochastic infinite binary sequences have constructive Hausdorff dimension 1. In this paper, we go even further, showing that from an infinite sequence of dimension less than H(\frac 12+d)\mathcal {H}(\frac {1}{2}+\delta) (ℋ being the Shannon entropy function) one can extract by an effective selection rule a biased subsequence with bias at least δ. We also prove an analogous result for finite strings.     

Optimality conditions for multiple objective fractional subset programming with (ρ,σ,θ)-type-I and related non-convex functions

In this paper, we introduce a class of generalized invex n-set functions, called (ρ,σ,θ)-type-I and related non-convex functions, and then establish a number of parametric and semi-parametric sufficient optimality conditions for the primal problem under the aforesaid assumptions. This work partially extends an earlier work of Mishra et al. (Math. Methods Oper. Res. 67, 493–504, 2008) to a wider class of functions.     

Quantum Erasure of Decoherence

We consider the classical algebra of observables that are diagonal in a given orthonormal basis, and define a complete decoherence process as a completely positive map that asymptotically converts any quantum observable into a diagonal one, while preserving the elements of the classical algebra. For quantum systems in dimension two and three any decoherence process can be undone by collecting classical information from the environment and using such an information to restore the initial system state. As a relevant example, we illustrate the quantum eraser of Scully et al. [Nature 351, 111 (1991)] as an example of environment-assisted correction, and present the generalization of the eraser setup for d-dimensional systems. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.     

On approximating the longest path in a graph

We consider the problem of approximating the longest path in undirected graphs. In an attempt to pin down the best achievable performance ratio of an approximation algorithm for this problem, we present both positive and negative results. First, a simple greedy algorithm is shown to find long paths in dense graphs. We then consider the problem of finding paths in graphs that are guaranteed to have extremely long paths. We devise an algorithm that finds paths of a logarithmic length in Hamiltonian graphs. This algorithm works for a much larger class of graphs (weakly Hamiltonian), where the result is the best possible. Since the hard case appears to be that of sparse graphs, we also consider sparse random graphs. Here we show that a relatively long path can be obtained, thereby partially answering an open problem of Broderet al. To explain the difficulty of obtaining better approximations, we also prove hardness results. We show that, for any ε<1, the problem of finding a path of lengthn-n ^ε in ann-vertex Hamiltonian graph isNP-hard. We then show that no polynomial-time algorithm can find a constant factor approximation to the longest-path problem unlessP=NP. We conjecture that the result can be strengthened to say that, for some constant δ>0, finding an approximation of ration ^δ is alsoNP-hard. As evidence toward this conjecture, we show that if any polynomial-time algorithm can approximate the longest path to a ratio of , for any ε>0, thenNP has a quasi-polynomial deterministic time simulation. The hardness results apply even to the special case where the input consists of bounded degree graphs. D. Karger was supported by an NSF Graduate Fellowship, NSF Grant CCR-9010517, and grants from the Mitsubishi Corporation and OTL. R. Motwani was supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Development Award, grants from Mitsubishi and OTL, NSF Grant CCR-9010517, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, the Schlumberger Foundation, the Shell Foundation, and the Xerox Corporation, G. D. S. Ramkumar was supported by a grant from the Toshiba Corporation. Communicated by M. X. Goemans.     

Solving Some Discrepancy Problems in NC

We show that several discrepancy-like problems can be solved in NC nearly achieving the discrepancies guaranteed by a probabilistic analysis and achievable sequentially. For example, we describe an NC algorithm that given a set system (X, S) , where X is a ground set and S⊆2 ^X , computes a set R⊆X so that for each S∈ S the discrepancy ||R S|-|R^-∩S|| is . Whereas previous NC algorithms could only achieve discrepancies with ɛ>0 , ours matches the probabilistic bound within a multiplicative factor 1+o(1) . Other problems whose NC solution we improve are lattice approximation, ɛ -approximations of range spaces with constant VC-exponent, sampling in geometric configuration spaces, approximation of integer linear programs, and edge coloring of graphs. Received June 26, 1998; revised February 18, 1999.     

Optimal error estimates for linear parabolic problems under minimal regularity assumptions

We study the approximation of linear parabolic problems by means of Galerkin approximation in space and θ-method in time. The error is evaluated in norms of typeH _t ^δ (H ₁ ^x ) ⋂H _t ^δ+1/2 (L _x ² ) for |δ|≤1/2. We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation. Dedicated to Professor Aldo Ghizzetti on his 75th birthday Istituto di Analisi Numerica del C.N.R.     

From liveness to promptness

Liveness temporal properties state that something “good” eventually happens, e.g., every request is eventually granted. In Linear Temporal Logic (LTL), there is no a priori bound on the “wait time” for an eventuality to be fulfilled. That is, F θ asserts that θ holds eventually, but there is no bound on the time when θ will hold. This is troubling, as designers tend to interpret an eventuality F θ as an abstraction of a bounded eventuality F ^≤k θ, for an unknown k, and satisfaction of a liveness property is often not acceptable unless we can bound its wait time. We introduce here prompt-LTL, an extension of LTL with the prompt-eventually operator F _p . A system S satisfies a prompt-LTL formula φ if there is some bound k on the wait time for all prompt-eventually subformulas of φ in all computations of S. We study various problems related to prompt-LTL, including realizability, model checking, and assume-guarantee model checking, and show that they can be solved by techniques that are quite close to the standard techniques for LTL.     

