We introduce a novel framework for the stability analysis of discrete-time linear switching systems with switching sequences constrained by an automaton. The key element of the framework is the algebraic concept of multinorm, which associates a different norm per node of the automaton, and allows to exactly characterize stability. Building upon this tool, we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radius (rho) over cap, that is the exponential growth rate of a switching system with constrained switching sequences. More precisely, given a relative accuracy r > 0, the algorithms compute an estimate of (rho) over cap within the range [(rho) over cap, (1+r) (rho) over cap]. These algorithms amount to solve a well defined convex optimization program with known time-complexity, and whose size depends on the desired relative accuracy r > 0. (C) 2016 Elsevier Ltd. All rights reserved.