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This paper deals with the complexity of task graph scheduling with transient and fail-stop failures. While computing the reliability of a given schedule is easy in the absence of task replication, the problem becomes much more difficult when task replication is used. Our main result is that this problem is #P'-Complete (hence at least as hard as NP-Complete problems), with both transient and fails-stop processor failures. We also study the complexity of a restricted class of schedules, where a task cannot be scheduled before all replicas of all its predecessors have completed their execution.

We present in this paper a study on fault management in a grid middleware. The middleware is our home-grown software called P2P-MPI. This framework is MPJ compliant, allows users to execute message passing parallel programs, and its objective is to support environments using commodity hardware. Hence, running programs is failure prone and a particular attention must be paid to fault management. The fault management covers two issues: fault-tolerance and fault detection. Fault-tolerance deals with the program execution: P2P-MPI provides a transparent fault tolerance facility based on replication of computations. Fault detection concerns the monitoring of the program execution by the system. The monitoring is done through a distributed set of modules called failure detectors. The contribution of this paper is twofold. The first contribution is the evaluation of the failure probability of an application depending on the replication degree. The failure probability depends on the execution length, and we propose a model to evaluate the duration of a replicated parallel program. Then, we give an expression of the replication degree required to keep the failure probability of an execution under a given threshold. The second contribution is a study of the advantages and drawbacks of several fault detection systems found in the literature. The criteria of our evaluation are the reliability of the failure detection service and

We present general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Our results are based on the adversary method of Ambainis. † UMR 5668 ENS Lyon, CNRS, UCBL associéeà l'INRIA. Work done when Landes and Yao were visiting LIP with financial support from the Mathlogaps program.

It was recently shown that the theories of generic algebraic curves converge to a limit theory as their degrees go to inÿnity. In this paper we give quantitative versions of this result and other similar results. In particular, we show that generic curves of degree higher than 2 2r cannot be distinguished by a ÿrst-order formula of quantiÿer rank r. A decision algorithm for the limit theory then follows easily. We also show that in this theory all formulas are equivalent to boolean combinations of existential formulas, and give a quantitative version of this result.

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real τ -conjecture" which is inspired by this connection. The real τ -conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real-τ conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.

We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap theorem which allows to test whether P(X) = ∑ k j=1 a j X α j (1 + X) β j is identically zero in polynomial time. The algorithm we obtain is more elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.

The resultant of a square system of homogeneous polynomials is a polynomial in their coefficients which vanishes whenever the system has a solution. Canny gave an algorithm running in polynomial space to compute it but no lower bound was known.

Let K be an infinite field. We give polynomial time constructions of families of r-dimensional subspaces of K n with the following transversality property: any linear subspace of K n of dimension n − r is transversal to at least one element of the family. We also give a new NP-completeness proof for the following problem: given two integers n and m with n m and a n × m matrix A with entries in Z, decide whether there exists a n × n sub-determinant of A which is equal to zero.

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant.

Dimitri Grigoriev has shown that for any family of N vectors in the ddimensional linear space E = (F 2 ) d , there exists a vector in E which is orthogonal to at least N/3 and at most 2N/3 vectors of the family. We show that the range [N/3, 2N/3] can be replaced by the much smaller range [N/2 − √ N/2, N/2 + √ N /2] and we give an efficient, deterministic parallel algorithm which finds a vector achieving this bound. The optimality of the bound is also investigated.

This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weights w. This results settles a long-standing open question, namely whether the well-known O(w log w) bound, known for hardthreshold nets, also held for more general sigmoidal nets. Implications for the number of samples needed for valid generalization are discussed.

We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.