Optimal Approximation for Submodular and Supermodular Optimization with Bounded Curvature

We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a monotone increasing submodular function or minimize a monotone decreasing supermodular function with a bounded total curvature c. Intuitively, the parameter c represents how nonlinear a function f is: when c = 0, f is linear, while for c = 1, f may be an arbitrary monotone increasing submodular function. For the case of submodular maximization with total curvature c, we obtain a (1 − c/e)-approximation—the first improvement over the greedy algorithm of of Conforti and Cornuéjols from 1984, which holds for a cardinality constraint, as well as a recent analogous result for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and nonoblivious local search, and allows us to approximately maximize the sum of a nonnegative, monotone increasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show a (1 − c)-approximation for maximization and a 1/(1 − c)-approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem

Budget Feasible Mechanisms for Experimental Design

In the classical experimental design setting, an experimenter E has access to a population of $n$ potential experiment subjects $i\in \{1,...,n\}$, each associated with a vector of features $x_i\in R^d$. Conducting an experiment with subject $i$ reveals an unknown value $y_i\in R$ to E. E typically assumes some hypothetical relationship between $x_i$'s and $y_i$'s, e.g., $y_i \approx \beta x_i$, and estimates $\beta$ from experiments, e.g., through linear regression. As a proxy for various practical constraints, E may select only a subset of subjects on which to conduct the experiment. We initiate the study of budgeted mechanisms for experimental design. In this setting, E has a budget $B$. Each subject $i$ declares an associated cost $c_i >0$ to be part of the experiment, and must be paid at least her cost. In particular, the Experimental Design Problem (EDP) is to find a set $S$ of subjects for the experiment that maximizes V(S) = \log\det(I_d+\sum_{i\in S}x_i\T{x_i}) under the constraint $\sum_{i\in S}c_i\leq B$; our objective function corresponds to the information gain in parameter $\beta$ that is learned through linear regression methods, and is related to the so-called $D$-optimality criterion. Further, the subjects are strategic and may lie about their costs. We present a deterministic, polynomial time, budget feasible mechanism scheme, that is approximately truthful and yields a constant factor approximation to EDP. In particular, for any small $\delta > 0$ and $\epsilon > 0$, we can construct a (12.98, $\epsilon$)-approximate mechanism that is $\delta$-truthful and runs in polynomial time in both $n$ and $\log\log\frac{B}{\epsilon\delta}$. We also establish that no truthful, budget-feasible algorithms is possible within a factor 2 approximation, and show how to generalize our approach to a wide class of learning problems, beyond linear regression

STUDY OF FATTY ACID COMPOSITION OF MILK FOR CHEESE PRODUCTION

The article shows that the studied samples of raw milk in terms of composition (mass fraction of fat, protein, lactose and milk solids-not-fat) met the criteria of suitability for cheese, and its physic-chemical properties (titratable acidity, density, freezing point) were within acceptable limits. Moreover, rennet sample corresponded to the first and second classes. It was proved that the number of spores of lactating fermenting microorganisms met the requirements for milk for the production of any kind of cheese. It was established that the fatty acid composition of raw cow’s milk used to produce cheeses at various enterprises varied significantly. The data indicate the variability of the content of all groups of fatty acids — low molecular weight, saturated, monounsaturated and polyunsaturated. As a result of gas chromatography studies using the chromatographic complex «Chromos GX-1000» with a flame ionization detector and a CP 88 quartz capillary column — Sil 88 for FAME100 m × 0.25 mm × 0.2 µm, it was found that the largest absolute fluctuations were for saturated (± 7.03 % of the average value) and monounsaturated fatty acids (± 3.77 % of the average value). Absolute fluctuations in the group of low molecular weight fatty acids amounted to ± 2.62 %, and in the group of polyunsaturated — ± 1.02 % of the average value. The calculation of the relative deviation showed that the most varied groups were the ones of low molecular weight fatty acids (± 28.40 rel.%) and polyunsaturated fatty acids (± 25.11 rel.%). At the same time, a relatively high content of certain fatty acids: myristoleic, palmitic, palmitoleic and low levels of stearic and oleic fatty acids, was revealed in individual milk samples.The article shows that the studied samples of raw milk in terms of composition (mass fraction of fat, protein, lactose and milk solids-not-fat) met the criteria of suitability for cheese, and its physic-chemical properties (titratable acidity, density, freezing point) were within acceptable limits. Moreover, rennet sample corresponded to the first and second classes. It was proved that the number of spores of lactating fermenting microorganisms met the requirements for milk for the production of any kind of cheese. It was established that the fatty acid composition of raw cow’s milk used to produce cheeses at various enterprises varied significantly. The data indicate the variability of the content of all groups of fatty acids — low molecular weight, saturated, monounsaturated and polyunsaturated. As a result of gas chromatography studies using the chromatographic complex «Chromos GX-1000» with a flame ionization detector and a CP 88 quartz capillary column — Sil 88 for FAME100 m × 0.25 mm × 0.2 µm, it was found that the largest absolute fluctuations were for saturated (± 7.03 % of the average value) and monounsaturated fatty acids (± 3.77 % of the average value). Absolute fluctuations in the group of low molecular weight fatty acids amounted to ± 2.62 %, and in the group of polyunsaturated — ± 1.02 % of the average value. The calculation of the relative deviation showed that the most varied groups were the ones of low molecular weight fatty acids (± 28.40 rel.%) and polyunsaturated fatty acids (± 25.11 rel.%). At the same time, a relatively high content of certain fatty acids: myristoleic, palmitic, palmitoleic and low levels of stearic and oleic fatty acids, was revealed in individual milk samples

