Abstract: We develop a new randomized rounding approach for fractional vectors defined on the edge-sets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include:---low congestion multi-path routing;---richer random-graph models for graphs with a given degree-sequence;---improved approximation algorithms for: (i) throughput-maximization in broadcast scheduling, (ii) delay-minimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and---fair scheduling of jobs on unrelated parallel machines.

Abstract: We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions. Our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least half that of the value of the fractional solution. This approximation ratio of 1/2 improves over an Ω(1/log m) ratio of Dobzinski, Nisan and Schapira [STOC 2005]. We also show an approximation ratio of 1 - 1/e when utility functions are fractionally subadditive. A result similar to this last result was previously obtained by Dobzinski and Schapira [Soda 2006], but via a different rounding technique that requires the use of a so called "XOS oracle".The randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation, but have no access to the actual utility functions of the players. This allows us to suggest new incentive compatible mechanisms for combinatorial auctions, extending previous work of Lavi and Swamy [FOCS 2005].

Abstract: We prove that the hit-and-run random walk is rapidly mixing for an arbitrary logconcave distribution starting from any point in the support. This extends the work of Lovasz and Vempala (2004), where this was shown for an important special case, and settles the main conjecture formulated there. From this result, we derive asymptotically faster algorithms in the general oracle model for sampling, rounding, integration and maximization of logconcave functions, improving or generalizing the main results of Lovasz and Vempala (2003), Applegate and Kannan (1990) and Kalai and Vempala respectively. The algorithms for integration and optimization both use sampling and are surprisingly similar

Abstract: The survivable network design problem (SNDP) is the following problem: given an undirected graph and values rij for each pair of vertices i and j, find a minimum-cost subgraph such that there are at least rij disjoint paths between vertices i and j. In the edge connected version of this problem (EC-SNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. The element connectivity problem (ELC-SNDP, or ELC) is a problem of intermediate difficulty. In this problem, the set of vertices is partitioned into terminals and nonterminals. The edges and nonterminals of the graph are called elements. The values rij are only specified for pairs of terminals i, j, and the paths from i to j must be element disjoint. Thus if rij-1 elements fail, terminals i and j are still connected by a path in the network.These variants of SNDP are all known to be NP-hard. The best known approximation algorithm for the EC-SNDP has performance guarantee of 2 and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primal-dual O(log k)-approximation algorithm, where k = maxi,j rij. Since this work first appeared as an extended abstract, it has been shown that it is hard to approximate VC-SNDP to factor 2log1-en.In this paper we investigate applying iterative rounding to ELC and VC-SNDP We show that iterative rounding will not yield a constant factor approximation algorithm for general VC-SNDP. On the other hand, we show how to extend the analysis of iterative rounding applied to EC-SNDP to yield 2-approximation algorithms for both general ELC, and for the case of VC-SNDP when rij ∈ {0, 1, 2}. The latter result improves on an existing 3-approximation algorithm. The former is the first constant factor approximation algorithm for a general survivable network design problem that allows node failures.

Abstract: We introduce a family of spectral partitioning methods. Edge separators of a graph are produced by iteratively reweighting the edges until the graph disconnects into the prescribed number of components. At each iteration a small number of eigenvectors with small eigenvalue are computed and used to determine the reweighting. In this way spectral rounding directly produces discrete solutions where as current spectral algorithms must map the continuous eigenvectors to discrete solutions by employing a heuristic geometric separator (e.g. k-means). We show that spectral rounding compares favorably to current spectral approximations on the Normalized Cut criterion (NCut). Results are given for natural image segmentation, medical image segmentation, and clustering. A practical version is shown to converge.

Abstract: We present a computational, simple and fast sufficient criterion to verify positive definiteness of a symmetric or Hermitian matrix. The criterion uses only standard floating-point operations in rounding to nearest, it is rigorous, it takes into account all possible computational and rounding errors, and is also valid in the presence of underflow. It is based on a floating-point Cholesky decomposition and improves a known result. Using the criterion an efficient algorithm to compute rigorous error bounds for the solution of linear systems with symmetric positive definite matrix follows. A computational criterion to verify that a given symmetric or Hermitian matrix is not positive definite is given as well. Computational examples demonstrate the effectiveness of our criteria.

Abstract: A rounding circuit is provided that includes an input for receiving a 2's complement number to be rounded. The 2's complement number has a format SXY, where S represents a sign bit, X represents one or more bits to round and keep, and Y represents one or more bits to be discarded. The rounding circuit also includes first logic for adding a rounding bias to the 2's complement number, and second logic for at least one of subtracting the sign bit S from the 2's complement number, or adding the inverted sign bit !S to the 2's complement number. Moreover, the rounding circuit includes third logic for truncating Y bits from a result produced by the first and second logic to produce a rounded 2's complement number having a format SX.