A classification of generic families of control-affine systems and their bifurcations

We study control-affine systems with n − 1 inputs evolving on an n-dimensional manifold for which the distribution spanned by the control vector fields is involutive and of constant rank (equivalently, they may be considered as 1-dimensional systems with n − 1 inputs entering nonlinearly). We provide a complete classification of such generic systems and their one-parameter families. We show that a generic family for n > 2 is equivalent (with respect to feedback or orbital feedback transformations) to one of nine canonical forms which differ from those for n = 2 by quadratic terms only. We also describe all generic bifurcations of 1-parameter families of systems of the above form.     

Effective elimination of redundant association rules

It is well-recognized that the main factor that hinders the applications of Association Rules (ARs) is the huge number of ARs returned by the mining process. In this paper, we propose an effective solution that presents concise mining results by eliminating the redundancy in the set of ARs. We adopt the concept of δ tolerance to define the set of δ-Tolerance ARs (δ-TARs), which is a concise representation for the set of ARs. The notion of δ-tolerance is a relaxation on the closure defined on the support of frequent itemsets, thus allowing us to effectively prune the redundant ARs. We devise a set of inference rules, with which we prove that the set of δ-TARs is a non-redundant representation of ARs. In addition, we prove that the set of ARs that is derived from the δ-TARs by the inference rules is sound and complete. We also develop a compact tree structure called the δ-TAR tree, which facilitates the efficient generation of the δ-TARs and derivation of other ARs. Experimental results verify the efficiency of using the δ-TAR tree to generate the δ-TARs and to query the ARs. The set of δ-TARs is shown to be significantly smaller than the state-of-the-art concise representations of ARs. In addition, the approximation on the support and confidence of the ARs derived from the δ-TARs are highly accurate.     

Degree-Optimal Routing for P2P Systems

We define a family of Distributed Hash Table systems whose aim is to combine the routing efficiency of randomized networks—e.g. optimal average path length O(log ² n/δlog δ) with δ degree—with the programmability and startup efficiency of a uniform overlay—that is, a deterministic system in which the overlay network is transitive and greedy routing is optimal. It is known that Ω(log n) is a lower bound on the average path length for uniform overlays with O(log n) degree (Xu et al., IEEE J. Sel. Areas Commun. 22(1), 151–163, 2004). Our work is inspired by neighbor-of-neighbor (NoN) routing, a recently introduced variation of greedy routing that allows us to achieve optimal average path length in randomized networks. The advantage of our proposal is that of allowing the NoN technique to be implemented without adding any overhead to the corresponding deterministic network. We propose a family of networks parameterized with a positive integer c which measures the amount of randomness that is used. By varying the value c, the system goes from the deterministic case (c=1) to an “almost uniform” system. Increasing c to relatively low values allows for routing with asymptotically optimal average path length while retaining most of the advantages of a uniform system, such as easy programmability and quick bootstrap of the nodes entering the system. We also provide a matching lower bound for the average path length of the routing schemes for any c. This work was partially supported by the Italian FIRB project “WEB-MINDS” (Wide-scalE, Broadband MIddleware for Network Distributed Services), .     

An Ordered Upwind Method with Precomputed Stencil and Monotone Node Acceptance for Solving Static Convex Hamilton-Jacobi Equations

We define a δ-causal discretization of static convex Hamilton-Jacobi Partial Differential Equations (HJ PDEs) such that the solution value at a grid node is dependent only on solution values that are smaller by at least δ. We develop a Monotone Acceptance Ordered Upwind Method (MAOUM) that first determines a consistent, δ-causal stencil for each grid node and then solves the discrete equation in a single-pass through the nodes. MAOUM is suited to solving HJ PDEs efficiently on highly-nonuniform grids, since the stencil size adjusts to the level of grid refinement. MAOUM is a Dijkstra-like algorithm that computes the solution in increasing value order by using a heap to sort proposed node values. If δ>0, MAOUM can be converted to a Dial-like algorithm that sorts and accepts values using buckets of width δ. We present three hierarchical criteria for δ-causality of a node value update from a simplex of nodes in the stencil.     

Energy-Efficient Paths in Radio Networks

We consider a radio network consisting of n stations represented as the complete graph on a set of n points in the Euclidean plane with edge weights ω(p,q)=|pq| ^δ +C _p , for some constant δ>1 and nonnegative offset costs C _p . Our goal is to find paths of minimal energy cost between any pair of points that do not use more than some given number k of hops.     

