Rounding Procedure Research Articles

Semidefinite Relaxation for Two Mixed Binary Quadratically Constrained Quadratic Programs: Algorithms and Approximation Bounds

This paper develops new semidefinite programming (SDP) relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. The first class of problem finds two minimum norm vectors in $N$-dimensional real or complex Euclidean space, such that $M$ out of $2M$ concave quadratic functions are satisfied. By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of this model and its SDP relaxation is upper bounded by $\frac{54M^2}{\pi}$ in the real case and by $\frac{24M}{\sqrt{\pi}}$ in the complex case. The second class of problem finds a series of minimum norm vectors subject to a set of quadratic constraints and a cardinality constraint with both binary and continuous variables. We show that in this case the approximation ratio is also bounded and independent of problem dimension for both the real and the complex cases.

Canadian society of surgical oncology annual general meeting.

# Urinary metabolomics of gastric cancer {#article-title-2} Gastric adenocarcinoma (GC) has 70%–75% mortality owing to delayed diagnosis. There is no standard screening in North America. Metabolomics measures low molecular weight chemicals (metabolites) in body fluids/tissues to provide a

Improving spare parts inventory control at a repair shop

We study spare parts inventory control for an aircraft component repair shop. Inspection of a defective component reveals which spare parts are needed to repair it, and in what quantity. Spare part shortages delay repairs, while aircraft operators demand short component repair times. Current spare parts inventory optimization methods cannot guarantee the performance on the component level, which is desired by the operators. To address this shortfall, our model incorporates operator requirements as time-window fill rate requirements for the repair turnaround times for each component type. In alignment with typical repair shop policies, spare parts are allocated on a first come first served basis to repairs, and their inventory is controlled using (s, S) policies. Our solution approach applies column generation in an integer programming formulation. A novel method is developed to solve the related pricing problem. Paired with efficient rounding procedures, the approach solves real-life instances of the problem, consisting of thousands of spare parts and components, in minutes.A case study at a repair shop reveals how data may be obtained in order to implement the approach as an automated method for decision support. We show that the implementation ensures that inventory decisions are aligned with performance targets.

Uniform sampling of steady states in metabolic networks: heterogeneous scales and rounding.

The uniform sampling of convex polytopes is an interesting computational problem with many applications in inference from linear constraints, but the performances of sampling algorithms can be affected by ill-conditioning. This is the case of inferring the feasible steady states in models of metabolic networks, since they can show heterogeneous time scales. In this work we focus on rounding procedures based on building an ellipsoid that closely matches the sampling space, that can be used to define an efficient hit-and-run (HR) Markov Chain Monte Carlo. In this way the uniformity of the sampling of the convex space of interest is rigorously guaranteed, at odds with non markovian methods. We analyze and compare three rounding methods in order to sample the feasible steady states of metabolic networks of three models of growing size up to genomic scale. The first is based on principal component analysis (PCA), the second on linear programming (LP) and finally we employ the Lovazs ellipsoid method (LEM). Our results show that a rounding procedure dramatically improves the performances of the HR in these inference problems and suggest that a combination of LEM or LP with a subsequent PCA perform the best. We finally compare the distributions of the HR with that of two heuristics based on the Artificially Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good agreement with the results of the HR for the small network, while on genome scale models present inconsistencies.

Constructive Discrepancy Minimization by Walking on the Edges

Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer [Trans. Amer. Math. Soc., 289 (1985), pp. 679--706]: In any system of $n$ sets in a universe of size $n$, there always exists a coloring which achieves discrepancy $6\sqrt{n}$. The original proof of Spencer was existential in nature and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal [Proceedings of FOCS, 2010, pp. 3--10] gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is truly constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

Selection, identification, and application of dual DNA aptamers against Shigella sonnei

An illustration of the aptamer selection round procedure based on whole-bacterium SELEX.