Abstract: A specialized processing block for a programmable logic device includes circuitry for performing multiplications and sums thereof, as well as circuitry for rounding the result. The rounding circuitry can selectably perform round-to-nearest and round-to-nearest-even operations. In addition, the bit position at which rounding occurs is preferably selectable. The specialized processing block preferably also includes saturation circuitry to prevent overflows and underflows, and the bit position at which saturation occurs also preferably is selectable. The selectability of both the rounding and saturation positions provides control of the output data word width. The rounding and saturation circuitry may be selectably located in different positions based on timing needs. Similarly, rounding may be speeded up using a look-ahead mode in which both rounded and unrounded results are computed in parallel, with the rounding logic selecting between those results.

Abstract: The floating-point unit (FPU) in the synergistic processor element (SPE) of a CELL processor is a fully pipelined 4-way single-instruction multiple-data (SIMD) unit designed to accelerate media and data streaming with 128-bit operands. It supports 32-bit single-precision floating-point and 16-bit integer operands with two different latencies, six-cycle and seven-cycle, with 11 FO4 delay per stage. The FPU optimizes the performance of critical single-precision multiply-add operations. Since exact rounding, exceptions, and de-norm number handling are not important to multimedia applications, IEEE correctness on the single-precision floating-point numbers is sacrificed for performance and simple design. It employs fine-grained clock gating for power saving. The design has 768K transistors in 1.3 mm/sup 2/, fabricated SOI in 90-nm technology. Correct operations have been observed up to 5.6 GHz with 1.4 V and 56/spl deg/C, delivering 44.8 GFlops. Architecture, logic, circuits, and integration are codesigned to meet the performance, power, and area goals.

Abstract: We present a probabilistic analysis of a large class of combinatorial optimization problems containing all binary optimization problems defined by linear constraints and a linear objective function over $\{0,1\}^n$. Our analysis is based on a semirandom input model that preserves the combinatorial structure of the underlying optimization problem by parameterizing which input numbers are of a stochastic and which are of an adversarial nature. This input model covers various probability distributions for the choice of the stochastic numbers and includes smoothed analysis with Gaussian and other kinds of perturbation models as a special case. In fact, we can exactly characterize the smoothed complexity of binary optimization problems in terms of their worst-case complexity: A binary optimization problem has polynomial smoothed complexity if and only if it admits a (possibly randomized) algorithm with pseudo-polynomial worst-case complexity. Our analysis is centered around structural properties of binary optimization problems, called winner, loser, and feasibility gap. We show that if the coefficients of the objective function are stochastic, then the gap between the best and second best solution is likely to be of order $\Omega(1/n)$. Furthermore, we show that if the coefficients of the constraints are stochastic, then the slack of the optimal solution with respect to this constraint is typically of order $\Omega(1/n^2)$. We exploit these properties in an adaptive rounding scheme that increases the accuracy of calculation until the optimal solution is found. The strength of our techniques is illustrated by applications to various pc-hard optimization problems from mathematical programming, network design, and scheduling for which we obtain the first algorithms with polynomial smoothed/average-case complexity.

Abstract: An analysis based on the L-moments theory suggests of adopting the generalized Pareto distribution to interpret daily rainfall depths recorded by the rain-gauge network of the Hydrological Survey of the Sardinia Region. Nevertheless, a big problem, not yet completely resolved, arises in the estimation of a left-censoring threshold able to assure a good fitting of rainfall data with the generalized Pareto distribution. In order to detect an optimal threshold, keeping the largest possible number of data, we chose to apply a “failure-to-reject” method based on goodness-of-fit tests, as it was proposed by Choulakian and Stephens [Choulakian, V., Stephens, M.A., 2001. Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43, 478–484]. Unfortunately, the application of the test, using percentage points provided by Choulakian and Stephens (2001), did not succeed in detecting a useful threshold value in most analyzed time series. A deeper analysis revealed that these failures are mainly due to the presence of large quantities of rounding off values among sample data, affecting the distribution of goodness-of-fit statistics and leading to significant departures from percentage points expected for continuous random variables. A procedure based on Monte Carlo simulations is thus proposed to overcome these problems.

Abstract: We study a transportation problem with the minimum quantity commitment (MQC), which is faced by a famous international company. The company has a large number of cargos for carriers to ship to the United States. However, the U.S. Marine Federal Commission stipulates that when shipping cargos to the United States, shippers must engage their carriers with an MQC. With such a constraint of MQC, the transportation problem becomes intractable. To solve it practically, we provide a mixed-integer programming model defined by a number of strong facets. Based on this model, a branch-and-cut search scheme is applied to solve small-size instances and a linear programming rounding heuristic for large ones. We also devise a greedy approximation method, whose solution quality depends on the scale of the minimum quantity if the transportation cost forms a distance metric. Extensive experiments have been conducted to measure the performance of the formulations and the algorithms and have shown that the linear rounding heuristic behaves best.

