We discuss the connection between computational social choice (comsoc) and computational complexity. We stress the work so far on, and urge continued focus on, two less-recognized aspects of this connection. Firstly, this is very much a two-way street: Everyone knows complexity classification is used in comsoc, but we also highlight benefits to complexity that have arisen from its use in comsoc...

Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a typical system state or history starting from simple initial conditions. The properties of depth are discussed and it is compared with other complexity measures. ...

2 Existential Lower Bounds Theorem 1 (Lupanov, Shannon) For each ε > 0 and for each sufficiently large n, ∃f : {0, 1}n → {0, 1} with circuit complexity (over the full basis B2) at least (1− ε)2n n . Proof: We will prove the required result using a counting argument. In particular, we will show that the number of functions mapping {0, 1}n to {0, 1} is much larger than the number of ’distinct’ ci...

Computational Complexity Theory is the field that studies the inherent costs of algorithms for solving mathematical problems. Its major goal is to identify the limits of what is efficiently computable in natural computational models. Computational complexity ranges from quantum computing to determining the minimum size of circuits that compute basic mathematical functions to the foundations of ...

The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix, up to polynomial factors. Despite much research, we still know very little about this conjecture. Recently, there has been renewed interest in this conjecture...