Abstract: In this paper, we study the non-orthogonal dynamic spectrum sharing for device-to-device (D2D) communications in the D2D underlaid cellular network. Our design aims to maximize the weighted system sum-rate under the constraints that: 1) each cellular or active D2D link is assigned one subband and 2) the required minimum rates for cellular and active D2D links are guaranteed. To solve this problem, we first characterize the optimal power allocation solution for a given subband assignment. Based on this result, we formulate the subband assignment problem by using the graph-based approach, in which each link corresponds to a vertex and each subband assignment is represented by a hyper-edge. We then propose an iterative rounding algorithm and an optimal branch-and-bound (BnB) algorithm to solve the resulting graph-based problem. We prove that the iterative rounding algorithm achieves at least 1/2 of the optimal weighted sum-rate. Extensive numerical studies illustrate that the proposed iterative rounding algorithm significantly outperforms the conventional spectrum sharing algorithms and attains almost the same system sum-rate as the optimal BnB algorithm.

Abstract: We present an efficient implementation for the evaluation of Wigner $3j$, $6j$, and $9j$ symbols. These represent numerical transformation coefficients that are used in the quantum theory of angular momentum. They can be expressed as sums and square roots of ratios of integers. The integers can be very large due to factorials. We avoid numerical precision loss due to cancellation through the use of multiword integer arithmetic for exact accumulation of all sums. A fixed relative accuracy is maintained as the limited number of floating-point operations in the final step incur rounding errors only in the least significant bits. Time spent to evaluate large multiword integers is in turn reduced by using explicit prime factorization of the ingoing factorials, thereby improving execution speed. Comparison with existing routines shows the efficiency of our approach, and we therefore provide a computer code based on this work.

Abstract: Power has become a key constraint in nanoscale integrated circuit design due to the increasing demands for mobile computing and higher integration density. As an emerging computational paradigm, an inexact circuit offers a promising approach to significantly reduce both dynamic and static power dissipation for error-tolerant applications. In this paper, an inexact floating-point adder is proposed by approximately designing an exponent subtractor and mantissa adder. Related operations such as normalization and rounding are also dealt with in terms of inexact computing. An upper bound error analysis for the average case is presented to guide the inexact design; it shows that the inexact floating-point adder design is dependent on the application data range. High dynamic range images are then processed using the proposed inexact floating-point adders to show the validity of the inexact design; comparison results show that the proposed inexact floating-point adders can improve the power consumption and power-delay product by 29.98 and 39.60 percent, respectively.

Abstract: Herein we report microfluidic droplet formation in flow-focusing geometries possessing varying degrees of rounding. Rounding is incorporated in all four corners (symmetric) or only in the two exit corners (asymmetric). The ratios of the radius of curvature, R, to channel width, w, are varied where R/w = 0, 0.5 and 1. In all cases, monodisperse droplets are produced, with the largest droplets being produced at the junctions with the largest rounding. Junctions without rounding are shown to produce droplets at higher frequencies than those with rounding. Droplet pinch-off position is found to be dependent on both geometry and volumetric flow rates; the location shifts toward the interior of the rounded junctions with increasing oil-to-water flow rate ratios. Accordingly, we find that rounding within microfluidic flow-focusing junctions strongly influences droplet formation. Junction rounding may be deliberate due to the selected fabrication method or occur as an unintended result of microfabrication processes not held to strict tolerances. Indeed, understanding droplet characteristics for those formed in such structures is critical for microfluidic applications where droplet volume or reagent mass must be well controlled. Thus, rounding can be a valuable design parameter when tuning the size and production frequency for emulsion collection or ensuing downstream operations such as chemical reactions.

Abstract: We provide concrete evidence that floating-point computations in C programs can be verified in a homogeneous verification setting based on Coq only, by evaluating the practicality of the combination of the formal semantics of CompCert Clight and the Flocq formal specification of IEEE 754 floating-point arithmetic for the verification of properties of floating-point computations in C programs. To this end, we develop a framework to automatically compute real-number expressions of C floating-point computations with rounding error terms along with their correctness proofs. We apply our framework to the complete analysis of an energy-efficient C implementation of a radar image processing algorithm, for which we provide a certified bound on the total noise introduced by floating-point rounding errors and energy-efficient approximations of square root and sine.