On a class of covering problems with variable capacities in wireless networks

We consider the problem of allocating clients to base stations in wireless networks. Two design decisions are the location of the base stations, and the power levels of the base stations. We model the interference, due to the increased power usage resulting in greater serving radius, as capacities that are non-increasing with respect to the covering radius. Clients have demands that are not necessarily uniform and the capacity of a facility limits the total demand that can be served by the facility. We consider three models. In the first model, the location of the base stations and the clients are fixed, and the problem is to determine the serving radius for each base station so as to serve a set of clients with maximum total profit subject to the capacity constraints of the base stations. In the second model, each client has an associated demand in addition to its profit. A fixed number of facilities have to be opened from a candidate set of locations. The goal is to serve clients so as to maximize the profit subject to the capacity constraints. In the third model, the location and the serving radius of the base stations are to be determined. There are costs associated with opening the base stations, and the goal is to open a set of base stations of minimum total cost so as to serve the entire demand subject to the capacity constraints at the base stations. We show that for the first model the problem is NP-complete even when there are only two choices for the serving radius, and the capacities are 1,2. For the second model, we give a 1/2 approximation algorithm. For the third model, we give a column generation procedure for solving the standard linear programming model, and a randomized rounding procedure. We establish the efficacy of the column generation based rounding scheme on randomly generated instances.

Specifying ICD9, ICPC and ATC codes for the STOPP/START criteria: a multidisciplinary consensus panel.

the STOPP/START criteria are a promising framework to increase appropriate prescribing in the elderly in clinical practice. However, the current definitions of the STOPP/START criteria are rather non-specific, allowing undesirable variations in interpretation and thus application. The aim of this study was to design specifications of the STOPP/START criteria into international disease and medication codes to facilitate computerised extraction from medical records and databases. a three round consensus procedure with a multidisciplinary expert panel was organised to prepare, judge and agree on the design of the STOPP/START criteria specifications in corresponding international disease codes (ICD9 and ICPC) and medication codes (ATC). after two rounds consensus was reached for 74% of the STOPP criteria and for 73% of the START criteria. After three rounds full consensus was reached resulting in a specification of 61 out of 62 STOPP criteria and 26 START criteria with their corresponding codes. One criterion could not be specified and for some criteria corresponding disease codes were lacking or imperfect. this study showed the necessity of a consensus procedure as even experts frequently differed on how to specify the STOPP/START criteria. This specification enables next steps such as prognostic validation of these criteria on adverse outcomes and studying the impact of improving appropriate prescribing in the elderly.

Assigning sporadic tasks to unrelated machines

We study the problem of assigning sporadic tasks to unrelated machines such that the tasks on each machine can be feasibly scheduled. Despite its importance for modern real-time systems, this problem has not been studied before. We present a polynomial-time algorithm which approximates the problem with a constant speedup factor of $$8+2\sqrt{6}\approx 12.9$$ and show that any polynomial-time algorithm needs a speedup factor of at least $$2$$ , unless P $$=$$ NP. In the case of a constant number of machines we give a polynomial-time approximation scheme. Key to these results are two new relaxations of the demand bound function, the function that yields a sufficient and necessary condition for a task system on a single machine to be feasible. In particular, we present new methods to approximate this function to obtain useful structural properties while incurring only bounded loss in the approximation quality. For the constant speedup result we employ a very general rounding procedure for linear programs (LPs) which model assignment problems with capacity-type constraints. It ensures that the cost of the rounded integral solution is no more than the cost of the optimal fractional LP solution and the capacity constraints are violated only by a bounded factor, depending on the structure of the matrix that defines the LP. In fact, our rounding scheme generalizes the well-known 2-approximation algorithm for the generalized assignment problem due to Shmoys and Tardos.

Boosting the feasibility pump

The feasibility pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP infeasible solutions. The process attempts to minimize the distance between consecutive iterates, producing an integer feasible solution when closing the distance between them. We investigate the benefits of enhancing the rounding procedure with a clever integer line search that efficiently explores a large set of integer points. An extensive computational study on benchmark instances demonstrates the efficacy of the proposed approach.

