Submodular Stochastic Probing on Matroids

In a stochastic probing problem we are given a universe $E$, where each element $e \in E$ is active independently with probability $p_e$, and only a probe of e can tell us whether it is active or not. On this universe we execute a process that one by one probes elements --- if a probed element is active, then we have to include it in the solution, which we gradually construct. Throughout the process we need to obey inner constraints on the set of elements taken into the solution, and outer constraints on the set of all probed elements. This abstract model was presented by Gupta and Nagarajan (IPCO '13), and provides a unified view of a number of problems. Thus far, all the results falling under this general framework pertain mainly to the case in which we are maximizing a linear objective function of the successfully probed elements. In this paper we generalize the stochastic probing problem by considering a monotone submodular objective function. We give a $(1 - 1/e)/(k_{in} + k_{out}+1)$-approximation algorithm for the case in which we are given $k_{in}$ matroids as inner constraints and $k_{out}$ matroids as outer constraints. Additionally, we obtain an improved $1/(k_{in} + k_{out})$-approximation algorithm for linear objective functions

Genetics of human sleep EEG

Sleep characteristics are candidates for predictive biological markers in patients with severe psychiatric diseases, in particular affective disorder and schizophrenia. The genetic components of sleep determination in humans remain, to a large degree, unelucidated. In particular, the heritability of rapid eye movement (REM) sleep and EEG bursts of oscillatory brain activity in Non-REM sleep, i.e. sleep spindles, are of interest. In addition, recent findings suggest a strong role of distinct sleep spindle types in memory consolidation, making it important to identify sleep spindles in slow wave sleep (SWS) and to separate slow and fast spindle localization in the frequency range. However, predictive sleep biomarker research requires large sample sizes of healthy and affected human individuals. Therefore, the present work addressed two questions. The first aim was to optimize data analysis by developing algorithms that allow an efficient and reliable identification of rapid eye movements (REMs) and sleep EEG spindles. In the second part, developed methods were applied to sleep EEG data from a classical twin study to identify genetic effects on tonic and phasic REM sleep parameters, sleep spindles, and their trait-like characteristics. The algorithm for REM detection was developed for standard clinical two channel electrooculographic montage. The goal was to detect REMs visible above the background noise, and in the case of REM saccades to classify each movement separately. In order to achieve a high level of sensitivity, detection was based on a first derivative of electrooculogram (EOG) potentials and two detection thresholds. The developed REM detector was then validated in n=12 polysomnographic recordings from n=7 healthy subjects who had been previously scored visually by two human experts according to standard guidelines. Comparison of automatic REM detection with human scorers revealed mean correlations of 0.94 and 0.90, respectively (mean correlation between experts was 0.91). The developed automatic sleep spindle detector assessed individualized signal amplitude for each channel as well as slow and fast spindle frequency peaks based on the spectral analysis of the EEG signal. The spindle detection was based on Continuous Wavelet Transform (CWT); it localized the exact length of sleep spindles and was sensitive also for detection of sleep spindles intermingled in high amplitude slow wave EEG activity. The automatic spindle detector was validated in n=18 naps from n=10 subjects, where EEG data were scored both visually and by a commercial automatic algorithm (SIESTA). Comparison of our own spindle detector with results from the SIESTA algorithm and visual scoring revealed the correlations of 0.97 and 0.92, respectively (correlation between SIESTA algorithm and visual scoring was 0.90). In the second part of the work, the similarity of given sleep EEG parameters in n=32 healthy monozygotic (MZ) twins was compared with the similarity in n=14 healthy same-gender dizygotic (DZ) twins. The author of the current work did not participate in acquisition of twin study sample. EEG sleep recordings used for the heritability study were collected and already described by Ambrosius et al. (2008). Investigation of REM sleep included the absolute EEG spectral power, the shape of REM power spectrum, the amount and the structural organization of REMs; parameters of Non-REM sleep included slow and fast sleep spindle characteristics as well as the shape of the Non-REM power spectrum in general. In addition to estimating genetic effects, differences in within-pair similarity and night-to-night stability of given parameters were illustrated by intraclass correlation coefficients (ICC) and cluster analysis. A substantial genetic influence on both spectral composition and phasic parameters of REM sleep was observed. A significant genetic variance in spectral power affected delta to high sigma and high beta to gamma EEG frequency bands, as well as all phasic REM parameters with the exception of the REMs occurring outside REM bursts. Furthermore, MZ and DZ twins differed significantly in their within-pair similarity of non-REM and REM EEG spectra morphology. Regarding sleep spindles, statistical analysis revealed a significant genetic influence on localization in frequency range as well as on basic spindle characteristics (amplitude, length, quantity), except in the quantity of fast spindles in stage 2 and whole Non-REM sleep. Basic spindle parameters showed trait-like characteristics and significant differences in within-pair similarity between the twin groups. In summary, the developed algorithms for automatic REM and sleep spindle detection provide several advantages: the elimination of human scorer biases and intra-rater variability, investigation of structural organization of REMs, exact determination of fast and slow spindle frequency for each individual. Algorithms are fully automated and therefore well suited to score REM density and sleep spindles in large patient samples. In the second part of the study, sleep EEG analysis in MZ and DZ twins revealed a substantial genetic determination of both tonic and phasic REM sleep parameters. This complements previous findings of a high genetic determination of the Non-REM sleep power spectrum. Interestingly, smaller genetic effects and lower night-to-night stability were observed for fast spindles, especially in SWS. This is in line with recent hypotheses on the differential function of sleep spindle types for memory consolidation. The results from the presented studies strongly support the application of sleep EEG to identify clinically relevant biomarkers for psychiatric disorders

