An O(n3 [square root of] log n) algorithm for the optimal stable marriage problem

We give an O(n^3 √logn) time algorithm for the optimal stable marriage problem. This algorithm finds a stable marriage that minimizes an objective function defined over all stable marriages in a given problem instance.Irving, Leather, and Gusfield have previously provided a solution to this problem that runs in O(n^4) time [ILG87]. In addition, Feder has claimed that an O(n^3 log n) time algorithm exists [F89]. Our result is an asymptotic improvement over both cases.As part of our solution, we solve a special blue-red matching problem, and illustrate a technique for simulating Hopcroft and Karp's maximum-matching algorithm [HK73] on the transitive closure of a graph

Lower bounds for the stable marriage problem and its variants

In an instance of the stable marriage problem of size n, n men and n women each ranks members of the opposite sex in order of preference. A stable marriage is a complete matching M = {(m_1, w_i_1), (m_2, w_i_2), ..., (m_n, w_i_n)} such that no unmatched man and woman prefer each other to their partners in M.A pair (m_i, w_j) is stable if it is contained in some stable marriage. In this paper, we prove that determining if an arbitrary pair is stable requires Ω(n^2) time in the worst case. We show, by an adversary argument, that there exists instances of the stable marriage problem such that it is possible to find at least one pair that exhibits the Ω(n^2) lower bound.As corollaries of our results, the lower bound of Ω(n^2) is established for several stable marriage related problems. Knuth, in his treatise on stable marriage, asks if there is an algorithm that finds a stable marriage in less than Θ(n^2) time. Our results show that such an algorithm does not exist

Complexity of the stable marriage and stable roommate problems in three dimensions

The stable marriage problem is a matching problem that pairs members of two sets. The objective is to achieve a matching that satisfies all participants based on their preferences. The stable roommate problem is a variant involving only one set, which is partitioned into pairs with a similar objective. There exist asymptotically optimal algorithms that solve both problems.In this paper, we investigate the complexity of three dimensional extensions of these problems. This is one of twelve research directions suggested by Knuth in his book on the stable marriage problem. We show that these problems are NP-complete, and hence it is unlikely that there exist efficient algorithms for their solutions.Applying the polynomial tranformation developed in this paper, we extend the NP-completeness result to include the problem of matching couples - who are both medical school graduates - to pairs of hospital resident positions. This problem is important in practice and is dealth with annually by NRMP, the centralized program that matches all medical school graduates in the United States to available resident positions

Bayesian Semi-supervised Learning with Graph Gaussian Processes

We propose a data-efficient Gaussian process-based Bayesian approach to the semi-supervised learning problem on graphs. The proposed model shows extremely competitive performance when compared to the state-of-the-art graph neural networks on semi-supervised learning benchmark experiments, and outperforms the neural networks in active learning experiments where labels are scarce. Furthermore, the model does not require a validation data set for early stopping to control over-fitting. Our model can be viewed as an instance of empirical distribution regression weighted locally by network connectivity. We further motivate the intuitive construction of the model with a Bayesian linear model interpretation where the node features are filtered by an operator related to the graph Laplacian. The method can be easily implemented by adapting off-the-shelf scalable variational inference algorithms for Gaussian processes.Comment: To appear in NIPS 2018 Fixed an error in Figure 2. The previous arxiv version contains two identical sub-figure

Hayman's classical conjecture on some nonlinear second order algebraic ODEs

In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second order nonlinear algebraic ordinary differential equations. We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the {\it classical conjecture} proposed by Hayman in 1996.Comment: 15 pages, to appear, Complex variables and elliptic equation

DYNAMIC RESPONSES OF FLOATING OFFSHORE PLATFORMS WITH LARGE HULLS

Spar and semi-submersible are the most common types of floating offshore platforms used for deepwater operations. The spar consists of a hollow cylindrical deep-draft floating hull that provides buoyancy, with strake surrounding the hull to reduce vortex induce vibration and to held in place by mooring lines. To remain stable, it is important to maintain the centre of gravity always below the centre of buoyancy. The semi-submersible comprises of two horizontal water tight pontoons and number of column units that stand on the pontoons to provide support to the deck structure. It is held in place by mooring lines and dynamic positioning system. Both these types of platforms are made up of large-sized hull for providing buoyancy. As the ratio of the diameter of these structures to the wave length is above 0.2, the wave diffraction theory is the correct theory to be applied for the calculation of wave forces and wave damping, according to the literature. However, the application of diffraction theory, even linear one, is very much complicated and requires very costly commercial software. Hence, many research papers have reported results of dynamic analysis, using Morison equation for such cases, reasoning that for a considerable part of the frequency range, the ratio of diameter to wave length is still below 0.2. This is because of the ease of using Morison equation in programming and the possibility of incorporating the various non-linearity in the analysis. Yet, it has been established that the consultants are using only diffraction analysis for the analysis and design of such platforms. The aim of this study was to determine and compare the responses by both Morison equation and diffraction theory to the model test responses, and to suggest nonlinear multiple regression curves to estimate the structure responses. Model tests were conducted for spar and semi-submersible platform models in the wave tank at the Offshore Engineering Laboratory of Universiti Teknologi PETRONAS and the responses were measured. The respective prototypes were analyzed using a numerical Newmark Beta time domain integration method that was developed by using Matlab program. The platforms were designed as rigid bodies and three degree of freedom; surge, heave and pitch were considered. Linear wave theory and Morison equation were used for wave force determination in time domain analysis. A commercial software was employed to determine responses of the structures by Linear Wave Diffraction module. These results proved that the diffraction theory results were much closer to the actual model test results, thereby proving that using Morison equation for such platforms is not justified. Using the results of the diffraction analysis for a large number of platforms and conducting a non-linear multiple regression analysis, this thesis also suggests formulae to obtain suitable regression curves for predicting the diffraction responses of the spar and semi-submersible for any dimension and draft within the range suggested

Production of high stellar-mass primordial black holes in trapped inflation

Trapped inflation has been proposed to provide a successful inflation with a steep potential. We discuss the formation of primordial black holes in the trapped inflationary scenario. We show that primordial black holes are naturally produced during inflation with a steep trapping potential. In particular, we have given a recipe for an inflaton potential with which particle production can induce large non-Gaussian curvature perturbation that leads to the formation of high stellar-mass primordial black holes. These primordial black holes could be dark matter observed by the LIGO detectors through a binary black-hole merger. At the end, we have given an attempt to realize the required inflaton potential in the axion monodromy inflation, and discussed the gravitational waves sourced by the particle production.Comment: 6 pages, 5 figures, match the version accepted by JHE