Worst-case analysis of restarted primal-dual hybrid gradient on totally unimodular linear programs

Abstract: We analyze restarted PDHG on totally unimodular linear programs. In particular, we show that restarted PDHG finds an $\epsilon$-optimal solution in $O( H m_1^{2.5} \sqrt{\textbf{nnz}(A)} \log(H m_2 /\epsilon) )$ matrix-vector multiplies where $m_1$ is the number of constraints, $m_2$ the number of variables, $\textbf{nnz}(A)$ is the number of nonzeros in the constraint matrix, $H$ is the largest absolute coefficient in the right hand side or objective vector, and $\epsilon$ is the distance to optimality of the outputted solution.