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Graph-based systems are models wherein the nodes represent the components and the edges represent the fault propagation between the components. For critical systems, some components are equipped with smart sensors for on-board system health management. When an abnormal situation occurs, alarms will be triggered from these sensors. This paper considers the problem of identifying the set of potential failure sources from the set of ringing alarms in graph-based systems. However, the computational complexity of solving the optimal multiple fault diagnosis (MFD) problem is exponential. Based on Lagrangian relaxation and subgradient optimization, we present a heuristic algorithm to find approximately the most likely candidate fault set. A computationally cheaper heuristic algorithm - primal heuristic - has also been applied to the problem so that real-time MFD in systems with several thousand failure sources becomes feasible in a fraction of a second. This paper also considers systems with asymmetric and multivalued alarms (tests).