A basic problem in the quality-of-service (QoS) analysis of multiagent distributed systems is to find optimal routes for the mobile agents that incrementally fuse the data as they visit hosts in the distributed system. The system is modeled as a directed acyclic graph in which the nodes represent hosts and the edges represent links between them. Each edge is assigned a cost (or benefit) and weights that represent link delay, reliability, or other QoS parameters. The agent scheduling problem is viewed as a constrained routing problem in which a maximum-benefit (or minimum-cost) route connecting the source and the destination subject to QoS constraints is to be found. We study approximation algorithms called 'fully polynomial time approximation schemes' (FPTAS) for solving the problem. We suggest an accelerating technique that improves known FPTAS, e.g., Hassin's (1992); Camponogara & Shima's (2010); and Elalouf et al. (2011) algorithms, and present new FPTASs.