Depletion potentials in highly size-asymmetric binary hard-sphere mixtures: Comparison of accurate simulation results with theory

We report a detailed study, using state-of-the-art simulation and theoretical methods, of the depletion potential between a pair of big hard spheres immersed in a reservoir of much smaller hard spheres, the size disparity being measured by the ratio of diameters q=\sigma_s/\sigma_b. Small particles are treated grand canonically, their influence being parameterized in terms of their packing fraction in the reservoir, \eta_s^r. Two specialized Monte Carlo simulation schemes --the geometrical cluster algorithm, and staged particle insertion-- are deployed to obtain accurate depletion potentials for a number of combinations of q\leq 0.1 and \eta_s^r. After applying corrections for simulation finite-size effects, the depletion potentials are compared with the prediction of new density functional theory (DFT) calculations based on the insertion trick using the Rosenfeld functional and several subsequent modifications. While agreement between the DFT and simulation is generally good, significant discrepancies are evident at the largest reservoir packing fraction accessible to our simulation methods, namely \eta_s^r=0.35. These discrepancies are, however, small compared to those between simulation and the much poorer predictions of the Derjaguin approximation at this \eta_s^r. The recently proposed morphometric approximation performs better than Derjaguin but is somewhat poorer than DFT for the size ratios and small sphere packing fractions that we consider. The effective potentials from simulation, DFT and the morphometric approximation were used to compute the second virial coefficient B_2 as a function of \eta_s^r. Comparison of the results enables an assessment of the extent to which DFT can be expected to correctly predict the propensity towards fluid fluid phase separation in additive binary hard sphere mixtures with q\leq 0.1.Comment: 16 pages, 9 figures, revised treatment of morphometric approximation and reordered some materia

Online Computation with Untrusted Advice

The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. This paradigm allows to establish tradeoffs between the amount of this additional information and the performance of the online algorithm. However, unlike real life in which advice is a recommendation that we can choose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm treats it as infallible. This means that if the advice is corrupt or, worse, if it comes from a malicious source, the algorithm may perform poorly. In this work, we study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner. To this end, we focus on well- studied online problems such as ski rental, online bidding, bin packing, and list update. For ski-rental and online bidding, we show how to obtain algorithms that are Pareto-optimal with respect to the competitive ratios achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in which Pareto-optimality is not necessarily guaranteed. For bin packing and list update, we give online algorithms with worst-case tradeoffs in their competitiveness, depending on whether the advice is trusted or not; this is motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging problem, but in which the competitiveness depends on the reliability of the advice. Furthermore, we demonstrate how to prove lower bounds, within this model, on the tradeoff between the number of advice bits and the competitiveness of any online algorithm. Last, we study the effect of randomization: here we show that for ski-rental there is a randomized algorithm that Pareto-dominates any deterministic algorithm with advice of any size. We also show that a single random bit is not always inferior to a single advice bit, as it happens in the standard model

On largest volume simplices and sub-determinants

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand, we show that there exists a constant $c>1$ such that this problem cannot be approximated with a factor of $c^d$, unless $P=NP$. % This improves over the $1.09$ inapproximability that was previously known. Our hardness result holds even if $n = O(d)$, in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix

Semidefinite optimization in discrepancy theory

Recently, there have been several new developments in discrepancy theory based on connections to semidefinite programming. This connection has been useful in several ways. It gives efficient polynomial time algorithms for several problems for which only non-constructive results were previously known. It also leads to several new structural results in discrepancy itself, such as tightness of the so-called determinant lower bound, improved bounds on the discrepancy of the union of set systems and so on. We will give a brief survey of these results, focussing on the main ideas and the techniques involved

Tagged-particle dynamics in a hard-sphere system: mode-coupling theory analysis

The predictions of the mode-coupling theory of the glass transition (MCT) for the tagged-particle density-correlation functions and the mean-squared displacement curves are compared quantitatively and in detail to results from Newtonian- and Brownian-dynamics simulations of a polydisperse quasi-hard-sphere system close to the glass transition. After correcting for a 17% error in the dynamical length scale and for a smaller error in the transition density, good agreement is found over a wide range of wave numbers and up to five orders of magnitude in time. Deviations are found at the highest densities studied, and for small wave vectors and the mean-squared displacement. Possible error sources not related to MCT are discussed in detail, thereby identifying more clearly the issues arising from the MCT approximation itself. The range of applicability of MCT for the different types of short-time dynamics is established through asymptotic analyses of the relaxation curves, examining the wave-number and density-dependent characteristic parameters. Approximations made in the description of the equilibrium static structure are shown to have a remarkable effect on the predicted numerical value for the glass-transition density. Effects of small polydispersity are also investigated, and shown to be negligible.Comment: 20 pages, 23 figure