How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time $(1+\epsilon)$-approximation algorithm that yields an $(e+\epsilon)$-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with \emph{optimal }cost at $1+\epsilon$ speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most $\log n$ many classes. This algorithm allows the jobs even to have up to $\log n$ many distinct release dates.Comment: 2 pages, 1 figur

Research on the possibility of extending the shelf life of cheese raw material and heat-treated cheese by their freezing for further use in HoReCa

The article presents the results of a study of the regularities of changes in the functional properties and quality indicators of heat-treated cheeses made from frozen cheese raw material or frozen after thermomechanical processing for further use in HoReCa. The objects of the study were: Caliatta cheese — a semi-hard ripening cheese intended as the main raw material in the production of heat-treated cheese, as well as heat-treated «pizzacheese», subjected to freezing at temperatures of minus 14 ±2 °Cand minus 55 ±2 °Cand low-temperature storage at a temperature of minus 14 ±2 °Cfor 270 days, followed by defrosting at a temperature of 20 ±2 °C. To confirm the possibility of using the freezing technique in order to increase the shelf life of both the original cheese raw material and heat-treated cheese, their microbiological and physicochemical indicators were determined by standardized methods. Studies of structural and mechanical (rheological) properties were carried out on a Weissenberg rheogoniometer, recording changes in the elastic modulus (G’) and dynamic viscosity (h’). The length of the cheese thread, as one of the main functional properties of the «pizza-cheese», was assessed with a fork test after baking. Organoleptic characteristics were assessed by flavor, texture and appearance. Research results have shown that low-temperature storage of frozen cheese can be considered as a way to retard biological and physicochemical changes, which is a safe way to increase shelf life. Freezing cheese raw material increases the length of the cheese thread in proportion to the temperature and duration of the low-temperature storage. When obtaining heat-treated cheese from both unfrozen and frozen cheese raw material, a significant deterioration in the desired functional properties is observed. Thus, the receipt of heat-treated cheese from the original cheese raw material for further use in the production of pizza is justified only by economic feasibility. Freezing «pizza-cheese» at a temperature of minus 55 ±2 °C, made from unfrozen cheese raw material, ensures the preservation of functional properties and increases the shelf life up to 150 days

Thresholded Covering Algorithms for Robust and Max-Min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".Comment: 24 page

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201

Experimental demonstration of near-infrared negative-index metamaterials

Metal-based negative refractive-index materials have been extensively studied in the microwave region. However, negative-index metamaterials have not been realized at near-IR or visible frequencies due to difficulties of fabrication and to the generally poor optical properties of metals at these wavelengths. In this Letter, we report the first fabrication and experimental verification of a transversely structured metal-dielectricmetal multilayer exhibiting a negative refractive index around 2 mu m. Both the amplitude and the phase of the transmission and reflection were measured experimentally, and are in good agreement with a rigorous coupled wave analysis