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 = {aijk}n i,j,k=1 such that for all i, j, k ∈ {1, . . . , n} we have ai jk = aik j = ak ji = ajik = aki j = ajki and aiik = aijj = aiji = 0, computes a number Alg(A) which satisfies with probability at least 1 2 , Ω   log n n   · max x∈{−1,1}n n i,j,k=1 ai jk xixjxk ≤ Alg(A) ≤ max x∈{−1,1}n n i, j,k=1 ai jk xixjxk. On the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [22] that under the assumption NP DTIME n(log n)O(1) , for every ε > 0 there is no algorithm that approximates maxx∈{−1,1}n n i,j,k=1 aijk xixjxk within a factor of 2(log n)1−ε 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 Rn with respect to the L1 norm. We show that it is possible to do so up to a multiplicative error of O n log n , while no randomized...

• Type: Conference
• Year: 2007
• Where: FOCS
• Authors: Subhash Khot, Assaf Naor
• Keywords: Algorithm, FOCS 2007, Polynomial Time Algorithm, Randomized Polynomial Time, Theoretical Computer Science