Abstract: In this paper, we investigate the issue of minimizing data center energy usage. In particular, we formulate a problem of virtual machine placement with the objective of minimizing the total power consumption of all the servers. To do this, we examine a CPU power consumption model and then incorporate the model into an mixed integer programming formulation. In order to find optimal or near-optimal solutions fast, we resolve two difficulties: non-linearity of the power model and integer decision variables. We first show how to linearize the problem, and then give a relaxation and iterative rounding algorithm. Computation experiments have shown that the algorithm can solve the problem much faster than the standard integer programming algorithms, and it consistently yields near-optimal solutions. We also provide a heuristic min-cost algorithm, which finds less optimal solutions but works even faster.

Abstract: We consider a generalized version of the correlation clustering problem, defined as follows. Given a complete graph G whose edges are labeled with + or -, we wish to partition the graph into clusters while trying to avoid errors: + edges between clusters or - edges within clusters. Classically, one seeks to minimize the total number of such errors. We introduce a new framework that allows the objective to be a more general function of the number of errors at each vertex (for example, we may wish to minimize the number of errors at the worst vertex) and provide a rounding algorithm which converts "fractional clusterings" into discrete clusterings while causing only a constant-factor blowup in the number of errors at each vertex. This rounding algorithm yields constant-factor approximation algorithms for the discrete problem under a wide variety of objective functions.

Abstract: In this paper we consider the classic scheduling problem of minimizing total weighted completion time on unrelated machines when jobs have release times, i.e, R|rij| Σj wjCj using the three-field notation. For this problem, a 2-approximation is known based on a novel convex programming (J. ACM 2001 by Skutella). It has been a long standing open problem if one can improve upon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman and Woeginger). We answer this question in the affirmative by giving a 1.8786-approximation. We achieve this via a surprisingly simple linear programming, but a novel rounding algorithm and analysis. A key ingredient of our algorithm is the use of random offsets sampled from non-uniform distributions. We also consider the preemptive version of the problem, i.e, R|rij, pmtn|ΣjwjCj. We again use the idea of sampling offsets from non-uniform distributions to give the first better than 2-approximation for this problem. This improvement also requires use of a configuration LP with variables for each job's complete schedules along with more careful analysis. For both non-preemptive and preemptive versions, we break the approximation barrier of 2 for the first time.

Abstract: Floating-point arithmetic is a very efficient solution to perform computations in the real field. However, it induces rounding errors making results computed in floating-point differ from what would be computed with reals. Although numerical analysis gives tools to bound such differences, the proofs involved can be painful, hence error prone. We thus investigate the ability of a proof assistant like Coq to mechanically check such proofs. We demonstrate two different results involving matrices, which are pervasive among numerical algorithms, and show that a large part of the development effort can be shared between them.

Abstract: We design the first truthful-in-expectation, constant-factor approximation mechanisms for NP-hard cases of the welfare maximization problem in combinatorial auctions with nonidentical items and in combinatorial public projects. Our results apply to bidders with valuations that are nonnegative linear combinations of gross-substitute valuations, a class that encompasses many of the most well-studied subclasses of submodular functions, including coverage functions and weighted matroid rank functions. Our mechanisms have an expected polynomial runtime and achieve an approximation factor of 1 − 1/e. This approximation factor is the best possible for both problems, even for known and explicitly given coverage valuations, assuming P ≠ NP. Recent impossibility results suggest that our results cannot be extended to a significantly larger valuation class.Both of our mechanisms are instantiations of a new framework for designing approximation mechanisms based on randomized rounding algorithms. The high-level idea of this framework is to optimize directly over the (random) output of the rounding algorithm, rather than the usual (and rarely truthful) approach of optimizing over the input to the rounding algorithm. This framework yields truthful-in-expectation mechanisms, which can be implemented efficiently when the corresponding objective function is concave. For bidders with valuations in the cone generated by gross-substitute valuations, we give novel randomized rounding algorithms that lead to both a concave objective function and a (1 − 1/e)-approximation of the optimal welfare.

