On the complexity of Submap Isomorphism

Transcription

1 On the compleit of Sbmap Isomorphism Christine Solnon 1,2, Gillame Damiand 1,2, Colin de la Higera 3, and Jean-Christophe Janodet 4 1 INSA de Lon, LIRIS, UMR 5205 CNRS, Villerbanne, France 2 Université de Lon, France 3 LINA, UMR CNRS 6241, Université de Nantes, France 4 IBISC, Université d Evr, France Abstract. Generalied maps describe the sbdivision of objects in cells, and incidence and adjacenc relations beteen cells, and the are idel sed to model 2D and 3D images. Recentl, e have defined sbmap isomorphism, hich involves deciding if a cop of a pattern map ma be fond in a target map, and e have described a polnomial time algorithm for solving this problem hen the pattern map is connected. In this paper, e sho that sbmap isomorphism becomes NP-complete hen the pattern map is not connected, b redcing the NP-complete problem Planar-4 3-SAT to it. 1 Motivations Combinatorial maps and generalied maps [1] are ver nice data strctres to model the topolog of nd objects sbdivided in cells (e.g., 0D vertices, 1D edges, 2D faces, 3D volmes,... ) b means of incidence and adjacenc relationships beteen these cells. In 2D, maps ma be sed to model the topolog of an embedding of a planar graph in a plane. In particlar, these models are ver ell sited for scene modeling [2], and for 2D and 3D image segmentation [3]. In [4], e have defined a basic tool for comparing 2D maps, i.e., sbmap isomorphism (hich involves deciding if a cop of a pattern map ma be fond in a target map), and e have proposed an efficient polnomial-time algorithm for solving this problem hen the pattern map is connected. This ork has been generalied to nd maps in [5]. The sbisomorphism defined in [5] is based on indced sbmap relations, sch that sbmaps are obtained b removing some darts and all their seams, jst like indced sbgraphs are obtained b removing some vertices and all their incident edges. In [6], e have introdced a ne kind of sbmap relation, called partial sbmap: partial sbmaps are obtained b removing not onl some darts (and all their seams), bt also some other seams, jst like partial sbgraphs are obtained b removing not onl some vertices (and their incident edges), bt also some other edges. The polnomial time algorithm described in [5] for solving the indced sbmap isomorphism problem ma be etended to the partial case in a ver straightforard a. Hoever, this algorithm still assmes that the pattern map is connected. In this paper, e sho that the sbmap isomorphism problem becomes NP-complete hen the pattern map is not connected, both for partial and indced sbmaps.

2 v1 v5 f1 v2 a b c d e f g h i j k l m n α 0 h c b e d g f a j i l k n m f2 v3 α 1 b a d c f e h g n k j m l i v4 α 2 a b c i j f g h d e k l m n (a) (b) (c) Fig. 1. (a) A plane graph. (b) The corresponding 2G-map. (c) Its graphical representation: darts are represented b segments labeled ith letters, consective darts separated ith a little segment are 0-sen (e.g., α 0(b) = c and α 0(c) = b), consective darts separated ith a dot are 1-sen (e.g., α 1(a) = b and α 1(b) = a), parallel adjacent darts are 2-sen (e.g., α 2(d) = i and α 2(i) = d). Otline of the paper. In Section 2, e recall definitions related to generalied maps. In Section 3, e define the sbmap isomorphism problem and recall some compleit reslts abot this problem. In Section 4, e describe the planar-4 3-SAT problem, hich is NP-complete. In Section 5, e describe a polnomialtime redction of planar-4 3-SAT to sbmap isomorphism, ths shoing that sbmap isomorphism is NP-complete. 2 Recalls and basic definitions on generalied maps In this ork e consider generalied maps, hich are more general than combinatorial maps, and e refer the reader to [1] for more details. Definition 1. (ng-map) Let n 0. An n-dimensional generalied map (or ngmap) is defined b a tple G = (D, α 0,..., α n ) sch that (i) D is a finite set of darts; (ii) i {0,..., n}, α i is an involtion 5 on D; and (iii) i, j {0,..., n} sch that i + 2 j, α i α j is an involtion. 2G-maps ma be sed to model the embedding of a planar graph into a plane. For eample, Fig. 1 displas a plane graph and the corresponding 2G-map. We sa that a dart d is i-sen ith a dart d henever d = α i (d ) and d d, hereas it is i-free henever d = α i (d). A seam is a tple (d, i, d ) sch that d is i-sen to d. For eample, (a, 0, h) is a seam of the map displaed in Fig. 1 becase α 0 (a) = h. Definition 2. (seams of a set of darts in an ng-map) Let G = (D, α 0,..., α n ) be an ng-map and E D be a set of darts. The set of seams associated ith E in G is: seams G (E) = {(d, i, α i (d)) d E, i {0,..., n}, α i (d) E, α i (d) d}. A map is connected if an pair of darts is connected b a path of sen darts. Definition 3 (Connected map). A generalied map G = (D, α 0,..., α n ) is connected if d D, d D, there eists a path beteen d and d, i.e., a seqence of darts (d 1,..., d k ) sch that d 1 = d, d k = d and i {1,..., k 1}, j i {0,..., n}, d i+1 = α ji (d i ). 5 An involtion f on D is a bijective mapping from D to D sch that f = f 1.

