CSC 2429 – Approaches to the P versus NP Question Lecture

Warm-up. It is an easy exercise to prove that for all A we have RigA(r) ≤ (n− r)2. This is nearly tight by a direct counting argument: Valiant [Val77, Theorem 6.4] proves that RigA(r) is at least about (n− r)2/ log n for a random A (and all fields F). This shows that rigid matrices exist in great numbers—however, as always, the challenge is to exhibit an explicit such matrix. Currently, the best explicit construction over finite fields is due to Friedman [Fri93] (see also Jukna [Juk12, §13.8]) that achieves rigidity of about ≥ n2/r log(n/r). Here we restrict to giving a simpler construction1 due to Midrijānis [Mid05] that comes close to this: Suppose n is divisible by 2r and construct A as the block matrix having n2/(2r)2 blocks, each containing a copy of the 2r× 2r identity matrix. Then RigA(r) ≥ n2/4r since in order to reduce the rank of A to at most r we need to reduce the rank of each 2r × 2r identity block to at most r and this requires modifying at least r entries in each block.