Tensor network complexity of multilinear maps

We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. This model turns out to be strong enough to capture the current best algorithms for a variety of problems, e.g., O(n ) time matrix multiplication, O(n logn) time discrete Fourier transform, O(n ) time 3t-clique counting and O∗(2n) time for computing the permanent of a matrix. For counting homomorphisms of a general pattern graph P into a host graph on n vertices we obtain an upper bound of O(n ) bw(P ) where bw(P ) is the branchwidth of P . This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of P . While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various linear maps, including: (a) an Ω(n ) time lower bound for counting homomorphisms from P to an n-vertex graph, matching the upper bound if ω = 2. In particular for P a v-clique this yields an Ω(nd2v/3e) time lower bound for counting v-cliques. (b) an Ω(2) time lower bound for the permanent of an n× n matrix. (c) an Ω(n) time lower bound for the trilinear map Dijk = ∑ lAijlBiklCjkl generalizing matrix multiplication, taking three n× n× n tensors A,B,C and producing an n× n× n tensor D, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-3-CSP problem. ar X iv :1 71 2. 09 63 0v 2 [ cs .C C ] 7 A pr 2 01 8