In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A ∈ R n×m and B ∈ R n×p be two matrices and ε > 0. We approximate the product A ⊤ B using two sketches A ∈ R t×m and B ∈ R t×p , where t ≪ n, such thatwith high probability. We analyze two different sampling procedures for constructing A and B; one of them is done by i.i.d. non-uniform sampling rows from A and B and the other by taking random linear combinations of their rows. We prove bounds on t that depend only on the intrinsic dimensionality of A and B, that is their rank and their stable rank.For achieving bounds that depend on rank when taking random linear combinations we employ standard tools from high-dimensional geometry such as concentration of measure arguments combined with elaborate ε-net constructions. For bounds that depend on the smaller parameter of stable rank this technology itself seems weak. However, we show that in combination with a simple truncation argument it is amenable to provide such bounds. To handle similar bounds for row sampling, we develop a novel matrix-valued Chernoff bound inequality which we call low rank matrixvalued Chernoff bound. Thanks to this inequality, we are able to give bounds that depend only on the stable rank of the input matrices.We highlight the usefulness of our approximate matrix multiplication bounds by supplying two applications. First we give an approximation algorithm for the ℓ2-regression problem that returns an approximate solution by randomly projecting the initial problem to dimensions linear on the rank of the constraint matrix. Second we give improved approximation algorithms for the low rank matrix approximation problem with respect to the spectral norm.

We present a new method for proving rank lower bounds for the cutting planes procedures of Gomory and Chvátal (GC) and Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove near-optimal rank bounds for GC and LS proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of GC or LS procedures when applied to the standard MAXSAT linear relaxation does not reduce the integrality gap. Second, we give unsatisfiable examples that have constant rank GC and LS proofs but that require linear rank Resolution proofs. Third, we give examples where the GC rank is O(log n) but the LS rank is linear. Finally, we address the question of size

We consider a model (BT) for backtracking algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the strength of the model, we then show its limitations by providing lower bounds for algorithms in this model for several classical problems such as interval scheduling, knapsack and satisfiability.

We propose a model called priority branching trees (pBT ) for backtracking and dynamic programming algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the strength of the model, we then show its limitations by providing lower bounds for algorithms in this model for several classical problems such as Interval Scheduling, Knapsack and Satisfiability.

We show how to determine whether the edit distance between two given strings is small in sublinear time. Specifically, we present a test which, given two n-character strings A and B, runs in time o(n) and with high probability returns "CLOSE" if their edit distance is O(n α ), and "FAR" if their edit distance is Ω(n), where α is a fixed parameter less than 1. Our algorithm for testing the edit distance works by recursively subdividing the strings A and B into smaller substrings and looking for pairs of substrings in A, B with small edit distance. To do this, we query both strings at random places using a special technique for economizing on the samples which does not pick the samples independently and provides better query and overall complexity. As a result, our test runs in timeÕ nfor any fixed α < 1. Our algorithm thus provides a trade-off between accuracy and efficiency that is particularly useful when the input data is very large.We also show a lower bound of Ω(n α/2 ) on the query complexity of every algorithm that distinguishes pairs of strings with edit distance at most n α from those with edit distance at least n/6.