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We give efficient algorithms for volume sampling, i.e., for picking k-subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section 7.4 of [15], also implicit in [1,5]).Our first algorithm for volume sampling k-subsets of rows from an m-by-n matrix runs in O(kmn ω log n) arithmetic operations and a second variant of it for (1 + ǫ)-approximate volume sampling runs in O(mn log m · k 2 /ǫ 2 + m log ω m · k 2ω+1 /ǫ 2ω · log(kǫ −1 log m)) arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small k.Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation:1. Given A ∈ R m×n , in O(kmn ω log n) arithmetic operations we can find k of its rows such that projecting onto their span gives a √ k + 1-approximation to the matrix of rank k closest to A under the Frobenius norm. This improves the O(k √ log k)-approximation of Boutsidis, Drineas and Mahoney [1] and matches the lower bound shown in [5]. The method of conditional expectations gives a deterministic algorithm with the same complexity. The running time can be improved to O(mn log m·k 2 /ǫ 2 +m log ω m·k 2ω+1 /ǫ 2ω ·log(kǫ −1 log m)) at the cost of losing an extra (1 + ǫ) in the approximation factor.2. The same rows and projection as in the previous point give a (k + 1)(n − k)-approximation to the matrix of rank k closest to A under the spectral norm. In this paper, we show an almost matching lower bound of √ n, even for k = 1.

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Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree nvertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n). For the random graph G n,p , for p = Ω(log n/n), we give a randomized algorithm for constructing two spanning trees whose union is an expander. This is suggested by the case of the complete graph, where we prove that two random spanning trees give an expander. The construction of the splicer is elementary; each spanning tree can be produced independently using an algorithm by Aldous and Broder: A random walk in the graph with edges leading to previously unvisited vertices included in the tree. Splicers also turn out to have applications to graph cut-sparsification where the goal is to approximate every cut using only a small subgraph of the original graph. For random graphs, splicers provide simple algorithms for sparsifiers of size O(n) that approximate every cut to within a factor of O(log n).

Consider the problem of computing the centroid of a convex body in R n . We prove that if the body is a polytope given as an intersection of half-spaces, then computing the centroid exactly is #P -hard, even for order polytopes, a special case of 0-1 polytopes. We also prove that if the body is given by a membership oracle, then for any deterministic algorithm that makes a polynomial number of queries there exists a body satisfying a roundedness condition such that the output of the algorithm is outside a ball of radius σ/100 around the centroid, where σ 2 is the minimum eigenvalue of the inertia matrix of the body.