Georgia Institute of Technology College of Sciences

Spielman and Teng (2004) showed that linear systems in Laplacian matrices can be solved in nearly linear time. Since then, a major research goal has been to develop fast solvers for linear systems in other classes of matrices. Recently, this has led to fast solvers for directed Laplacians (CKPPRSV'17) and connection Laplacians (KLPSS'16).Connection Laplacians are a special case of PSD-Graph-Structured Block Matrices (PGSBMs), block matrices whose non-zero structure correspond to a graph, and which additionally can be expressed as a sum of positive semi-definite matrices each corresponding to an edge in the graph. A major open question is whether fast solvers can be obtained for all PGSBMs (Spielman, 2016). Fast solvers for Connection Laplacians provided some hope for this. Other important families of matrices in the PGSBM class include truss stiffness matrices, which have many applications in engineering, and multi-commodity Laplacians, which arise when solving multi-commodity flow problems. In this talk, we show that multi-commodity and truss linear systems are unlikely to be solvable in nearly linear time, unless general linear systems (with integral coefficients) can be solved in nearly linear time. Joint work with Rasmus Kyng.