Matching Is as Easy as the Decision Problem, in the NC Model

Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [KUW85, MVV87]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. All known efficient matching algorithms for general graphs follow one of two approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based algorithm follows a new approach and uses many of the ideas discovered in the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs.Comment: Appeared in ITCS 202

Automatic Finding Trapezoidal Membership Functions in Mining Fuzzy Association Rules Based on Learning Automata

Association rule mining is an important data mining technique used for discovering relationships among all data items. Membership functions have a significant impact on the outcome of the mining association rules. An important challenge in fuzzy association rule mining is finding an appropriate membership functions, which is an optimization issue. In the most relevant studies of fuzzy association rule mining, only triangle membership functions are considered. This study, as the first attempt, used a team of continuous action-set learning automata (CALA) to find both the appropriate number and positions of trapezoidal membership functions (TMFs). The spreads and centers of the TMFs were taken into account as parameters for the research space and a new approach for the establishment of a CALA team to optimize these parameters was introduced. Additionally, to increase the convergence speed of the proposed approach and remove bad shapes of membership functions, a new heuristic approach has been proposed. Experiments on two real data sets showed that the proposed algorithm improves the efficiency of the extracted rules by finding optimized membership functions

Surface Modification of PVDF Membranes for Treating Produced Waters by Direct Contact Membrane Distillation

Direct contact membrane distillation is a promising unit operation for treating hydraulic fracturing flow back and produced water. However, while a hydrophobic membrane is essential to prevent the passage of water from the feed to the permeate side, fouling by dissolved organic species can compromise membrane performance and result in wetting of the membrane pores. Here four monomers, hydroxyethylmethacrylate, acrylic acid, 1-vinyl-3-allylimidazolium bromide, and 1-vinyl-3-hexylimidazolium bromide have been grafted from the surface of a PVDF membrane. The modified and base membranes were tested in a direct contact membrane distillation system. All membranes were challenged with real produced water. In addition, base membranes and membranes modified by grafting 1-vinyl-3-allylimidazolium bromide were challenged with produced water that was pretreated by electrocoagulation. These membranes were also challenged with a synthetic wastewater made by adding to DI water the major inorganic compounds present in the produced water. The highest fluxes were obtained for the membrane grafted with 1-vinyl-3-allylimidazolium bromide chains. The membrane surface after membrane distillation was analyzed by scanning electron microscopy and energy-dispersive X-ray (EDX) spectroscopy. For all membranes, the interaction between adsorbed organic and inorganic species determines the degree of fouling and hence the loss in flux and membrane stability. Polyionic liquid chains that contain a repeating charged species and hydrophobic segments minimized fouling by organic species and improved the flux and membrane stability. The results suggest that by carefully tuning the properties of the monomer units in the polymer chains, membrane stability and performance can be improved

Approximating the Largest Root and Applications to Interlacing Families

We study the problem of approximating the largest root of a real-rooted polynomial of degree $n$ using its top $k$ coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in $k$ that use the top $k$ coefficients to approximate the maximum root within a factor of $n^{1/k}$ and $1+O(\tfrac{\log n}{k})^2$ when $k\leq \log n$ and $k>\log n$ respectively. We also prove corresponding information-theoretic lower bounds of $n^{\Omega(1/k)}$ and $1+\Omega\left(\frac{\log \frac{2n}{k}}{k}\right)^2$, and show strong lower bounds for noisy version of the problem in which one is given access to approximate coefficients. This problem has applications in the context of the method of interlacing families of polynomials, which was used for proving the existence of Ramanujan graphs of all degrees, the solution of the Kadison-Singer problem, and bounding the integrality gap of the asymmetric traveling salesman problem. All of these involve computing the maximum root of certain real-rooted polynomials for which the top few coefficients are accessible in subexponential time. Our results yield an algorithm with the running time of $2^{\tilde O(\sqrt[3]n)}$ for all of them

Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{\gamma+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices