Near-optimal deterministic algorithms for volume computation via M-ellipsoids

This bound is achieved by the maximum volume ellipsoid contained in K. Ellipsoids have also been critical to the design and analysis of efficient algorithms. The most notable example is the ellipsoid algorithm (1, 2) for linear (3) and convex optimization (4), which represents a frontier of polynomial-time solvability. For the basic problems of sampling and integration in high dimension, the inertial ellipsoid defined by the covariance matrix of a distribution is an important ingredient of efficient algorithms (5–7). This ellipsoid also achieves the bounds of John’s theorem for general convex bodies (for centrally symmetric convex bodies, the maximum volume ellipsoid achieves the best possible sandwiching ratio of ffiffiffi n p whereas the inertial ellipsoid could still have a ratio of n). Another ellipsoid that has played a critical role in the development of modern convex geometry is the M-ellipsoid (Milman’s ellipsoid). This object was introduced by Milman as a tool to prove fundamental inequalities in convex geometry (e.g., ref. 8, chap. 7). An M-ellipsoid E of a convex body K has small covering numbers with respect to K. We let NðA;BÞ denote the number of translations of a set B required to cover the set A. Then, as shown by Milman, every convex body K in Rn has an ellipsoid E for which NðK ;EÞNðE;KÞ is bounded by 2OðnÞ. This is the best possible bound up to a constant in the exponent. In contrast, the John ellipsoid can have this covering bound as high as nΩðnÞ. Intuitively, an M-ellipsoid for K is the largest ellipsoid with the property that roughly 1=2n fraction of its volume is inside K (as opposed to the entire ellipsoid being in K). The existence of M-ellipsoids now has several proofs in the literature: by Milman (9), multiple proofs by Pisier (8), and, more recently, by Klartag (10). The complexity of computing these ellipsoids is interesting for its own sake, but also due to several important consequences that we discuss presently. John ellipsoids are hard to compute, but their sandwiching bounds can be approximated deterministically to within Oð ffiffiffi n p Þ in polynomial time (4). Inertial ellipsoids can be approximated to arbitrary accuracy by random sampling in polynomial time. Algorithms for M-ellipsoids have been considered more recently. The proof of Klartag (10) gives a randomized polynomial-time algorithm (11). This approach is based on estimating a covariance matrix from random samples and seems inherently difficult to derandomize. It has been open to give a deterministic algorithm for constructing an M-ellipsoid that achieves optimal covering bounds. The extent to which randomness is essential for efficiency is a very interesting question in general and specifically for problems on convex bodies where separations between randomized and deterministic complexity are known in the general oracle model (12, 13). Here we address the question of deterministic M-ellipsoid construction and consider its algorithmic consequences for volume estimation and also for fundamental lattice problems, namely the shortest vector problem (SVP) and the bounded distance decoding (BDD) problem. The first discovery of this paper is a deterministic 2OðnÞ algorithm for computing an M-ellipsoid of a convex body in the oracle model (4). This is the best possible up to a constant in the exponent as there is a 2ΩðnÞ lower bound for deterministic algorithms. We state this result formally and then proceed to its extensions and consequences. For all our algorithmic problems with convex bodies, we need the body to be specified only by a standard well-guaranteed membership oracle; i.e., the algorithm has access to a membership oracle for the convex body of interest K, a point x0 in K, and numbers r, and R s.t. balls of these radii sandwich K; i.e., x0 + rB2 ⊆K⊆RB n 2 (4). By time complexity of an algorithm, we refer to the total number of oracle calls and additional arithmetic operations {we focus on the dependence of the complexity on the dimension and suppress factors that depend polynomially on the size of the input [in particular, logðR=rÞ]}.