Abstract: Max-cut, clustering, and many other partitioning problems that are of significant importance to machine learning and other scientific fields are NP-hard, a reality that has motivated researchers to develop a wealth of approximation algorithms and heuristics. Although the best algorithm to use typically depends on the specific application domain, a worst-case analysis is often used to compare algorithms. This may be misleading if worst-case instances occur infrequently, and thus there is a demand for optimization methods which return the algorithm configuration best suited for the given application's typical inputs. We address this problem for clustering, max-cut, and other partitioning problems, such as integer quadratic programming, by designing computationally efficient and sample efficient learning algorithms which receive samples from an application-specific distribution over problem instances and learn a partitioning algorithm with high expected performance. Our algorithms learn over common integer quadratic programming and clustering algorithm families: SDP rounding algorithms and agglomerative clustering algorithms with dynamic programming. For our sample complexity analysis, we provide tight bounds on the pseudodimension of these algorithm classes, and show that surprisingly, even for classes of algorithms parameterized by a single parameter, the pseudo-dimension is superconstant. In this way, our work both contributes to the foundations of algorithm configuration and pushes the boundaries of learning theory, since the algorithm classes we analyze consist of multi-stage optimization procedures and are significantly more complex than classes typically studied in learning theory.

Abstract: The following paradigm is often used for handling NP-hard combinatorial optimization problems. One first formulates the problem as an integer program, then one relaxes it to a linear program (LP, or more generally, a convex program), then one solves the LP relaxation in polynomial time, and finally one rounds the optimal LP solution, obtaining a feasible solution to the original problem. Many of the commonly used rounding schemes (such as randomized rounding, threshold rounding and others) are "oblivious" in the sense that the rounding is performed based on the LP solution alone, disregarding the objective function. The goal of our work is to better understand in which cases oblivious rounding suffices in order to obtain approximation ratios that match the integrality gap of the underlying LP. Our study is information theoretic - the rounding is restricted to be oblivious but not restricted to run in polynomial time. In this information theoretic setting we characterize the approximation ratio achievable by oblivious rounding. It turns out to equal the integrality gap of the underlying LP on a problem that is the closure of the original combinatorial optimization problem. We apply our findings to the study of the approximation ratios obtainable by oblivious rounding for the maximum welfare problem, showing that when valuation functions are submodular oblivious rounding can match the integrality gap of the configuration LP (though we do not know what this integrality gap is), but when valuation functions are gross substitutes oblivious rounding cannot match the integrality gap (which is 1).

Abstract: We suggest a method to construct a homomorphic encryption scheme for approximate arithmetic. It supports an approximate addition and multiplication of encrypted messages, together with a new rescaling procedure for managing the magnitude of plaintext. This procedure truncates a ciphertext into a smaller modulus, which leads to rounding of plaintext. The main idea is to add a noise following significant figures which contain a main message. This noise is originally added to the plaintext for security, but considered to be a part of error occurring during approximate computations that is reduced along with plaintext by rescaling. As a result, our decryption structure outputs an approximate value of plaintext with a predetermined precision.

Abstract: We consider positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. Srinivasan (2006) gave a rounding algorithm based on the FKG inequality for systems which are "column-sparse." This algorithm may return an integer solution in which the variables get assigned large (integral) values; Kolliopoulos & Young (2005) modified this algorithm to limit the solution size, at the cost of a worse approximation ratio. We develop a new rounding scheme based on the Partial Resampling variant of the Lovasz Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + ln([EQUATION]), where amin is the minimum covering constraint and Δ1 is the maximum e1-norm of any column of the covering matrix (whose entries are scaled to lie in [0, 1]); we also show nearly-matching inapproximability and integrality-gap lower bounds.Our approach improves asymptotically, in several different ways, over known results. First, it replaces Δ0, the maximum number of nonzeroes in any column (from the result of Srinivasan) by Δ1 which is always - and can be much - smaller than Δ0; this is the first such result in this context. Second, our algorithm automatically handles multi-criteria programs; we achieve improved approximation ratios compared to the algorithm of Srinivasan, and give, for the first time when the number of objective functions is large, polynomial-time algorithms with good multi-criteria approximations. We also significantly improve upon the upper-bounds of Kolliopoulos & Young when the integer variables are required to be within (1 + e) of some given upper-bounds, and show nearly-matching inapproximability.

