In recent years, approximation algorithms based on randomized rounding of fractional optimal solutions have been applied to several classes of discrete optimization problems. In this paper, we describe a class of rounding methods that exploits the structure and geometry of the underlying problem to round fractional solution to 0–1 solution. This is achieved by introducing dependencies in the ro...

We show that any real matrix can be rounded to an integer matrix in such a way that the rounding errors of all row sums are less than one, and the rounding errors of all column sums as well as all sums of consecutive row entries are less than two. Such roundings can be computed in linear time. This extends and improves previous results on rounding sequences and matrices in several directions. I...

Abstract Given a high dimensional convex body K ⊆ IR by a separation oracle, we can approximate its volume with relative error ε, using O∗(n5) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use “rounding” followed by a multiphase Monte-Carlo (product estimator) technique. Both parts rely on sampling (generating random p...

Interval arithmetic is hardly feasible without directed rounding as provided, for example, by the IEEE floating-point standard. Equally essential for interval methods is directed rounding for conversion between the external decimal and internal binary numerals. This is not provided by the standard I/O libraries. Conversion algorithms exist that guarantee identity upon conversion followed by its...

Training of large-scale deep neural networks is often constrained by the available computational resources. We study the effect of limited precision data representation and computation on neural network training. Within the context of low-precision fixed-point computations, we observe the rounding scheme to play a crucial role in determining the network’s behavior during training. Our results s...

Suppose that we have a timetable of a round-robin tournament with a number of teams, and distances among their homes. The home-away assignment problem is to find a home-away assignment that minimizes the total traveling distance of the teams. We propose a formulation of the home-away assignment problem as an integer program, and a rounding algorithm based on Bertsimas, Teo and Vohra’s dependent...

A new deterministic floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid the excessive rounding error propagation of conventional floating-point arithmetic. Unlike interval arithmetic, its uncertainty tracking is based on statistics and the central limit theorem, with a much tighter bounding r...

We present methods for constructing approximation algorithms to NP-hard optimization problems using linear programming (LP). We typically obtain an LP relaxation of the problem by formulating it as an integer linear program and dropping the integrality constraints. We then solve the LP relaxation exactly, and apply various rounding strategies (including randomized rounding and iterative roundin...

In power law graphs, cut quality varies inversely with cut balance. Using some million node social graphs as a test bed, we empirically investigate this property and its implications for graph partitioning. We use six algorithms, including Metis and MQI (state of the art methods for finding bisections and quotient cuts) and four relaxation/rounding methods. We find that an SDP relaxation avoids...

The rounding of the charge density wave depinning transition by thermal noise is examined. Hops by localized modes over small barriers trigger “avalanches”, resulting in a creep velocity much larger than that expected from comparing thermal energies with typical barriers. For a field equal to the T = 0 depinning field, the creep velocity is predicted to have a power-law dependence on the temper...