Numerical Study of a Modified Time-Stepping θ-Scheme for Incompressible Flow Simulations

In [Turek (1996). Int. J. Numer. Meth. Fluids 22, 987–1011], we had performed numerical comparisons for different time stepping schemes for the incompressible Navier–Stokes equations. In this paper, we present the numerical analysis in the context of the Navier–Stokes equations for a modified time-stepping θ-scheme which has been recently proposed by Glowinski [Glowinski (2003). In: Ciarlet, P. G., and Lions, J. L. (eds.), Handbook of Numerical Analysis, Vol. IX, North-Holland, Amsterdam, pp. 3–1176]. Like the well-known classical Fractional-Step-θ-scheme which had been introduced by Glowinski [Glowinski (1985). In Murman, E. M. and Abarbanel, S. S. (eds.), Progress and Supercomputing in Computational Fluid Dynamics, Birkh?user, Boston MA; Bristeau et al. (1987). Comput. Phys. Rep. 6, 73–187], too, and which is still one of the most popular time stepping schemes, with or without operator splitting techniques, this new scheme consists of 3 substeps with nonequidistant substepping to build one macro time step. However, in contrast to the Fractional-Step-θ-scheme, the second substep can be formulated as an extrapolation step for previously computed data only, and the two remaining substeps look like a Backward Euler step so that no expensive operator evaluations for the right hand side vector with older solutions, as for instance in the Crank–Nicolson scheme, have to be performed. This modified scheme is implicit, strongly A-stable and second order accurate, too, which promises some advantageous behavior, particularly in implicit CFD simulations for the nonstationary Navier–Stokes equations. Representative numerical results, based on the software package FEATFLOW [Turek (2000). FEATFLOW Finite element software for the incompressible Navier–Stokes equations: User Manual, Release 1.2, University of Dortmund] are obtained for typical flow problems with benchmark character which provide a fair rating of the solution schemes, particularly in long time simulations.Dedicated to David Gottlieb on the occasion of his 60th anniversary     

Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators

This article is the second part continuing Part I [16]. We apply the -matrix techniques combined with the Kronecker tensor-product approximation to represent integral operators as well as certain functions F(A) of a discrete elliptic operator A in a hypercube (0,1) ^d ∈ ℝ ^d in the case of a high spatial dimension d. We focus on the approximation of the operator-valued functions A ^{− σ} , σ>0, and sign (A) for a class of finite difference discretisations A ∈ ℝ ^{N × N} . The asymptotic complexity of our data-sparse representations can be estimated by (n ^p log ^q n), p = 1, 2, with q independent of d, where n=N ^{1/ d} is the dimension of the discrete problem in one space direction.     

On the Longest Common Rigid Subsequence Problem

The longest common subsequence problem (LCS) and the closest substring problem (CSP) are two models for finding common patterns in strings, and have been studied extensively. Though both LCS and CSP are NP-Hard, they exhibit very different behavior with respect to polynomial time approximation algorithms. While LCS is hard to approximate within n ^δ for some δ>0, CSP admits a polynomial time approximation scheme. In this paper, we study the longest common rigid subsequence problem (LCRS). This problem shares similarity with both LCS and CSP and has an important application in motif finding in biological sequences. We show that it is NP-hard to approximate LCRS within ratio n ^δ , for some constant δ>0, where n is the maximum string length. We also show that it is NP-Hard to approximate LCRS within ratio Ω(m), where m is the number of strings.     

On Empirical Meaning of Randomness with Respect to?Parametric Families of Probability Distributions

We study the a priori semimeasure of sets of P _θ -random infinite sequences, where P _θ is a family of probability distributions depending on a real parameter θ. In the case when for a computable probability distribution P _θ an effectively strictly consistent estimator exists, we show that Levin’s a priory semimeasure of the set of all P _θ -random sequences is positive if and only if the parameter θ is a computable real number. We show that the a priory semimeasure of the set è_qI_q\bigcup_{\theta}I_{\theta}, where I _θ is the set of all P _θ -random sequences and the union is taken over all algorithmically non-random θ, is positive.     

Approximability of Sparse Integer Programs

The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx:Ax≥b,0≤x≤d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A,b,c,d are nonnegative.) For any k≥2 and ε>0, if P≠NP this ratio cannot be improved to k−1−ε, and under the unique games conjecture this ratio cannot be improved to k−ε. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx:Ax≤b,0≤x≤d} where A has at most k nonzeroes per column, we give a (2k ²+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A _ij is small compared to b _i . Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.     

Finite volume discretization and multilevel methods in flow problems

This article is intended as a preliminary report on the implementation of a finite volume multilevel scheme for the discretization of the incompressible Navier-Stokes equations. As is well known the use of staggered grids (e.g. MAC grids, Perićet al. Comput. Fluids,16(4), 389–403, (1988)) is a serious impediment for the implementation of multilevel schemes in the context of finite differences. This difficulty is circumvented here by the use of a colocated finite volume discretization (Faureet al. (2004a) Submitted, Perićet al. Comput. Fluids,16(4), 389–403, (1988)), for which the algebra of multilevel methods is much simpler than in the context of MAC type finite differences. The general ideas and the numerical simulations are presented in this article in the simplified context of a two-dimensional Burgers equations; the two-, and three-dimensional Navier-Stokes equations introducing new difficulties related to the incompressibility condition and the time discretization, will be considered elsewhere (see Faureet al. (2004a) Submitted and Faureet al. (2004b), in preparation).