Abstract: The accuracy analysis of complex floating-point multiplication done by Brent, Percival, and Zimmermann [{\it Math.~ Comp.}, 76:1469--1481, 2007] is extended to the case where a fused multiply-add (FMA) operation is available. Considering floating-point arithmetic with rounding to nearest and unit roundoff $u$, we show that their bound $\sqrt 5 \, u$ on the normwise relative error $|\hat z/z-1|$ of a complex product $z$ can be decreased further to $2u$ when using the FMA in the most naive way. Furthermore, we prove that the term $2u$ is asymptotically optimal not only for this naive FMA-based algorithm, but also for two other algorithms, which use the FMA operation as an efficient way of implementing rounding error compensation. Thus, although highly accurate in the componentwise sense, these two compensated algorithms bring no improvement to the normwise accuracy $2u$ already achieved using the FMA naively. Asymptotic optimality is established for each algorithm thanks to the explicit construction of floating-point inputs for which we prove that the normwise relative error then generated satisfies $|\hat z/z-1| \to 2u$ as $u\to 0$. All our results hold for IEEE floating-point arithmetic, with radix $\beta \ge 2$, precision $p \ge 2$, and rounding to nearest; it is only assumed that underflows and overflows do not occur and, when bounding errors from below, that $\beta^{p-1} \ge 12$.

Abstract: In the current era of big data, researchers routinely collect and analyze data of super-large sample sizes. Data-oriented statistical methods have been developed to extract information from super-large data. Smoothing spline ANOVA (SSANOVA) is a promising approach for extracting information from noisy data; however, the heavy computational cost of SSANOVA hinders its wide application. In this paper, we propose a new algorithm for fitting SSANOVA models to super-large sample data. In this algorithm, we introduce rounding parameters to make the computation scalable. To demonstrate the benefits of the rounding parameters, we present a simulation study and a real data example using electroencephalography data. Our results reveal that (using the rounding parameters) a researcher can fit nonparametric regression models to very large samples within a few seconds using a standard laptop or tablet computer.

Abstract: Modern control is implemented with digital microcontrollers, embedded within a dynamical plant that represents physical components. We present a new algorithm based on counter-example guided inductive synthesis that automates the design of digital controllers that are correct by construction. The synthesis result is sound with respect to the complete range of approximations, including time discretization, quantization effects, and finite-precision arithmetic and its rounding errors. We have implemented our new algorithm in a tool called DSSynth, and are able to automatically generate stable controllers for a set of intricate plant models taken from the literature within minutes.

Abstract: Recently there has been significant interest in training machine-learning models at low precision: by reducing precision, one can reduce computation and communication by one order of magnitude. We examine training at reduced precision, both from a theoretical and practical perspective, and ask: is it possible to train models at end-to-end low precision with provable guarantees? Can this lead to consistent order-of-magnitude speedups? We present a framework called ZipML to answer these questions. For linear models, the answer is yes. We develop a simple framework based on one simple but novel strategy called double sampling. Our framework is able to execute training at low precision with no bias, guaranteeing convergence, whereas naive quantization would introduce significant bias. We validate our framework across a range of applications, and show that it enables an FPGA prototype that is up to 6.5x faster than an implementation using full 32-bit precision. We further develop a variance-optimal stochastic quantization strategy and show that it can make a significant difference in a variety of settings. When applied to linear models together with double sampling, we save up to another 1.7x in data movement compared with uniform quantization. When training deep networks with quantized models, we achieve higher accuracy than the state-of-the-art XNOR-Net. Finally, we extend our framework through approximation to non-linear models, such as SVM. We show that, although using low-precision data induces bias, we can appropriately bound and control the bias. We find in practice 8-bit precision is often sufficient to converge to the correct solution. Interestingly, however, in practice we notice that our framework does not always outperform the naive rounding approach. We discuss this negative result in detail.

Abstract: We provide a simple and novel algorithmic design technique, for which we call iterative partial rounding, that gives a tight rounding-based approximation for vertex cover with hard capacities (VC-HC). In particular, we obtain an $f$-approximation for VC-HC on hypergraphs, improving over a previous results of Cheung et al (SODA 2014) to the tight extent. This also closes the gap of approximation since it was posted by Chuzhoy and Naor in (FOCS 2002). We believe that our rounding technique is of independence interests when hard constraints are considered. Our main technical tool for establishing the approximation guarantee is a separation lemma that certifies the existence of a strong partition for solutions that are basic feasible in an extended version of the natural LP.

