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The multiple longest common subsequence (MLCS) problem, which is related to the measurement of sequence similarity, is one of the fundamental problems in many fields. As an NP-hard problem, finding a good approximate solution within a reasonable time is important for solving large-size problems in practice. In this paper, we present a new progressive algorithm, Pro-MLCS, based on the dominant point approach. Pro-MLCS can find an approximate solution quickly and then progressively generate better solutions until obtaining the optimal one. Pro-MLCS employs three new techniques: 1) a new heuristic function for prioritizing candidate points; 2) a novel $(d)$-index-tree data structure for efficient computation of dominant points; and 3) a new pruning method using an upper bound function and approximate solutions. Experimental results show that Pro-MLCS can obtain the first approximate solution almost instantly and needs only a very small fraction, e.g., 3 percent, of the entire running time to get the optimal solution. Compared to existing state-of-the-art algorithms, Pro-MLCS can find better solutions in much shorter time, one to two orders of magnitude faster. In addition, two parallel versions of Pro-MLCS are developed: DPro-MLCS for distributed memory architecture and DSDPro-MLCS for hierarchical distributed shared memory architecture. Both parallel algorithms can efficiently utilize parallel computing resources and achieve nearly linear speedups. They also have a desirable progressiveness property—finding better solutions in shorter time when given more hardware resources.