Gábor Rudolf

On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction

AbstractnGiven a directed graph G=(V,A) with a non-negative weight (length) function on its arcs w:A→ℝ+ and two terminals s,t∈V, our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v∈V a fixed number k(v) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra’s algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c<2 the maximum s–t distance d(s,t) obtainable by removing (at most) k arcs fromxa0G. Furthermore, given d, it is NP-hard to approximate within a factor n

Optimization Problems with Second Order Stochastic Dominance Constraints: Duality, Compact Formulations, and Cut Generation Methods

c<10sqrt{5}-21approx1.36

Relaxations of linear programming problems with first order stochastic dominance constraints

nthe minimum number of arcs which has to be removed to guarantee d(s,t)≥d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination.n

Optimization with Multivariate Conditional Value-at-Risk Constraints

For stochastic optimization problems with second order stochastic dominance constraints we develop a new form of the duality theory featuring measures on the product of the probability space and the real line. We present two formulations involving small numbers of variables and exponentially many constraints: primal and dual. The dual formulation reveals connections between dominance constraints, generalized transportation problems, and the theory of measures with given marginals. Both formulations lead to two classes of cutting plane methods. Finite convergence of both methods is proved in the case of finitely many events. Numerical results for a portfolio problem are provided.

Methylome-wide comparison of human genomic DNA extracted from whole blood and from EBV-transformed lymphocyte cell lines

Linear stochastic programming problems with first order stochastic dominance (FSD) constraints are non-convex. For their mixed 0-1 linear programming formulation we present two convex relaxations based on second order stochastic dominance (SSD). We develop necessary and sufficient conditions for FSD, used to obtain a disjunctive programming formulation and to strengthen one of the SSD-based relaxations.

Estimation of CpG coverage in whole methylome next-generation sequencing studies.

For many decision-making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces, we develop a cut generation algorithm, where each cut is obtained by solving a mixed-integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR co...

Arrival Rate Approximation by Nonnegative Cubic Splines

DNA from Epstein–Barr virus-transformed lymphocyte cell lines (LCLs) has proven useful for studies of genetic sequence polymorphisms. Whether LCL DNA is suitable for methylation studies is less clear. We conduct a genome-wide methylation investigation using an array set with 45 million probes to investigate the methylome of LCL DNA and technical duplicates of WB DNA from the same 10 individuals. We focus specifically on methylation sites that show variation between individuals and, therefore, are potentially useful as biomarkers. The sample correlations for the methylation variable probes ranged from 0.69 to 0.78 for the WB duplicates and from 0.27 to 0.72 for WB vs LCL. To compare the pattern of the methylation signals, we grouped adjacent probes based on their inter-correlations. These analyses showed ∼29u2009000 and ∼14u2009000 blocks in WB and LCL, respectively. Merely 31% of the methylated regions detected in WB were detectable in LCLs. Furthermore, we observed significant differences in mean difference between WB and LCL as compared with duplicates of WB (P-value =2.2 × 10−16). Our study shows that there are substantial differences in the DNA methylation patterns between LCL and WB. Thus, LCL DNA should not be used as a proxy for WB DNA in methylome-wide studies.

Bilinear optimality constraints for the cone of positive polynomials

BackgroundMethylation studies are a promising complement to genetic studies of DNA sequence. However, detailed prior biological knowledge is typically lacking, so methylome-wide association studies (MWAS) will be critical to detect disease relevant sites. A cost-effective approach involves the next-generation sequencing (NGS) of single-end libraries created from samples that are enriched for methylated DNA fragments. A limitation of single-end libraries is that the fragment size distribution is not observed. This hampers several aspects of the data analysis such as the calculation of enrichment measures that are based on the number of fragments covering the CpGs.ResultsWe developed a non-parametric method that uses isolated CpGs to estimate sample-specific fragment size distributions from the empirical sequencing data. Through simulations we show that our method is highly accurate. While the traditional (extended) read count methods resulted in severely biased coverage estimates and introduces artificial inter-individual differences, through the use of the estimated fragment size distributions we could remove these biases almost entirely. Furthermore, we found correlations of 0.999 between coverage estimates obtained using fragment size distributions that were estimated with our method versus those that were “observed” in paired-end sequencing data.ConclusionsWe propose a non-parametric method for estimating fragment size distributions that is highly precise and can improve the analysis of cost-effective MWAS studies that sequence single-end libraries created from samples that are enriched for methylated DNA fragments.

Generating minimal k -vertex connected spanning subgraphs

We describe an optimization method to approximate the arrival rate of data such as e-mail messages, Web site visits, changes to databases, and changes to Web sites mirrored by other servers. We model these arrival rates as non-homogeneous Poisson process based on observed arrival data. We estimate the arrival function by cubic splines using the maximum likelihood principle. A critical feature of the model is that the splines are constrained to be everywhere nonnegative. We formulate this constraint using a characterization of nonnegative polynomials by positive semidefinite matrices. We also describe versions of our model that allow for periodic arrival rate functions and input data of limited precision. We formulate the estimation problem as a convex program related to semidefinite programming and solve it with a standard nonlinear optimization package called KNITRO. We present numerical results using both an actual record of e-mail arrivals over a period of sixty weeks, and artificially generated data sets. We also present a cross-validation procedure for determining an appropriate number of spline knots to model a set of arrival observations

Kusuoka representations of coherent risk measures in general probability spaces

For a proper cone