3 Map isomorphism [1] allos s decide of the eqivalence of to maps. Definition 4. (ng-map isomorphism [1]) To ng-maps G = (D, α 0,..., α n ) and G = (D, α 0,..., α n) are isomorphic if there eists a bijection f : D D, sch that d D, i [0, n], f(α i (d)) = α i (f(d)). In [4], indced sbmaps have been defined: G is an indced sbmap of G if G preserves all seams of G, i.e, for ever cople of darts (d 1, d 2 ) of G, d 1 is i-sen to d 2 in G if and onl if d 1 is i-sen to d 2 in G. Definition 5. (indced sbmap) A map G = (D, α 0,..., α n) is an indced sbmap of G = (D, α 0,..., α n ) if D D and seams G (D ) = seams G (D ). In [6], e have introdced another sbmap relation, called partial sbmap b analog ith eisting ork on graphs. Indeed, indced sbgraphs are obtained b removing some nodes (and all their incident edges) hereas partial sbgraphs are obtained b removing not onl some nodes (and all their incident edges) bt also some edges. In or map contet, partial sbmaps are obtained b removing not onl some darts (and all their seams) bt also some other seams. Definition 6. (partial sbmap) A map G = (D, α 0,..., α n) is a partial sbmap of G = (D, α 0,..., α n ) if D D and seams G (D ) seams G (D ). 3 The sbmap isomorphism problem The sbmap isomorphism problem involves deciding if a pattern map is isomorphic to a sbmap of a target map, and it is formall defined as follos: Problem: Partial (resp. indced) sbmap isomorphism Instance: A triple (n, G, G ) sch that n > 0, and G and G are ng-maps. Qestion: Does there eist a partial (resp. indced) sbmap of G hich is isomorphic to G? We note G p G (resp. G i G ) hen the anser is es. Note that G i G G p G. Fig. 2 displas eamples of sbmap isomorphisms. The compleit of the sbmap isomorphism problem depends on the connectedness of the pattern map. For eample, the map G 1 of Fig. 2 is not connected, and is composed of to connected components, hereas the maps G 2, G 3 and G 4 are connected. In [5], e have described a polnomial-time algorithm hich solves the sbmap isomorphism problem hen the pattern map G is connected. When the pattern map G is not connected, e ma se this algorithm to search for all occrrences of each connected component of G in the target map G. Let s consider, for eample, the map G 1 of Fig. 2. Its left hand side component occrs once in G 2 and tice in G 3 and G 4, and its right hand side component occrs tice in G 2, G 3 and G 4 (as it is atomorphic). To solve the sbmap isomorphism problem from these occrrence lists, e have to solve the folloing combinatorial problem: Can e select one occrrence in G of each connected component of G so that the selected occrrences do not overlap in G? Theorem 1 claims that this combinatorial problem is NP-complete.

