Almost Optimal Stochastic Weighted Matching With Few Queries

We consider the {\em stochastic matching} problem. An edge-weighted general (i.e., not necessarily bipartite) graph $G(V, E)$ is given in the input, where each edge in $E$ is {\em realized} independently with probability $p$; the realization is initially unknown, however, we are able to {\em query} the edges to determine whether they are realized. The goal is to query only a small number of edges to find a {\em realized matching} that is sufficiently close to the maximum matching among all realized edges. This problem has received a considerable attention during the past decade due to its numerous real-world applications in kidney-exchange, matchmaking services, online labor markets, and advertisements. Our main result is an {\em adaptive} algorithm that for any arbitrarily small $\epsilon > 0$, finds a $(1-\epsilon)$-approximation in expectation, by querying only $O(1)$ edges per vertex. We further show that our approach leads to a $(1/2-\epsilon)$-approximate {\em non-adaptive} algorithm that also queries only $O(1)$ edges per vertex. Prior to our work, no nontrivial approximation was known for weighted graphs using a constant per-vertex budget. The state-of-the-art adaptive (resp. non-adaptive) algorithm of Maehara and Yamaguchi [SODA 2018] achieves a $(1-\epsilon)$-approximation (resp. $(1/2-\epsilon)$-approximation) by querying up to $O(w\log{n})$ edges per vertex where $w$ denotes the maximum integer edge-weight. Our result is a substantial improvement over this bound and has an appealing message: No matter what the structure of the input graph is, one can get arbitrarily close to the optimum solution by querying only a constant number of edges per vertex. To obtain our results, we introduce novel properties of a generalization of {\em augmenting paths} to weighted matchings that may be of independent interest

Reentrant Adhesion Behavior in Nanocluster Deposition

We simulate the collision of atomic clusters with a weakly attractive surface using molecular dynamics in a regime between soft-landing and fragmentation, where the cluster undergoes large deformation but remains intact. As a function of incident kinetic energy, we find a transition from adhesion to reflection at low kinetic energies. We also identify a second adhesive regime at intermediate kinetic energies, where strong deformation of the cluster leads to an increase in contact area and adhesive energy.Comment: 7 pages, 6 figure

Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries

The stochastic matching problem deals with finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic $k$-set packing, where the problem is to find a maximum packing of sets, each of which exists with some probability. In this paper, we provide edge and set query algorithms for these two problems, respectively, that provably achieve some fraction of the omniscient optimal solution. Our main theoretical result for the stochastic matching (i.e., $2$-set packing) problem is the design of an \emph{adaptive} algorithm that queries only a constant number of edges per vertex and achieves a $(1-\epsilon)$ fraction of the omniscient optimal solution, for an arbitrarily small $\epsilon>0$. Moreover, this adaptive algorithm performs the queries in only a constant number of rounds. We complement this result with a \emph{non-adaptive} (i.e., one round of queries) algorithm that achieves a $(0.5 - \epsilon)$ fraction of the omniscient optimum. We also extend both our results to stochastic $k$-set packing by designing an adaptive algorithm that achieves a $(\frac{2}{k} - \epsilon)$ fraction of the omniscient optimal solution, again with only $O(1)$ queries per element. This guarantee is close to the best known polynomial-time approximation ratio of $\frac{3}{k+1} -\epsilon$ for the \emph{deterministic} $k$-set packing problem [Furer and Yu, 2013] We empirically explore the application of (adaptations of) these algorithms to the kidney exchange problem, where patients with end-stage renal failure swap willing but incompatible donors. We show on both generated data and on real data from the first 169 match runs of the UNOS nationwide kidney exchange that even a very small number of non-adaptive edge queries per vertex results in large gains in expected successful matches

Molecular dynamics simulations of reflection and adhesion behavior in Lennard-Jones cluster deposition

We conduct molecular dynamics simulations of the collision of atomic clusters with a weakly-attractive surface. We focus on an intermediate regime, between soft-landing and fragmentation, where the cluster undergoes deformation on impact but remains largely intact, and will either adhere to the surface (and possibly slide), or be reflected. We find that the outcome of the collision is determined by the Weber number, We i.e. the ratio of the kinetic energy to the adhesion energy, with a transition between adhesion and reflection occurring as We passes through unity. We also identify two distinct collision regimes: in one regime the collision is largely elastic and deformation of the cluster is relatively small but in the second regime the deformation is large and the adhesion energy starts to depend on the kinetic energy. If the transition between these two regimes occurs at a similar kinetic energy to that of the transition between reflection and adhesion, then we find that the probability of adhesion for a cluster can be bimodal. In addition we investigate the effects of the angle of incidence on adhesion and reflection. Finally we compare our findings both with recent experimental results and with macroscopic theories of particle collisions.Comment: 18 pages, 13 figure

Break-down of the single-active-electron approximation for one-photon ionization of the B 1Σu+^1\Sigma_u^+ state of H2_2 exposed to intense laser fields

Ionization, excitation, and de-excitation to the ground state is studied theoretically for the first excited singlet state B $^1\Sigma_u^+$ of H$_2$ exposed to intense laser fields with photon energies in between about 3 eV and 13 eV. A parallel orientation of a linear polarized laser and the molecular axis is considered. Within the dipole and the fixed-nuclei approximations the time-dependent Schr\"odinger equation describing the electronic motion is solved in full dimensionality and compared to simpler models. A dramatic break-down of the single-active-electron approximation is found and explained to be due to the inadequate description of the final continuum states.Comment: 9 pages, 4 figure

Biochemical composition of promising leaves genotypes of buckwheat grown in Himachal Pradesh

Buckwheat originated from China and being cultivated all over the world, and has become a prominent pseudocereal. Among the pseudocereals (amaranthus, buckwheat and quinoa), buckwheat plant is economically important primarily due to their carbohydrate and protein rich grains, short growth span; besides foliage being used as a green vegetable and commercial source of the glycoside rutin used in medicine. In the present study, an attempt was made to evaluate the biochemical constituents of nutritional and nutraceutical significance of fourteen promising leaves genotypes of common buckwheat grown in Sangla region by following standard procedures. Wide variations in moisture content, crude protein, fat, ash, crude fiber, carbohydrates, ascorbic acid, oxalate and in vitro protein digestibility were observed to range from 87.4 to 92.2 %, 22.4 to 30.4 %, 1.8 to 3.7 %, 10.6 to 15.4 %, 12.0 to 13.9 %, 34.8 to 42.4 %, 25.0 to 29.2 mg/100g, 1375 to 1390 mg/100g and 53.4 to 65.1 % in that order. The content of minerals such as potassium, phosphorus, calcium, magnesium, zinc, manganese and copper varied from 1767.5 to 2035.0 mg/100g, 808 to 910 mg/100g, 394 to 409 mg/100g, 232.0 to 248.2 mg/100g, 3.1 to 6.1 mg/100g, 20.4 to 29.8 mg/100g and 0.2 to 1.4 mg/100g respectively. Based on cumulative grading done in respect of nutritionally desirable quality i.e., protein, ash, crude fiber, carbohydrates, ascorbic acid, in vitro protein digestibility, calcium, phosphorus, iron and oxalate content, the genotype IC-323731 followed by Kullugangetri and VL-27 emerged out to be overall superior versatile cultivars for cultivation under dry temperate climate