Judgment skills, a missing component in health literacy: development of a tool for asthma patients in the Italian-speaking region of Switzerland

BackgroundHealth literacy has been recognized as an important factor influencing health behaviors and health outcomes. However, its definition is still evolving, and the tools available for its measurement are limited in scope. Based on the conceptualization of health literacy within the Health Empowerment Model, the present study developed and validated a tool to assess patient’s health knowledge use, within the context of asthma self-management.MethodsA review of scientific literature on asthma self-management, and several interviews with pulmonologists and asthma patients were conducted. From these, 19 scenarios with 4 response options each were drafted and assembled in a scenario-based questionnaire. Furthermore, a three round Delphi procedure was carried out, to validate the tool with the participation of 12 specialists in lung diseases.ResultsThe face and content validity of the tool were achieved by face-to-face interviews with 2 pulmonologists and 5 patients. Consensus among the specialists on the adequacy of the response options was achieved after the three round Delphi procedure. The final tool has a 0.97 intra-class correlation coefficient (ICC), indicating a strong level of agreement among experts on the ratings of the response options. The ICC for single scenarios, range from 0.92 to 0.99.ConclusionsThe newly developed tool provides a final score representing patient’s health knowledge use, based on the specialist’s consensus. This tool contributes to enriching the measurement of a more advanced health literacy dimension.

Batch scheduling with a rate-modifying maintenance activity to minimize total flowtime

We study a single machine batch scheduling problem with unit time jobs and an optional maintenance activity. The maintenance activity is assumed to be rate modifying, i.e. the processing times of the jobs processed after the maintenance are reduced. The objective function is minimum total flowtime. We focus first on the relaxed version of the problem, where batch sizes are not forced to be integers. For a given number of jobs, setup time, duration of the maintenance activity, and a rate-modifying factor, we show that the optimal solution has a unique property: the batch sizes of the jobs scheduled prior to the maintenance, and after it, form two decreasing arithmetic sequences. Based on this property, we introduce an optimal algorithm which is polynomial in the number of jobs. We propose a simple rounding procedure that guarantees an integer solution. Our numerical tests indicate that this procedure leads to very close-to-optimal schedules.

Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains

The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches.

Phase-unwrapping algorithm by a rounding-least-squares approach

A simple and efficient phase-unwrapping algorithm based on a rounding procedure and a global least-squares minimization is proposed. Instead of processing the gradient of the wrapped phase, this algorithm operates over the gradient of the phase jumps by a robust and noniterative scheme. Thus, the residue-spreading and over-smoothing effects are reduced. The algorithm’s performance is compared with four well-known phase-unwrapping methods: minimum cost network flow (MCNF), fast Fourier transform (FFT), quality-guided, and branch-cut. A computer simulation and experimental results show that the proposed algorithm reaches a high-accuracy level than the MCNF method by a low-computing time similar to the FFT phase-unwrapping method. Moreover, since the proposed algorithm is simple, fast, and user-free, it could be used in metrological interferometric and fringe-projection automatic real-time applications.

Efficient rounding for the noncommutative Grothendieck inequality

$ \newcommand{\cclass}[1]{{\textsf{#1}}} $The classical Grothendieck inequality has applications to the design of approximation algorithms for $\cclass{NP}$-hard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a polynomial-time constant-factor approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principal component analysis and the orthogonal Procrustes problem.

Semidefinite Approximation for Mixed Binary Quadratically Constrained Quadratic Programs

Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. We consider both a minimization and a maximization model of this problem. For the minimization model, the objective is to find a minimum norm vector in $N$-dimensional real or complex Euclidean space, such that $M$ concave quadratic constraints and a cardinality constraint are satisfied with both binary and continuous variables. {\color{blue}By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of the minimization model and its SDP relaxation is upper bounded by $\cO(Q^2(M-Q+1)+M^2)$ in the real case and by $\cO(M(M-Q+1))$ in the complex case.} For the maximization model, the goal is to find a maximum norm vector subject to a set of quadratic constraints and a cardinality constraint with both binary and continuous variables. We show that in this case the approximation ratio is bounded from below by $\cO(\epsilon/\ln(M))$ for both the real and the complex cases. Moreover, this ratio is tight up to a constant factor.