4 a c b d G 1 G 2 G 3 G 4 Fig. 2. Sbmap isomorphism eamples. G 1 is not isomorphic to a sbmap of G 2 (i.e., G 1 p G 2 and G 1 i G 2), thogh each connected component of G 1 is a sbmap of G 2. G 1 is isomorphic to a partial sbmap of G 3, bt not to an indced one (i.e., G 1 p G 3 and G 1 i G 3), becase the seams (a, 2, c) and (b, 2, d) of G 3 are not preserved in G 1. G 1 is isomorphic to an indced sbmap of G 4 and, therefore, it is also isomorphic to a partial sbmap of G 4 (i.e., G 1 p G 4 and G 1 i G 4). Theorem 1. The partial (resp. indced) sbmap isomorphism problem is N P- complete. The problem triviall belongs to NP since one can check that a given partial (resp. indced) sbmap of the target map G is isomorphic to the pattern map G in polnomial time. We ma se for eample the polnomial algorithm of [5], hich has been defined for non connected maps. To prove that it is NP-complete, e sho in Section 5 that Planar-4 3-SAT, hich is knon to be NP-complete, ma be redced to it in polnomial time. 4 Planar-4 3-SAT Planar-4 3-SAT is a special case of the SAT problem, hich involves deciding if there eists a trth assignment for a set X of variables sch that a boolean formla F over X is satisfied [7]. We assme that F is in Conjnctive Normal Form (CNF), i.e., it is a conjnction of clases sch that each clase is a disjnction of literals hich are either variables of X or negations of variables of X. The formla-graph associated ith a CNF formla F over a set of variables X is the bipartite graph G X,F = (V, E) sch that V associates a verte ith ever variable i X and ever clase c j of F, and E associates an edge ( i, c j ) ith ever variable/clase cople sch that variable i occrs in clase c j. The planar-4 3-SAT problem is formall defined as follos. Problem: Planar-4 3-SAT Instance: A cople (X, F ) sch that X is a set of boolean variables and F is a CNF formla over X sch that (i) ever clase of F is a disjnction of 3 literals, (ii) the formla-graph G X,F is planar, and (iii) the degree of ever verte of G X,F is bonded b 4 (i.e., each variable occrs in at most 4 different clases) Qestion: Does there eist a trth assignment for X hich satisfies F? Planar-4 3-SAT has been shon to be NP-complete in [8]. Fig. 3 displas an instance of Planar-4 3-SAT and its associated formla-graph.

5 X = {,,,, } F = ( ) ( ) (ȳ ) ( ) ( ū) C1 C2 C3 C4 C5 Fig. 3. An instance of Planar-4 3-SAT and its associated formla graph (clases correspond to circles, and variables to sqares). To redce an instance (X, F ) of Planar-4 3-SAT to an instance (n, G, G ) of sbmap isomorphism, e first perform a preprocessing: We iterativel eliminate from (X, F ) ever variable i X hich occrs in onl one clase c j of F (those hose degree is eqal to 1 in the formla-graph), set i to the trth vale hich satisfies c j, and eliminate c j from F, ntil either X and F become empt (ths shoing that the anser is triviall es), or all variables in X occr in 2, 3, or 4 clases of F. 5 Redction of Planar-4 3-SAT to sbmap isomorphism Let s first sho that planar-4 3-SAT can be redced to indced sbmap isomorphism in polnomial time: The partial case ill be stdied at the end of this section. We consider an instance (X, F ) of planar-4 3-SAT and e sho ho to bild an instance (n, G, G ) sch that G i G iff the anser to (X, F ) is es. We consider 2G-maps, so that n = 2, and the 2G-maps G and G are constrcted b assembling bilding blocks hich are 2G-maps. Fig. 4 displas bilding blocks associated ith variables: For each variable i X sch that the degree of i in the formla-graph G X,F is eqal to k ith 2 k 4 (as the preprocessing step has removed an variable hose degree is eqal to 1), e bild to variable patterns V k and V k hich ill respectivel occr in G and G. Each variable pattern V k (resp. V k) looks like a floer hose core is a 2k-edge face and hich have 2k petals (resp. k petals), here each petal is a 6-edge face. For each petal in each variable pattern V k, the edge opposite to the core of the floer is a connecting edge hich ma be 2-sen ith clase patterns to define G. For each clase, e bild to clase patterns C and C hich ill respectivel occr in G and G. The clase pattern C is composed of a 3-edge central face hich has 3 adjacent 4-edge faces, hereas the clase pattern C is composed of a 3-edge face hich has 1 adjacent 4-edge face, as displaed belo: Clase pattern C : Clase pattern C:

