This paper studies the multi-cascade influence maximization problem, which explores strategies for launching one information cascade in a social network with multiple existing cascades. With natural extensions to the classic models, we first propose the independent multi-cascade model where the diffusion process is governed by the so-called activation function. We show that the proposed model is sufficiently flexible as it generalizes most of the existing cascade-based models. We then study the multi-cascade influence maximization problem under the designed model and provide approximation hardness under common complexity assumptions, namely Exponential Time Hypothesis and $NP \subseteq DTIME(n^{\poly \log n})$. Given the hardness results, we build a framework for designing heuristic seed selection algorithms with a testable data-dependent approximation ratio. The designed algorithm leverages upper and lower bounds, which reveal the key combinatorial structure behind the multi-cascade influence maximization problem. The performance of the framework is theoretically analyzed and practically evaluated through extensive simulations. The superiority of the proposed solution is supported by encouraging experimental results, in terms of effectiveness and efficiency.