Abstract: Techniques and tools are described for compensating for rounding when estimating sample-domain distortion in the transform domain. For example, a video encoder estimates pixel-domain distortion in the transform domain for a block of transform coefficients after compensating for rounding in the DC coefficient of the block. In this way, the video encoder improves the accuracy of pixel-domain distortion estimation but retains the computational advantages of performing the estimation in the transform domain. Rounding compensation includes, for example, looking up an index (from a de-quantized transform coefficient) in a rounding offset table to determine a rounding offset, then adjusting the coefficient by the offset. Other techniques and tools described herein are directed to creating rounding offset tables and encoders that make encoding decisions after considering rounding effects that occur after an inverse frequency transform on de-quantized transform coefficient values.

Abstract: This article introduces an automatic design procedure for determining the sensitivity of outputs in a digital signal processing design to small errors introduced by rounding or truncation of internal variables. The proposed approach can be applied to both linear and nonlinear designs. By analyzing the resulting sensitivity values, the proposed procedure is able to determine an appropriate distinct word-length for each internal variable in a fixed-point hardware implementation. In addition, the power-optimizing capabilities of word-length optimization are studied. Application of the proposed procedure to adaptive filters and polynomial evaluation circuits realized in a Xilinx Virtex FPGA has resulted in area reductions of up to 80p (mean 66p) combined with power reductions of up to 98p (mean 87p) and speed-up of up to 36p(mean 20p) over common alternative design strategies.

Abstract: An integrated circuit (IC) for convergent rounding including: an adder circuit configured to produce a summation; a comparison circuit configured to bitwise compare the summation with an input pattern, bitwise mask the comparison using a mask, and combine the masked comparison to produce a comparison bit; and rounding circuitry for rounding the summation based at least in part on the comparison bit.

Abstract: Adder/rounder circuitry for use in a programmable logic device computes a rounded sum quickly, and ideally within one clock cycle. The rounding position is selectable within a range of bit positions. In an input stage, for each bit position in that range, bits from both addends and a rounding bit are processed, while for each bit position outside that range only bits from both addends are processed. The input stage processing aligns its output in a common format for bits within and outside the range. The input processing may include 3:2 compression for bit positions within the range and 2:2 compression for bit positions outside the range, so that further processing is performed for all bit positions on a sum vector and a carry vector. Computation of the sum proceeds substantially simultaneously with and without the rounding input, and rounding logic makes a selection later in the computation.

Abstract: Biproportional rounding of a matrix is the problem of assigning values to the elements of a matrix that are proportional to a given input matrix. The assignment should be integral and fulfill a set of rowand column-sum requirements. In a divisor-based method the problem is solved by computing appropriate rowand column-divisors, and by rounding the quotients. The only known divisor-based method that provably solves the problem is the tie-and-transfer algorithm by Balinski, Demange and Rachev. We analyze the complexity of this algorithm, and show that it is pseudo-polynomial. Two different approaches for reducing the complexity to (weakly) polynomial are presented. Finally, we give efficient algorithms for identifying ties, quotient intervals and divisor intervals.

Abstract: Rounding methods are common techniques in many statistical offices to protect sensitive information when publishing data in tabular form. Classical versions of these methods do not consider protection levels while searching patterns with minimum information loss, and therefore typically the so-called auditing phase is required to check the protection of the proposed patterns. This paper presents a mathematical model for the whole problem of finding a protected pattern with minimum loss of information, and proposes a branch-and-cut algorithm to solve it. It also describes a new methodology closely related to the classical Controlled Rounding methods but with several advantages. The new methodology is named Cell Perturbation and leads to a different optimization problem which is simpler to solve than the previous problem. This paper presents a cutting-plane algorithm for finding an exact solution of the new problem, which is a pattern guaranteeing the same protection level requirements but with smaller loss of information when compared with the classical Controlled Rounding optimal patterns. The auditing phase is unnecessary on the solutions generated by the two algorithms. The paper concludes with computational results on real-world instances and discusses a modification in the objective function to guarantee statistical properties in the solutions.

Abstract: , We provide a general method to generate randomized roundings that satisfy cardinality constraints. Our approach is different from the one taken by Srinivasan (FOCS 2001) and Gandhi et al. (FOCS 2002) for one global constraint and the bipartite edge weight rounding problem. Also for these special cases, our approach is the first that can be derandomized. For the bipartite edge weight rounding problem, in addition, we gain an O(|V|) factor run-time improvement for generating the randomized solution. We also improve the current best result on the general problem of derandomizing randomized roundings. Here we obtain a simple O(mn log n) time algorithm that works in the RAM model for arbitrary matrices with entries in Q ≥0 . This improves over the O(m n 2 log(mn)) time solution of Srivastav and Stangier.