Constant-Factor FPT Approximation for Capacitated k-Median

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon

Computing Equilibrium in Matching Markets

Market equilibria of matching markets offer an intuitive and fair solution for matching problems without money with agents who have preferences over the items. Such a matching market can be viewed as a variation of Fisher market, albeit with rather peculiar preferences of agents. These preferences can be described by piece-wise linear concave (PLC) functions, which however, are not separable (due to each agent only asking for one item), are not monotone, and do not satisfy the gross substitute property-- increase in price of an item can result in increased demand for the item. Devanur and Kannan in FOCS 08 showed that market clearing prices can be found in polynomial time in markets with fixed number of items and general PLC preferences. They also consider Fischer markets with fixed number of agents (instead of fixed number of items), and give a polynomial time algorithm for this case if preferences are separable functions of the items, in addition to being PLC functions. Our main result is a polynomial time algorithm for finding market clearing prices in matching markets with fixed number of different agent preferences, despite that the utility corresponding to matching markets is not separable. We also give a simpler algorithm for the case of matching markets with fixed number of different items

Genetics of human sleep EEG

Sleep characteristics are candidates for predictive biological markers in patients with severe psychiatric diseases, in particular affective disorder and schizophrenia. The genetic components of sleep determination in humans remain, to a large degree, unelucidated. In particular, the heritability of rapid eye movement (REM) sleep and EEG bursts of oscillatory brain activity in Non-REM sleep, i.e. sleep spindles, are of interest. In addition, recent findings suggest a strong role of distinct sleep spindle types in memory consolidation, making it important to identify sleep spindles in slow wave sleep (SWS) and to separate slow and fast spindle localization in the frequency range. However, predictive sleep biomarker research requires large sample sizes of healthy and affected human individuals. Therefore, the present work addressed two questions. The first aim was to optimize data analysis by developing algorithms that allow an efficient and reliable identification of rapid eye movements (REMs) and sleep EEG spindles. In the second part, developed methods were applied to sleep EEG data from a classical twin study to identify genetic effects on tonic and phasic REM sleep parameters, sleep spindles, and their trait-like characteristics. The algorithm for REM detection was developed for standard clinical two channel electrooculographic montage. The goal was to detect REMs visible above the background noise, and in the case of REM saccades to classify each movement separately. In order to achieve a high level of sensitivity, detection was based on a first derivative of electrooculogram (EOG) potentials and two detection thresholds. The developed REM detector was then validated in n=12 polysomnographic recordings from n=7 healthy subjects who had been previously scored visually by two human experts according to standard guidelines. Comparison of automatic REM detection with human scorers revealed mean correlations of 0.94 and 0.90, respectively (mean correlation between experts was 0.91). The developed automatic sleep spindle detector assessed individualized signal amplitude for each channel as well as slow and fast spindle frequency peaks based on the spectral analysis of the EEG signal. The spindle detection was based on Continuous Wavelet Transform (CWT); it localized the exact length of sleep spindles and was sensitive also for detection of sleep spindles intermingled in high amplitude slow wave EEG activity. The automatic spindle detector was validated in n=18 naps from n=10 subjects, where EEG data were scored both visually and by a commercial automatic algorithm (SIESTA). Comparison of our own spindle detector with results from the SIESTA algorithm and visual scoring revealed the correlations of 0.97 and 0.92, respectively (correlation between SIESTA algorithm and visual scoring was 0.90). In the second part of the work, the similarity of given sleep EEG parameters in n=32 healthy monozygotic (MZ) twins was compared with the similarity in n=14 healthy same-gender dizygotic (DZ) twins. The author of the current work did not participate in acquisition of twin study sample. EEG sleep recordings used for the heritability study were collected and already described by Ambrosius et al. (2008). Investigation of REM sleep included the absolute EEG spectral power, the shape of REM power spectrum, the amount and the structural organization of REMs; parameters of Non-REM sleep included slow and fast sleep spindle characteristics as well as the shape of the Non-REM power spectrum in general. In addition to