Linear Equations Modulo 2 and the L1 Diameter of Convex Bodies

We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = \left\{ {aijk} \right\}_{i,j,k = 1}^n such that for all i,j,k \in \left\{ {1, \ldots,n} \right\} we have a_{ijk}= a_{ikj}= a_{kji}= a_{jik}= a_{kij}= a_{jki} and a_{iik}= a_{ijj}= a_{iji}= 0, computes a number Alg(A) which satisfies with probability at least \frac{1} {2}, \Omega (\sqrt {\frac{{\log n}} {n}}\cdot _{\chi\in \left\{ { - 1,1} \right\}^n }^{\max} \sum\limits_{i,j,k = 1}^n {a_{ijk} x_i x_i } x_k\leqslant A\lg (A) \leqslant _{\chi\in \left\{ { - 1,1} \right\}^n }^{\max } \sum\limits_{i,j,k = 1}^n {a_{ijk} x_i x_j x_k }. On the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [22] that under the assumption {\rm N}{\rm P} \varsubsetneq DTIME(n^{(\log n){\rm O}(1)} ), for every \varepsilon\le 0 there is no algorithm that approximates \max _{\chi\in \left\{ { - 1,1} \right\}^n } \sum\limits_{}^{} {_{i,j,k = 1}^n } a_{ijk} x_i x_j x_k within a factor of 2^{(\log n)^{1 - \varepsilon } } in time 2^{(\log n)^{{\rm O}(1)} }. Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in \mathbb{R}^n with respect to the L_1 norm. We show that it is possible to do so up to a multiplicative error of {\rm O}(\sqrt {\frac{n} {{\log n}}} ), while no randomized polynomial time algorithm can achieve accuracy {\rm O}(\sqrt {\frac{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 \varepsilon of N linear equations modulo 2 in n \leqslant 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 \varepsilon minus \frac{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 {\rm O}(\sqrt N ). We obtain a {\rm O}(\sqrt {\frac{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.