Unifying Known Lower Bounds via Geometric Complexity Theory

Unifying and generalizing known lower bounds via geometric complexity theory

We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan–Wigderson), the results of Razborov and Smolensky on AC[p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Ba...

Strong computational lower bounds via parameterized complexity

We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O(nk) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely coll...

On Superlinear Lower Bounds in Complexity Theory

This paper first surveys the near-total lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the model allows non-local communication with memory at unit cost . We study a model that imposes a “fair...

Lower Bounds on the State Complexity of Geometric Goppa Codes

Some Geometric Lower Bounds

Dobkin and Lipton introduced the connected components argument to prove lower bounds in the linear decision tree model for membership problems, for example the element uniqueness problem. In this paper we apply the same idea to obtain lower bound statements for a variety of problems, each having the avor of element uniqueness. In fact one of these problems is a parametric version of element uni...