Topological value iteration (TVI) is an effective algorithm for solving Markov decision processes (MDPs) optimally, which 1) divides an MDP into strongly-connected components, and 2) solves these components sequentially. Yet, TVI’s usefulness tends to degrade if an MDP has large components, because the cost of the division process isn’t offset by gains during solution. This paper presents a new algorithm to solve MDPs optimally, focused topological value iteration (FTVI). FTVI addresses TVI’s limitations by restricting its attention to connected components that are relevant for solving the MDP. Specifically, FTVI uses a small amount of heuristic search to eliminate provably sub-optimal actions; this pruning allows FTVI to find smaller connected components, thus running faster. We demonstrate that our new algorithm outperforms TVI by an order of magnitude, averaged across several domains. Surprisingly, FTVI also significantly outperforms popular ‘heuristically-informed’ MDP algorithms such as LAO*, LRTDP, and BRTDP in many domains, sometimes by as much as two orders of magnitude. Finally, we characterize the type of domains where FTVI excels — suggesting a way to an informed choice of solver.