estimating genetic effects, differences in within-pair similarity and night-to-night stability of given parameters were illustrated by intraclass correlation coefficients (ICC) and cluster analysis. A substantial genetic influence on both spectral composition and phasic parameters of REM sleep was observed. A significant genetic variance in spectral power affected delta to high sigma and high beta to gamma EEG frequency bands, as well as all phasic REM parameters with the exception of the REMs occurring outside REM bursts. Furthermore, MZ and DZ twins differed significantly in their within-pair similarity of non-REM and REM EEG spectra morphology. Regarding sleep spindles, statistical analysis revealed a significant genetic influence on localization in frequency range as well as on basic spindle characteristics (amplitude, length, quantity), except in the quantity of fast spindles in stage 2 and whole Non-REM sleep. Basic spindle parameters showed trait-like characteristics and significant differences in within-pair similarity between the twin groups. In summary, the developed algorithms for automatic REM and sleep spindle detection provide several advantages: the elimination of human scorer biases and intra-rater variability, investigation of structural organization of REMs, exact determination of fast and slow spindle frequency for each individual. Algorithms are fully automated and therefore well suited to score REM density and sleep spindles in large patient samples. In the second part of the study, sleep EEG analysis in MZ and DZ twins revealed a substantial genetic determination of both tonic and phasic REM sleep parameters. This complements previous findings of a high genetic determination of the Non-REM sleep power spectrum. Interestingly, smaller genetic effects and lower night-to-night stability were observed for fast spindles, especially in SWS. This is in line with recent hypotheses on the differential function of sleep spindle types for memory consolidation. The results from the presented studies strongly support the application of sleep EEG to identify clinically relevant biomarkers for psychiatric disorders

When the Optimum is also Blind: a New Perspective on Universal Optimization

Consider the following variant of the set cover problem. We are given a universe U={1,...,n} and a collection of subsets C = {S_1,...,S_m} where each S_i is a subset of U. For every element u from U we need to find a set phi(u) from collection C such that u belongs to phi(u). Once we construct and fix the mapping phi from U to C a subset X from the universe U is revealed, and we need to cover all elements from X with exactly phi(X), that is {phi(u)}_{all u from X}. The goal is to find a mapping such that the cover phi(X) is as cheap as possible. This is an example of a universal problem where the solution has to be created before the actual instance to deal with is revealed. Such problems appear naturally in some settings when we need to optimize under uncertainty and it may be actually too expensive to begin finding a good solution once the input starts being revealed. A rich body of work was devoted to investigate such problems under the regime of worst case analysis, i.e., when we measure how good the solution is by looking at the worst-case ratio: universal solution for a given instance vs optimum solution for the same instance. As the universal solution is significantly more constrained, it is typical that such a worst-case ratio is actually quite big. One way to give a viewpoint on the problem that would be less vulnerable to such extreme worst-cases is to assume that the instance, for which we will have to create a solution, will be drawn randomly from some probability distribution. In this case one wants to minimize the expected value of the ratio: universal solution vs optimum solution. Here the bounds obtained are indeed smaller than when we compare to the worst-case ratio. But even in this case we still compare apples to oranges as no universal solution is able to construct the optimum solution for every possible instance. What if we would compare our approximate universal solution against an optimal universal solution that obeys the same rules as we do? We show that under this viewpoint, but still in the stochastic variant, we can indeed obtain better bounds than in the expected ratio model. For example, for the set cover problem we obtain $H_n$ approximation which matches the approximation ratio from the classic deterministic setup. Moreover, we show this for all possible probability distributions over $U$ that have a polynomially large carrier, while all previous results pertained to a model in which elements were sampled independently. Our result is based on rounding a proper configuration IP that captures the optimal universal solution, and using tools from submodular optimization. The same basic approach leads to improved approximation algorithms for other related problems, including Vertex Cover, Edge Cover, Directed Steiner Tree, Multicut, and Facility Location