Abstract: In one embodiment, the present invention includes a method for receiving a rounding instruction and an immediate value in a processor, determining if a rounding mode override indicator of the immediate value is active, and if so executing a rounding operation on a source operand in a floating point unit of the processor responsive to the rounding instruction and according to a rounding mode set forth in the immediate operand. Other embodiments are described and claimed.

Abstract: An apparatus for scaling numbers comprises register means for storing an operand to be scaled, bit shifting means (402) for performing a right shift operation on the operand, rounding means (404) , and decision means (406, 416) to test for the existence of at least one of an overflow and an underflow condition.

Abstract: Integer transform plays an important role in lossless signal compression. In order to achieve high accuracy to its corresponding theoretical transform, the rounding number should be as low as possible. At the same time, the matrix factorization for reversible integer mapping should be handled with care, especially when the processing block length is high (N>16). In this correspondence, a new method for realizing reversible integer discrete cosine transform type IV (DCT-IV) is proposed that is a key component used in audio compression. The proposed method exhibits low rounding number (2.5N) and low complexity O(Nlog/sub 2/N) as well as well as accurately represents its counterpart floating-point DCT-IV transformation.

Abstract: We consider a scheduling problem on unrelated parallel machines with the objective to minimize the makespan. In addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time. The more of the resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied by Shmoys and Tardos. On the basis of an integer linear programming formulation for (a relaxation of) the problem, we adopt a randomized LP rounding technique from Kumar et al. (FOCS 2005) in order to obtain a deterministic, integral LP solution that is close to optimum. We show how this rounding procedure can be used to derive a deterministic 3.75-approximation algorithm for the scheduling problem. This improves upon previous results, namely a deterministic 6.83-approximation, and a randomized 4-approximation. The improvement is due to the better LP rounding and a new scheduling algorithm that can be viewed as a restricted version of the harmonic algorithm for bin packing.

Abstract: Snap Rounding and its variant, Iterated Snap Rounding, are methods for converting arbitrary-precision arrangements of segments into a fixed-precision representation (we call them SR and ISR for short). Both methods approximate each original segment by a polygonal chain, and both may lead, for certain inputs, to rounded arrangements with undesirable properties: in SR the distance between a vertex and a non-incident edge of the rounded arrangement can be extremely small, inducing potential degeneracies. In ISR, a vertex and a non-incident edge are well separated, but the approximating chain may drift far away from the original segment it approximates. We propose a new variant, Iterated Snap Rounding with Bounded Drift, which overcomes these two shortcomings of the earlier methods. The new solution augments ISR with simple and efficient procedures that guarantee the quality of the geometric approximation of the original segments, while still maintaining the property that a vertex and a non-incident edge in the rounded arrangement are well separated. We investigate the properties of the new method and compare it with the earlier variants. We have implemented the new scheme on top of CGAL, the Computational Geometry Algorithms Library, and report on experimental results.

Abstract: This paper presents new algorithms for snap rounding an arrangement A of line segments in the plane. Snap rounding defines a set of hot pixels, which are unit squares centered on the integer grid points closest to the vertices of A. Snap rounding simplifies A by replacing every input segment by a piecewise linear curve connecting the centers of the hot pixels the segment intersects. Let H be the set of all hot pixels, and for each A∈H let (h) be the number of segments with an intersection or endpoint inside h. If A contains n input segments, the running time of the first new algorithm is O(Eh∈H is (h) log n). This improves previous input- and output-sensitive algorithms by a factor of Θ(n) in the worst case. The second algorithm has an even better running time of O(Eh∈H ed (h) log n); here ed(h) is the description complexity of the crossing pattern in h, which may be substantially less than is(h) and is never greater.

Abstract: Partitioning a permutation into a minimum number of monotone subsequences is ${\mathcal NP}$-hard. We extend this complexity result to minimum partitioning into k-modal subsequences, that is, subsequences having at most k internal extrema. Based on a network flow interpretation we formulate both, the monotone and the k-modal version, as mixed integer programs. This is the first proposal to obtain provably optimal partitions of permutations. From these models we derive an LP rounding algorithm which is a 2-approximation for minimum monotone partitions and a (k+1)-approximation for minimum (upper) k-modal partitions in general; this is the first approximation algorithm for this problem. In computational experiments we see that the rounding algorithm performs even better in practice. For the associated online problem, in which the permutation becomes known to an algorithm sequentially, we derive a logarithmic lower bound on the competitive ratio for minimum monotone partitions, and we analyze two (bin packing) online algorithms. These findings immediately apply to online cocoloring of permutation graphs; they are the first results concerning online algorithms for this graph theoretical interpretation.