Mapping instances to the Linked Open Data (LOD) cloud plays an important role for enriching information of instances, since the LOD cloud contains abundant amounts of interlinked instances describing the instances. Consequently, many techniques have been introduced for mapping instances to a LOD data set; however, most of them merely focus on tackling with the problem of heterogeneity. Unfortunately, the problem of the large number of LOD data sets has yet to be addressed. Owing to the number of LOD data sets, mapping an instance to a LOD data set is not sufficient because an identical instance might not exist in that data set. In this article, we therefore introduce a heuristic expansion based framework for mapping instances to LOD data sets. The key idea of the framework is to gradually expand the search space from one data set to another data set in order to discover identical instances. In experiments, the framework could successfully map instances to the LOD data sets by increasing the coverage to 90.36%. Experimental results also indicate that the heuristic function in the framework could efficiently limit the expansion space to a reasonable space. Based upon the limited expansion space, the framework could effectively reduce the number of candidate pairs to 9.73% of the baseline without affecting any performances.