Structural relaxation of polydisperse hard spheres: comparison of the mode-coupling theory to a Langevin dynamics simulation

We analyze the slow, glassy structural relaxation as measured through collective and tagged-particle density correlation functions obtained from Brownian dynamics simulations for a polydisperse system of quasi-hard spheres in the framework of the mode-coupling theory of the glass transition (MCT). Asymptotic analyses show good agreement for the collective dynamics when polydispersity effects are taken into account in a multi-component calculation, but qualitative disagreement at small $q$ when the system is treated as effectively monodisperse. The origin of the different small-$q$ behaviour is attributed to the interplay between interdiffusion processes and structural relaxation. Numerical solutions of the MCT equations are obtained taking properly binned partial static structure factors from the simulations as input. Accounting for a shift in the critical density, the collective density correlation functions are well described by the theory at all densities investigated in the simulations, with quantitative agreement best around the maxima of the static structure factor, and worst around its minima. A parameter-free comparison of the tagged-particle dynamics however reveals large quantiative errors for small wave numbers that are connected to the well-known decoupling of self-diffusion from structural relaxation and to dynamical heterogeneities. While deviations from MCT behaviour are clearly seen in the tagged-particle quantities for densities close to and on the liquid side of the MCT glass transition, no such deviations are seen in the collective dynamics.Comment: 23 pages, 26 figure

Derandomizing Concentration Inequalities with dependencies and their combinatorial applications

Both in combinatorics and design and analysis of randomized algorithms for combinatorial optimization problems, we often use the famous bounded differences inequality by C. McDiarmid (1989), which is based on the martingale inequality by K. Azuma (1967), to show positive probability of success. In the case of sum of independent random variables, the inequalities of Chernoff (1952) and Hoeffding (1964) can be used and can be efficiently derandomized, i.e. we can construct the required event in deterministic, polynomial time (Srivastav and Stangier 1996). With such an algorithm one can construct the sought combinatorial structure or design an efficient deterministic algorithm from the probabilistic existentce result or the randomized algorithm. The derandomization of C. McDiarmid's bounded differences inequality was an open problem. The main result in Chapter 3 is an efficient derandomization of the bounded differences inequality, with the time required to compute the conditional expectation of the objective function being part of the complexity. The following chapters 4 through 7 demonstrate the generality and power of the derandomization framework developed in Chapter 3. In Chapter 5, we derandomize the Maker's random strategy in the Maker-Breaker subgraph game given by Bednarska and Luczak (2000), which is fundamental for the field, and analyzed with the concentration inequality of Janson, Luczak and Rucinski. But since we use the bounded differences inequality, it is necessary to give a new proof of the existence of subgraphs in G(n,M)-random graphs (Chapter 4). In Chapter 6, we derandomize the two-stage randomized algorithm for the set-multicover problem by El Ouali, Munstermann and Srivastav (2014). In Chapter 7, we show that the algorithm of Bansal, Caprara and Sviridenko (2009) for the multidimensional bin packing problem can be elegantly derandomized with our derandomization framework of bounded differences inequality, while the authors use a potential function based approach, leading to a rather complex analysis. In Chapter 8, we analyze the constrained hypergraph coloring problem given in Ahuja and Srivastav (2002), which generalizes both the property B problem for the non-monochromatic 2-coloring of hypergraphs and the multidimensional bin packing problem using the bounded differences inequality instead of the Lovasz local lemma. We also derandomize the algorithm using our framework. In Chapter 9, we turn to the generalization of the well-known concentration inequality of Hoeffding (1964) by Janson (1994), to sums of random variables, that are not independent, but are partially dependent, or in other words, are independent in certain groups. Assuming the same dependency structure as in Janson (1994), we generalize the well-known concentration inequality of Alon and Spencer (1991). In Chapter 10, we derandomize the inequality of Alon and Spencer. The derandomization of our generalized Alon-Spencer inequality under partial dependencies remains an interesting, open problem

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn