Multi-constrained quality of service (QoS) routing finds a route in the network that satisfies multiple independent quality of service constraints. This problem is NP-hard and a number of heuristic algorithms have been proposed to solve the problem. This paper studies two heuristics, the limited granularity heuristic and the limited path heuristic, for solving general k-constrained problems. Analytical and simulation studies are conducted to compare the time/space requirements of the heuristics and the effectiveness of the heuristics in finding the paths that satisfy the QoS constraints. We prove analytically that for an N nodes and E edges network with k (a small constant) independent QoS constraints, the limited granularity heuristic must maintain a table of size O(|N|/sup k-1/) in each node to be effective, which results in a time complexity of O(|N|/sup k/|E|). We also prove that the limited path heuristic can achieve very high performance by maintaining O(|N|/sup 2/lg(|N|)) entries in each node, which indicates that the performance of the limited path heuristic is not sensitive to the number of constraints. We conclude that although both the limited granularity heuristic and the limited path heuristic can efficiently solve 2-constrained QoS routing problems, the limited path heuristic is superior to the limited granularity heuristic in solving k-constrained QoS routing problems when k>3. Our simulation study further confirms this conclusion.