On the Complexity of Submap Isomorphism

Abstract

Abstract. Generalized maps describe the subdivision of objects in cells, and incidence and adjacency relations between cells, and they are widely used to model 2D and 3D images. Recently, we have defined submap isomorphism, which involves deciding if a copy of a pattern map may be found in a target map, and we have described a polynomial time algorithm for solving this problem when the pattern map is connected. In this paper, we show that submap isomorphism becomes NP-complete when the pattern map is not connected, by reducing the NP-complete problem Planar-4 3-SAT to it. MotivationsCombinatorial maps and generalized maps [1] are very nice data structures to model the topology of nD objects subdivided in cells (e.g., 0D vertices, 1D edges, 2D faces, 3D volumes, . . . ) by means of incidence and adjacency relationships between these cells. In 2D, maps may be used to model the topology of an embedding of a planar graph in a plane. In particular, these models are very well suited for scene modeling [2], and for 2D and 3D image segmentation [3].In [4], we have defined a basic tool for comparing 2D maps, i.e., submap isomorphism (which involves deciding if a copy of a pattern map may be found in a target map), and we have proposed an efficient polynomial-time algorithm for solving this problem when the pattern map is connected. This work has been generalized to nD maps in [5]. The subisomorphism defined in [5] is based on induced submap relations, such that submaps are obtained by removing some darts and all their seams, just like induced subgraphs are obtained by removing some vertices and all their incident edges. In [6], we have introduced a new kind of submap relation, called partial submap: partial submaps are obtained by removing not only some darts (and all their seams), but also some other seams, just like partial subgraphs are obtained by removing not only some vertices (and their incident edges), but also some other edges. The polynomial time algorithm described in [5] for solving the induced submap isomorphism problem may be extended to the partial case in a very straightforward way. However, this algorithm still assumes that the pattern map is connected. In this paper, we show that the submap isomorphism problem becomes NP-complete when the pattern map is not connected, both for partial and induced submaps.