Approximability of the vertex cover problem in power-law graphs

In this paper we construct an approximation algorithm for the Minimum Vertex Cover (Min-VC) problem with an expected approximation ratio of 2−ζ(β)−1−12β2βζ(β−1)ζ(β) for random power-law graphs in the P(α,β) model due to Aiello et al. Here ζ(β) is the Riemann zeta function of β. We obtain this result by combining the Nemhauser and Trotter approach for Min-VC with a new deterministic rounding procedure which achieves an approximation ratio of 32 on a subset of low degree vertices for which the expected contribution to the cost of the associated linear program is sufficiently large.

Explicit Optimal Hardness via Gaussian Stability Results

The results of Raghavendra [2008] show that assuming Khot’s Unique Games Conjecture [2002], for every constraint satisfaction problem there exists a generic semidefinite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate exactly the Unique Games hardness of the problem. Obtaining an explicit optimal approximation scheme and the corresponding approximation factor is a difficult challenge for each specific approximation problem. Khot et al. [2004] established a general approach for determining the exact approximation factor and the corresponding optimal rounding algorithm for any given constraint satisfaction problem. However, this approach crucially relies on results explicitly proving optimal partitions in the Gaussian space. Until recently, Borell’s result [1985] was the only nontrivial Gaussian partition result known. In this article we derive the first explicit optimal approximation algorithm and the corresponding approximation factor using a new result on Gaussian partitions due to Isaksson and Mossel [2012]. This Gaussian result allows us to determine the exact Unique Games Hardness of MAX-3-EQUAL. In particular, our results show that Zwick’s algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is ≈ 0.796 as conjectured by Zwick [1998]. We further use the previously known optimal Gaussian partitions results to obtain a new Unique Games Hardness factor for MAX-k-CSP: Using the well-known fact that jointly normal pairwise independent random variables are fully independent, we show that the UGC hardness of Max-k-CSP is ⌈( k +1)/2⌉ 2 k−1 , improving on results of Austrin and Mossel [2009].

Exact and stable recovery of rotations for robust synchronization

The synchronization problem over the special orthogonal group SO(d) consists of estimating a set of unknown rotations from noisy measurements of a subset of their pairwise ratios ⁠. The problem has found applications in computer vision, computer graphics and sensor network localization, among others. Its least squares solution can be approximated by either spectral relaxation or semidefinite programming followed by a rounding procedure, analogous to the approximation algorithms of Max-Cut. The contribution of this paper is three-fold: first, we introduce a robust penalty function involving the sum of unsquared deviations and derive a relaxation that leads to a convex optimization problem; secondly, we apply the alternating direction method to minimize the penalty function. Finally, under a specific model of the measurement noise and for both complete and random measurement graphs, we prove that the rotations are exactly and stably recovered, exhibiting a phase transition behavior in terms of the proportion of noisy measurements. Numerical simulations confirm the phase transition behavior for our method as well as its improved accuracy compared with existing methods.

Batch scheduling of identical jobs with controllable processing times

In scheduling models with controllable processing times, the job processing times can be controlled (i.e. compressed) by allocating additional resources. In batch scheduling a large number of jobs may be grouped and processed as separate batches, where a batch processing time is identical to the total processing times of the jobs contained in the batch, and a setup time is incurred when starting a new batch.A model combining these two very popular and practical phenomena is studied. We focus on identical jobs and linear compression cost function. Two versions of the problem are considered: in the first we minimize the sum of the total flowtime and the compression cost, and in the second we minimize the total flowtime subject to an upper bound on the maximum compression. We study both problems on a single machine and on parallel identical machines. In all cases we introduce closed form solutions for the relaxed version (allowing non-integer batch sizes). Then, a simple rounding procedure is introduced, tested numerically, and shown to generate extremely close-to-optimal integer solutions. For a given number of machines, the total computational effort required by our proposed solution procedure is O(n), where n is the number of jobs.