x n ∈ {0, 1} so that the partial sums ¯ x 1 + · · · + ¯ x k and ¯ x σ1 + · · · + ¯ x σk differ from the unrounded values x 1 + · · · + x k and x σ1 + · · · + x σk by at most n/(n + 1), for 1 ≤ k ≤ n. The latter bound is best possible. The proof uses an elementary argument about flows in a certain network, and leads to a simple algorithm that finds an optimum way to round. Many combinatorial opt...

In this lecture we study a different randomized rounding method called pipage rounding method. The materials of this lecture are based on the work of Chekuri, Vondrák and Zenklusen [CVZ10]. This method is stronger than the maximum entropy rounding by sampling method in some aspects and is weaker in some other aspects. From a highlevel point of view, one can use this method to round a fractional...

• If m > n, the exact quotient of two n-bit numbers cannot be an m-bit number. • Let x, y ∈ Mn. x = y ⇒ |x/y − 1| ≥ 2−n. We call a breakpoint a value z where the rounding changes, that is, if t1 and t2 are real numbers satisfying t1 < z < t2 and ◦t is the rounding mode, then ◦t(t1) < ◦t(t2). For “directed” rounding modes (i.e., towards +∞, −∞ or 0), the breakpoints are the FP numbers. For round...

Rounding to odd is a non-standard rounding on floating-point numbers. By using it for some intermediate values instead of rounding to nearest, correctly rounded results can be obtained at the end of computations. We present an algorithm for emulating the fused multiply-and-add operator. We also present an iterative algorithm for computing the correctly rounded sum of a set floating-point number...

A watermark hidden in an image is retrieved differently from the original watermark due to the frequently used rounding approach. The simple rounding will cause numerous errors in the embedded watermark especially when it is large. A novel technique based on genetic algorithms (GAs) is presented in this paper to correct the rounding errors. The fundamental is to adopt a fitness function for cho...

integrality constraints of a description as an optimization problem over variables in Rn. Linear programming formulations can imply polynomial-time algorithms even if they have exponentially many variables or constraints (by the equivalence of optimization and separation). Linear relaxations can be strengthened by adding further linear constraints, called cutting planes. One can also consider n...

Recently, Yannakakis presented the rst 3 4-approximation algorithm for the Maximum Satissability Problem (MAX SAT). His algorithm makes non-trivial use of solutions to maximum ow problems. We present new, simple 3 4-approximationalgorithmsthat apply the probabilistic method/randomized rounding to the solution to a linear programming relaxation of MAX SAT. We show that although standard randomiz...

Recall that many combinatorial problems of interest can be encoded as integer linear programs. Solving integer linear programs is in general NP-hard, so we nearly always relax the integrality requirement into a linear constraint like nonnegativity during our analysis. Our previous algorithms for solving these problems never solved the relaxed program explicitly (e.g. using simplex). In LP round...

We searched for the worst cases for correct rounding of the exponential function in the IEEE 754r decimal64 format, and computed all the bad cases whose distance from a breakpoint (for all rounding modes) is less than 10 ulp, and we give the worst ones. In particular, the worst case for |x| ≥ 3 × 10 is exp(9.407822313572878 × 10) = 1.098645682066338 5 0000000000000000 278 . . .. This work can b...