We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {a i jk } n i, j,k=1 such that for all i, j, k ∈ {1, . . . , n} we have a i jk = a ik j = a k ji = a jik = a ki j = a jki and a iik = a i j j = a i ji = 0, computes a number Alg(A) which satisfies with probability at leastOn the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [22] that under the assumption NP DT I ME n (log n) O(1) , for every ε > 0 there is no algorithm that approximates max x∈{−1,1} n n i, j,k=1 a i jk x i x j x k within a factor of 2O(1) . Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in R n with respect to the L 1 norm. We show that it is possible to do so up to a multiplicative error of O n log n , while no randomized polynomial time algorithm can achieve accuracy o n log n . This resolves a question posed by Brieden, Gritzmann, Kannan, Klee, Lovász and Simonovits in [10].We apply our new algorithm to improve the algorithm of Håstad and Venkatesh [22] for the Max-E3-Lin-2 problem. Given an over-determined system E of N linear equations modulo 2 in n ≤ N Boolean variables, such that in each equation appear only three distinct variables, the goal is to approximate in polynomial time the maximum number of satisfiable equations in E minus N 2 (i.e. we subtract the expected number of satisfied equations in a random assignment). Håstad and Venkatesh [22] obtained an algorithm which approximates this value up to a factor of O √ N . We obtain a O n log n approximation algorithm. By relating this problem to the refutation problem for random 3 − CNF formulas we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult.

Discovery of microbial hydrocarbon degradation pathways has traditionally relied on laboratory isolation and characterization of microorganisms. Although many metabolic pathways for hydrocarbon degradation have been discovered, the absence of tools dedicated to their annotation makes it difficult to identify the relevant genes and predict the hydrocarbon degradation potential of microbial genomes and metagenomes. Furthermore, sequence homology between hydrocarbon degradation genes and genes with other functions often results in misannotation. A tool that systematically identifies hydrocarbon metabolic potential is therefore needed. We present the Calgary approach to ANnoTating HYDrocarbon degradation genes (CANT-HYD), a database containing HMMs of 37 marker genes involved in anaerobic and aerobic degradation pathways of aliphatic and aromatic hydrocarbons. Using this database, we show that hydrocarbon metabolic potential is widespread in the tree of life and identify understudied or overlooked hydrocarbon degradation potential in many phyla. We also demonstrate scalability by analyzing large metagenomic datasets for the prediction of hydrocarbon utilization in diverse environments. To the best of our knowledge, CANT-HYD is the first comprehensive tool for robust and accurate identification of marker genes associated with aerobic and anaerobic hydrocarbon degradation.

For every > 0, and integer ≥ 3, we show that given an -vertex graph that has an induced -colorable subgraph of size (1− ) , it is NP-hard to find an independent set of size 2 .