Linear Equations Modulo 2 and the $L_1$ Diameter of Convex Bodies

We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers $A=\{a_{ijk}\}_{i,j,k=1}^n$ such that for all $i,j,k\in\{1,\dots,n\}$ 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 $\operatorname{Alg}(A)$ which satisfies with probability at least $\frac12$, $\Omega(\sqrt{\frac{\log n}{n}}t)\cdot\max_{x\in \{-1,1\}^n}\sum_{i,j,k=1}^n a_{ijk}x_ix_jx_k\le\operatorname{Alg}(A)\le\max_{x\in \{-1,1\}^n}\sum_{i,j,k=1}^n a_{ijk}x_ix_jx_k$. On the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [Random Structures Algorithms, 25 (2004), pp. 117-149] that under the assumption $NP\not\subseteq DTIME(n^{(\log n)^{O(1)}})$, for every $\epsilon>0$ there is no algorithm that approximates $\max_{x\in \{-1,1\}^n}\sum_{i,j,k=1}^n a_{ijk}x_ix_jx_k$ within a factor of $2^{(\log n)^{1-\epsilon}}$ in time $2^{(\log n)^{O(1)}}$. 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 $O(\sqrt{\frac{n}{\log n}})$, while no randomized polynomial time algorithm can achieve accuracy $o(\sqrt{\frac{n}{\log n}})$. This resolves a question posed by Brieden et al. in [Mathematika, 48 (2001), pp. 63-105]. We apply our new algorithm to improve the algorithm of Håstad and Venkatesh for the Max-E3-Lin-2 problem. Given an overdetermined system $\mathcal{E}$ of $N$ linear equations modulo 2 in $n\le N$ Boolean variables such that in each equation only three distinct variables appear, the goal is to approximate in polynomial time the maximum number of satisfiable equations in $\mathcal{E}$ minus $\frac{N}{2}$ (i.e., we subtract the expected number of satisfied equations in a random assignment). Håstad and Venkatesh obtained an algorithm which approximates this value up to a factor of $O(\sqrt{N})$. We obtain an $O(\sqrt{\frac{n}{\log n}})$ 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.