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We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete black-box de-randomization of Identity Testing problem for depth four circuits with… (More)

We consider the problem of partitioning a set of m points in the n-dimensional Euclidean space into k clusters (usually m and n are variable, while k is fixed), so as to minimize the sum of squared distances between each point and its cluster center. This formulation is usually called k-means clustering (KMN + 00). We prove that this problem in NP-hard even… (More)

We investigate the phenomenon of depth-reduction in commutative and non-commutative arithmetic circuits. We prove that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting.… (More)

We extend the lower bound techniques of [17], to the unbounded-error probabilistic model. A key step in the argument is a generalization of Nepom-njašči˘ ı's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspace-uniform TC 0 circuits for iterated multiplication [11]. Here is… (More)

We consider the problem of dividing a set of m points in Euclidean n?space into k clusters (m; n are variable while k is xed), so as to minimize the sum of distances squared of each point to its \cluster center". This formulation diiers in two ways from the most frequently considered clustering problems in the literature, namely, here we have k xed and m; n… (More)

Recent experience suggests that branching algorithms are among the most attractive for solving propositional satissability problems. A key factor in their success is the rule they use to decide on which variable to branch next. We attempt to explain and improve the performance of branching rules with an empirical model-building approach. One model is based… (More)

In this paper we approach the problem of computing the characteristic polynomial of a matrix from the combinatorial viewpoint. We present several combinatorial characterizations of the coefficients of the characteristic polynomial in terms of walks and closed walks of different kinds in the underlying graph. We develop algorithms based on these… (More)