Abstract: Floating point number can co-occurrently develop a prominent range of numbers and a high level of precision. Multiplication of floating point numbers found extensive use in wider range of technological and commercial calculations. It is needed to implement faster multipliers involving limited area and consuming reduced power. An IEEE-754 format established multiplier applying Vedic Urdhva — Tiryagbhyam mathematics will be cultivated to cover both single precision and double precision format floating point numbers in the paper. This paper proposes a floating point multiplier which manages overflow, underflow and rounding. The proposed and conventional floating point multipliers based on Vedic mathematics would be coded in Verilog, Synthesized and Simulated using ISE Simulator. Xilinx Virtex VI FPGA will be used for Hardware realization and Verification. It is proposed to compare resource utilization and timing performance of the proposed multiplier with that of existing as of now.

Abstract: Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations. However, to achieve this communication hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the influence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare to the traditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG, while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems.

Abstract: "Questions on income in surveys are prone to two sources of errors that can cause bias if not addressed adequately at the analysis stage. On the one hand, income is considered sensitive information and response rates on income questions generally tend to be lower than response rates for other non-sensitive questions. On the other hand respondents usually don't remember their exact income and thus tend to provide a rounded estimate. The negative effects of item nonresponse are well studied and most statistical agencies have developed sophisticated imputation methods to correct for this potential source of bias. However, to our knowledge the effects of rounding are hardly ever considered in practice, despite the fact that several studies have found strong evidence that most of the respondents round their reported income values. In this paper we illustrate the substantial impact that rounding can have on important measures derived from the income variable such as the poverty rate. To obtain unbiased estimates, we propose a two stage imputation strategy that estimates the posterior probability for rounding given the observed income values at the first stage and re-imputes the observed income values given the rounding probabilities at the second stage. A simulation study shows that the proposed imputation model can help overcome the possible negative effects of rounding. We also present results based on the household income variable from the German panel study 'Labour Market and Social Security.'" (Author's abstract, IAB-Doku) ((en))

Abstract: This article presents a new algorithm called Snap Rounding with Restore (SRR), which aims to make geometric datasets robust and to increase the quality of geometric approximation and the preservation of topological structure. It is based on the well-known Snap Rounding algorithm but improves it by eliminating from the snap rounded arrangement the configurations in which the distance between a vertex and a nonincident edge is smaller than half the width of a pixel of the rounding grid. Therefore, the goal of SRR is exactly the same as the goal of another algorithm, Iterated Snap Rounding (ISR), and of its evolution, Iterated Snap Rounding with Bounded Drift (ISRBD). However, SRR produces an output with a quality of approximation that is on average better than ISRBD, under the viewpoint both of the distance from the original segments and of the conservation of their topological structure. The article also reports some cases where ISRBD, notwithstanding the bounded drift, produces strong topological modifications while SRR does not. A statistical analysis on a large collection of input datasets confirms these differences. It follows that the proposed Snap Rounding with Restore algorithm is suitable for applications that require robustness, a guaranteed geometric approximation, and a good topological approximation.

Abstract: We introduce computable a priori and a posteriori error bounds for optimality and feasibility of a point generated as the rounding of an optimal point of the LP relaxation of a mixed integer linear optimization problem. Treating the mesh size of integer vectors as a parameter allows us to study the effect of different "granularities" in the discrete variables on the error bounds. Our analysis mainly bases on a global error bound for mixed integer linear problems constructed via a so-called grid relaxation retract. Relations to proximity results, the integer rounding property, and binary analytic problems are highlighted.

Abstract: In 2009, Gentry proposed the first Fully Homomorphic Encryption (FHE) scheme, an extremely powerful cryptographic primitive that enables to perform computations, i.e., to evaluate circuits, on encrypted data without decrypting them first. This has many applications, particularly in cloud computing.In all currently known FHE schemes, encryptions are associated with some (non-negative integer) noise level. At each evaluation of an AND gate, this noise level increases. This increase is problematic because decryption succeeds only if the noise level stays below some maximum level L at every gate of the circuit. To ensure that property, it is possible to perform an operation called bootstrapping to reduce the noise level. Though critical, boostrapping is a time-consuming operation. This expense motivates a new problem in discrete optimization: minimizing the number of bootstrappings in a circuit while still controlling the noise level.In this paper, we (1) formally define the bootstrap problem, (2) design a polynomial-time L-approximation algorithm using a novel method of rounding of a linear program, and (3) show a matching hardness result: (L − e)-inapproximability for any e > 0.