Simulation Theorems via Pseudo-random Properties
Arkadev Chattopadhyay, Michal Koucký, Bruno Loff, Sagnik Mukhopadhyay · Computational Complexity · 2019
We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999 ), to any gadget which satisfies a certain hitting property. We prove that inner product and gap-Hamming satisfy this property, and as a corollary, we obtain a deterministic simulation theorem for these gadgets, where the gadget’s input size is logarithmic in the input size of the outer function. This yields the first deterministic simulation theorem with a logarithmic gadget size, answering an open question posed by Göös, Pitassi & Watson (in: Proceedings of the 56th FOCS, 2015 ). Our result also implies the previous results for the indexing gadget, with better parameters than was previously known. Moreover, a simulation theorem with logarithmic-sized gadget implies a quadratic separation in the deterministic communication complexity and the logarithm of the 1-partition number, no matter how high the 1-partition number is with respect to the input size—something which is not achievable by previous results of Göös, Pitassi & Watson ( 2015 ).