6 Patterns associated ith variables in G : Patterns associated ith variables in G: V2 : V4 : V2: V4: V3 : V3: Fig. 4. Variable patterns sed as bilding blocks to define G and G. Connecting edges in G are displaed in bold. Edges of C displaed in bold are connecting edges hich are 2-sen ith variable patterns to define G. Definition of the 2G-map G. For each variable i X sch that the degree of i in the formla-graph G X,F is eqal to k, G contains an occrrence of the variable pattern V k. Each petal of this occrrence of V k is alternativel labeled ith i and i. For each clase c j of F, G contains an occrrence of the clase pattern C. Each 4-edge face of this occrrence of C is labeled ith a different literal of c j. Variable and clase patterns are 2-sen to define a connected 2Gmap: ever connecting edge of each clase pattern is 2-sen ith a different connecting edge of a variable pattern sch that the to faces hich become adjacent b this seam are labeled ith the same literal. We can easil check that this 2G-map can alas be bilt in polnomial time as the formla-graph G X,F is planar, and there eist polnomial-time algorithms for embedding a planar graph in a plane [9]: We can se the same embedding for constrcting G. Fig. 5 displas the 2G-map associated ith the formla displaed in Fig. 3. Definition of the 2G-map G. If the SAT instance has n variables and c clases, then G is composed of n + c different components: a component V k is associated ith ever variable i X, here k is the degree of i in G X,F ; a component C is associated ith ever clase. For eample, the 2G-map G associated ith the formla displaed in Fig. 3 contains 10 components: 3 occrrences of V 3, 1 occrrence of V 4, 1 occrrence of V 2, and 5 occrrences of C. Proof of (G i G ) ( trth assignment of X hich satisfies F ). Let s first assme that there eists an indced sbmap G of G hich is isomorphic to G, and let s sho that there eists a trth assignment of X hich satisfies F. If G is isomorphic to G then, according to Def. 4, there eists a bijection f hich matches darts of G ith darts of G and hich preserves all seams. B etension, e sa that f matches faces of G ith faces of G. As e consider

7 C1 C2 C3 C4 C5 Fig. 5. 2G-map G associated ith the SAT instance displaed in Fig. 3. Note that this map contains holes (corresponding to hite parts in the figre): each dart d adjacent to these holes is 2-free so that α 2(d) = d. indced sbmap isomorphism, to faces of G hich belong to to different connected components cannot be matched b f ith faces hich are 2-sen in G (according to Def. 5). Fig. 6 displas an eample of sch a soltion for the instance (2, G, G ) of the indced sbmap isomorphism problem associated ith the instance (X, F ) of Planar-4 3-SAT displaed in Fig. 3. G contains c occrrences of C, here c is the nmber of clases of F. Each occrrence of C has a 3-edge face adjacent to a 4-edge face. These faces can onl be matched ith faces hich belong to occrrences of C in G as 3-edge faces in G onl come from C patterns. As there are c occrrences of C in G, each occrrence of C in G is matched ith a different occrrence of C in G. For the same reasons, each occrrence of a variable pattern V k in G is matched ith a different occrrence of a variable pattern V k in G : Petal and core faces in G can onl be matched ith petal and core faces in G, and an occrrence of V i cannot be matched ith faces of an occrrence of V j if i j. For each variable pattern V k, the label of the petals of V k hich are not matched ith petals of variable patterns of G gives the trth assignment for the corresponding variable. For each clase pattern C, the label of the 4-edge face of C hich is matched ith a 4-edge face of C corresponds to a literal hich satisfies the clase associated ith C. As to faces of G hich belong to to different connected components cannot be matched b f ith faces hich are 2-sen in G, e ensre that hen a 4-edge face of a clase pattern is matched, then the adjacent petal is not

8 C1 C2 C3 C4 C5 Fig. 6. Soltion of the indced sbmap isomorphism instance (2, G, G ) associated ith the Planar-4 3-SAT instance displaed in Fig. 3. The indced sbmap of G hich is isomorphic to G is displaed in dark gre. Note that to different components of this sbmap cannot be 2-sen in G as e consider indced sbmap isomorphism. matched, i.e., hen a clase is satisfied b a literal l, then no other clase can be satisfied b the negation of this literal so that the trth assignment dedced from the floer matching actall satisfies all clases of F. For eample, the trth assignment corresponding to the soltion displaed in Fig. 6 is {,,,, }. Proof of ( trth assignment of X hich satisfies F ) (G i G ). Let s assme that there eists a trth assignment of X hich satisfies F and let s sho that there eists an indced sbmap G of G hich is isomorphic to G, i.e., that there eists a dart matching hich preserves all seams of G. For each variable pattern V k in G associated ith a variable i, e match the darts of the core face ith the darts of the core face of the variable pattern associated ith i in G and e match the darts of the k 6-edge petals of V k ith the darts of the k 6-edge petals hich are labeled ith the negation of the trth vale of i. For each clase pattern C in G associated ith a clase c j, e match the darts of the 3-edge face of C ith the darts of the 3-edge face of the clase pattern associated ith c j in G and e match the darts of the 4-edge face of C ith the darts of one of the three 4-edge faces: We choose a 4-edge face hich is labeled ith a literal hich is satisfied b the trth assignment (this 4-edge face cannot be 2-sen ith a matched 6-edge petal). Proof for the partial case. Let s no consider the partial case: We consider an instance (X, F ) of planar-4 3-SAT and e sho ho to bild an instance

