To better explore the robustness against cascading failures on complex networks, a local congestion-aware routing strategy with a tunable parameter is proposed, and a cascading model is constructed. According to new measures to the robustness of a network against cascading failures, cascading behaviors on complex heterogeneous networks triggered by deliberate attacks are investigated. Simulation results show that, under the local routing, the existence of an optimal value of the routing parameter makes the networks achieve a maximum level of robustness and, the robustness of the networks has a...