Tensor Network Complexity of Multilinear Maps

Per Austrin, Petteri Kaski, Kaie Kubjas, Michael Wagner
2018 Innovations in Theoretical Computer Science  
We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as O(n ω+ ) time matrix multiplication, and in addition many other algorithms such as O(n log n) time discrete Fourier transform and O * (2 n ) time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the
more » ... t known O(n (ω+ )t ) time algorithms for counting 3t-cliques can be implemented with tensor networks, even though the underlying tensor has border rank n 3t for all t ≥ 2. For counting homomorphisms of a general pattern graph P into a host graph on n vertices we obtain an upper bound of O(n (ω+ ) bw(P )/2 ) 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 multilinear maps, including: (a) an Ω(n bw(P ) ) 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(n 2v/3 ) time lower bound for counting v-cliques, and for P a k-uniform v-hyperclique we obtain an Ω(n v ) time lower bound for k ≥ 3, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-3-CSP problem. (b) an Ω(2 0.918n ) time lower bound for the permanent of an n × n matrix. ACM Subject Classification Theory of computation → Models of computation, Theory of computation → Computational complexity and cryptography, Theory of computation → Design and analysis of algorithms
doi:10.4230/lipics.itcs.2019.7 dblp:conf/innovations/AustrinKK19 fatcat:t4xmqd4rxfb27lgqzslpxtduli