9 C1 C2 C3 C4 C5 Fig. 7. Soltion of the partial sbmap isomorphism instance (2, G, G ) associated ith the Planar-4 3-SAT instance displaed in Fig. 3. The partial sbmap of G hich is isomorphic to G is displaed in dark gre. (n, G, G ) sch that G p G iff the anser to (X, F ) is es. The proof is similar to the indced case. The difference beteen the indced and the partial cases is that, hen considering indced sbmap isomorphism, to faces hich belong to to different components in G cannot be matched ith faces of G hich are 2-sen hereas, hen considering partial sbmap isomorphism, to faces hich belong to to different components in G ma be matched ith faces of G hich are 2-sen. Therefore, e modif the clase pattern C so that the 4-edge face is adjacent to a 3-edge face, on one side, and to a 6-edge face on the opposite side, as displaed belo: These 6-edge faces can onl be matched ith petals.the label of the petal hich is matched ith the 6-edge face of a clase pattern corresponds to the literal hich satisfies the clase. Fig. 7 displas an eample of soltion for partial sbmap isomorphism. 6 Conclsion We have shon that sbmap isomorphism is NP-complete hen the pattern map G is not connected. This implies that there does not eist a polnomialtime algorithm for this problem, nless P=NP. The practical tractabilit of this

10 problem actall depends on the nmber of different connected components of G. Indeed, if G contains k different connected components, e can se the polnomial-time algorithm of [5] to search for all occrrences of each component of G in the target map G. Let m be the maimm nmber of occrrences of a connected component of G in G (m is bonded b the nmber of darts of G ). The nmber of candidate soltions to eplore is bonded b m k so that the problem remains tractable if k is small enogh. A conseqence of or NP-completeness proof is that the maimm common sbmap problem introdced in [10] is NP-hard in the general case, i.e., if the common sbmap is not necessaril connected (as searching for a common sbmap is more general than deciding of sbmap isomorphism). Hoever, the compleit of the maimm common sbmap problem in the particlar case here the common sbmap mst be connected is still an open qestion: We haven t fond a polnomial-time algorithm for solving this problem, neither have e fond a polnomial-time redction from a knon NP-complete problem to this problem. Hence, frther ork ill mainl concern the anser to this qestion. Acknoledgments. The athors old like to thank Daniel Goncalves (Universit of Montpellier) for his pointer to problem Planar-3SAT, and fritfl remarks. References 1. Lienhardt, P.: N-dimensional generalied combinatorial maps and celllar qasimanifolds. Comptational Geometr and Applications 4(3) (1994) Fradin, D., Menevea, D., Lienhardt, P.: A hierarchical topolog-based model for handling comple indoor scenes. Compter Graphics Form 25(2) (Jne 2006) Braqelaire, J.P., Brn, L.: Image segmentation ith topological maps and interpiel representation. Visal Commnication and Image representation 9(1) (1998) Damiand, G., De La Higera, C., Janodet, J.C., Samel, E., Solnon, C.: Polnomial algorithm for sbmap isomorphism: Application to searching patterns in images. In: GbR. Volme 5534 of LNCS., Springer (2009) Damiand, G., Solnon, C., de la Higera, C., Janodet, J.C., Samel, E.: Polnomial algorithms for sbisomorphism of nd open combinatorial maps. Compter Vision and Image Understanding (CVIU) 115(7) (Jl 2011) Combier, C., Damiand, G., Solnon, C.: From maimm common sbmaps to edit distances of generalied maps. Pattern Recognition Letters 33(15) (2012) Cook, S.A.: The compleit of theorem-proving procedres. In: ACM Smposim on Theor of Compting. (1971) 151? Jansen, K., Müller, H.: The minimm broadcast time problem for several processor netorks. Theoretical Compter Science 147(1-2) (1995) Mohar, B.: A linear time algorithm for embedding graphs in an arbitrar srface. SIAM Jornal on Discrete Mathematics 12(1) (1999) Combier, C., Damiand, G., Solnon, C.: Measring the distance of generalied maps. In: GbR. LNCS, Springer (2011) 82 91