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We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state |φ) whose maximum overlap with a product state is 1- ε, the test passes with probability 1-Θ(ε), regardless of n or the local dimensions of the individual systems. The test uses two copies of |φ). We prove correctness of this test as a special case...
We say an algorithm on n × n matrices with entries in [-M, M] (or n-node graphs with edge weights from [-M, M]) is truly subcubic if it runs in O(n3-δ - poly(log M)) time for some δ > 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n3) time are equivalent under subcubic reductions. Namely, the following weighted problems...
The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of "bad" events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is non-constructively guaranteed by the LLL, culminating in the recent breakthrough of Moser & Tardos. We show that the output distribution...
We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor. For strings of length n and every fixed ε >; 0, the algorithm computes a (log n)O(1/ε) approximation in n1+ε time. This is an exponential improvement over the previously known approximation factor, 2Õ(√log n), with a comparable running time [Ostrovsky and Rabani, J....
We prove that planar graphs have poly-logarithmic queue number, thus improving upon the previous polynomial upper bound. Consequently, planar graphs admit 3D straight-line crossing-free grid drawings in small volume.
The satisfiability problem (SAT) is shown to be the first decision NP-complete problem. It is central in complexity theory. A CNF formula usually contains an interesting number of symmetries. That is, the formula remains invariant under some variable permutations. Such permutations are the symmetries of the formula, their elimination can lead to make a short proof for a satisfiability proof procedure...
The face recognition problem is made difficult by the great variability in head rotation and tilt, lighting intensity and angle, facial expression, aging, partial occlusion (e.g. Wearing Hats, scarves, glasses etc.), etc. In this paper two multi scale techniques Discrete Cosine Transform and Discrete Wavelet Transform are used. Discrete Cosine Transform is applied by retaining various levels of DCT...
This paper considers a size constrained version of the undirected feedback vertex set problem motivated by placing wavelength converters on a WDM network efficiently, and proves that this problem is NP-complete even in several special cases. Moreover, the paper presents a simple approximation algorithm for a minimization version of the problem using an algorithm for the original minimum undirected...
Given a set system (V, S), V = {1,..., n} and S = {S1,...,Sm}, the minimum discrepancy problem is to find a 2-coloring X : V → {-1,+1}, such that each set is colored as evenly as possible, i.e. find X to minimize maxj∈|m] Σi∈sj X(i)|· In this paper we give the first polynomial time algorithms for discrepancy minimization that achieve bounds similar to those known existentially using the so-called...
The Coin Problem is the following problem: a coin is given, which lands on head with probability either 1/2 + β or 1/2 - β. We are given the outcome of n independent tosses of this coin, and the goal is to guess which way the coin is biased, and to answer correctly with probability ≥ 2/3. When our computational model is unrestricted, the majority function is optimal, and succeeds when β ≥ c/√n for...
Complexity theory typically studies the complexity of computing a function h(x) : {0, 1}m → {0,1}n of a given input x. We advocate the study of the complexity of generating the distribution h(x) for uniform x, given random bits. Our main results are: (1) Any function f : {0, 1}ℓ → {0,1}n such that (i) each output bit fi depends on o(log n) input bits, and (ii) ℓ ≤ log2 (αnn) + n0.99, has output distribution...
We present an all-pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0,1] is O(n2), in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis...
Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertex-disjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m√n) time, where m and n are the number of edges and vertices in the graph. Surprisingly, restricted versions of the problem,...
Ensemble pruning is concerned with the reduction of the size of an ensemble prior to its combination. Its purpose is to reduce the space and time complexity of the ensemble and/or to increase the ensemble's accuracy. This paper focuses on instance-based approaches to ensemble pruning, where a different subset of the ensemble may be used for each different unclassified instance. We propose modeling...
We consider the following general scheduling problem: The input consists of n jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as...
In this paper we show how the complexity of performing nearest neighbor (NNS) search on a metric space is related to the expansion of the metric space. Given a metric space we look at the graph obtained by connecting every pair of points within a certain distance r. We then look at various notions of expansion in this graph relating them to the cell probe complexity of NNS for randomized and deterministic,...
We give sub linear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions of these problems, such as SVDD, hard margin SVM, and L2-SVM, for which sub linear-time algorithms were not known before. These new algorithms use a combination...
Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We give an algorithm for this problem that has running time and data requirements polynomial in the dimension and the inverse of the desired accuracy, with provably minimal assumptions on the Gaussians. As a simple consequence of our learning algorithm, we we give the first...
It has been shown by Indyk and Sidiropoulos that any graph of genus g > 0 can be stochastically embedded into a distribution over planar graphs with distortion 2O(g). This bound was later improved to O(g2) by Borradaile, Lee and Sidiropoulos. We give an embedding with distortion O(log g), which is asymptotically optimal. Apart from the improved distortion, another advantage of our embedding is...
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