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={a
_ijk
}
_ij,k=1

ⁿ
such that for all i,j,kisin{1,...,n} we have a
_ijk
=a
_ikj
=a
_kji
=a
_jik
=a
_kij
=a
_kji
and a
_iik
=a
_ijj
=a
_iji
=0, computes a number Alg(A) which satisfies with probability at least 1/2, Omega(radic(logn/n))ldrmax
_xisin{-1,1}

ⁿ
Sigma
_i,j,k=1

ⁿ
a
_ijk
x
_i
x
_j
x
_k
lesAlg(A)lesmax
_xisin{-1,1}

ⁿ
Sigma
_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 Hastad and Venkatesh that under the assumption NPnsubeDTIME(n
<sup>(logn)

O(1)</sup>
),for every epsiv>0 there is no algorithm that approximates max
_xisin{-1,1}

ⁿ
Sigma
_i,j,k=1

ⁿ
a
_ijk
x
_i
x
_j
x
_k
within a factor of 2(logn)
^t-epsiv
in time 2
<sup>(logn)

O(1)</sup>
. 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(radic(n/logn)), while no randomized polynomial time algorithm can achieve accuracy O(radic(n/logn)). This resolves a question posed by Brieden, Gritzmann, Kantian, Klee, Lovasz and Simonos. We apply our new algorithm improve the algorithm of Hastad and Venkatesh or the Max-E3-Lin-2 problem. Given an over-determined system epsiv of N linear equations modulo 2 in nlesN 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 epsiv minus N/2 (i.e. we subtract the expected number of satisfied equations in a random assignment). Hastad and Venkatesh obtained an algorithm which approximates this value up to a factor of O(radicN). We obtain a O(radic(n/logn)) 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.