A Bound for Non-Subgraph Isomorphism

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1 A Bound for Non-Sub Isomorphism Christian Schellewald School of Computing, Dublin City University, Dublin 9, Ireland cschellewald/ Abstract. In this paper we propose a new lower bound to a sub isomorphism problem. This bound can provide a proof that no sub isomorphism between two s can be found. The computation is based on the SDP relaxation of a to the best of our knowledge new combinatorial optimisation formulation for sub isomorphism. We consider problem instances where only the structures of the two instances are given and therefore we deal with simple s in the first place. The idea is based on the fact that a sub isomorphism for such problem instances always leads to 0 as lowest possible optimal objective value for our combinatorial optimisation problem formulation. Therefore, a lower bound that is larger than 0 represents a proof that a sub isomorphism don t exist in the problem instance. But note that conversely, a negative lower bound does not imply that a sub isomorphism must be present and only indicates that a sub isomorphism is still possible. 1 Introduction The isomorphism problem is a well known problem in computer science and usually involves also the problem of finding the appropriate matching. Therefore it is also of interest in computer vision. If an object is represented by a the object could be identified as sub within a possibly larger scene. Error-correcting matching [1] also known as error-tolerant matching is a quite general and appropriate approach to calculate an assignment between the nodes of two s. It is based on the minimisation of so called edit costs which appear when one is turned into the other by some predefined edit operations. Commonly introduced edit operations are deletion, insertion, and substitution of nodes and edges. Each edit operation has a cost assigned which is application dependent. The minimal edit cost defines the so called edit distance between two s. The idea to define the edit distance for matching goes back to Sanfeliu and Fu [2] in Before that the edit distance was mainly used for string matching. Several algorithms for error correcting matching have been proposed that are based on different methods like tree search [3], genetic algorithms [4] and others (see e.g. This research was supported by Marie Curie Intra-European Fellowships within the 6th European Community Framework Programme.

2 2 [1]). In this paper we first propose an combinatorial optimisation formulation for the sub isomorphism problem that can be seen as a error-correcting matching approach. The integer optimisation problem we end up with is generally an indefinite quadratic integer optimisation problem which is NP-hard [5]. For example Pardalos and Vavasis showed in that indefinite quadratic programs are NP-hard problems, even if the quadratic program is very simple (see [6]). Then we compute a (convex) SDP relaxation of the combinatorial problem to obtain a lower bound to the sub isomorphism problem. The bound can be computed with standard methods for semidefinite programs. Finally we show that the bound can indeed be used to proof that no sub isomorphism between two s can be found. Several approaches have been proposed to tackle the sub isomorphism problem [7, 8,3,9]. Our approach differs to a more recent proposed approach that is based on a reformulation to a largest clique problem [10,11]. Our approach intends to find the full first as an sub isomorphism in the second where the largest clique represents the largest common sub isomorphism. 2 Preliminaries In this work we consider simple s G = (V, E) with nodes V = {1,...,n} and edges E V V. We denote the first possibly smaller with G K and the second with G L. The corresponding sets V K and V L contain K = V K and L = V L nodes respectively. We assume that L K. We make extensive use of the direct product C = A B, which is also known as Kronecker product [12]. It is the product of every matrix element A ij of A IR n m with the whole matrix B IR p q resulting in the larger matrix C IR np mq. The sub isomorphism is a mapping m : V K V V L of all nodes in the G K to a subset V of V L with K nodes of the G L such that the structure is preserved. That means that any two nodes i and j from G K that are adjacent must be mapped to nodes m(i) and m(j) in G L that are adjacent too. The same has to be true for the inverse mapping m 1 : V V K which maps the nodes V of the sub to nodes V K of G K. 3 Combinatorial Objective Function In this section we propose and proof a formulation of the combinatorial problem of finding a sub- isomorphism. The general idea is to find a bipartite matching between the set of nodes from the smaller to the set of nodes of the larger. The bipartite matching is evaluated by an objective function that can be interpreted as a comparison of the structure between all possible node pairs in the first and the structure of the node pairs to which the nodes are matched in the second. A matching that leads to no structural differences has no costs and represents a sub- isomorphism. Mathematically the evaluation can be performed by a simple quadratic objective function

3 3 x Qx. The full task of finding a sub- isomorphism results in the following combinatorial quadratic optimisation problem, which details are explained below: min x Qx x s.t. A K x = e K, A L x e L (1) x {0, 1} KL The constraints that make use of the matrices A K = I K e L and A L = e K I L ensure that the vector x is a 0,1-indicator vector which represents a bipartite matching between the two node sets of the s. Here e n IR n represents a vector with all elements 1 and I n IR n n denotes the unit matrix. A vector element x ji = 1 indicates that the node i of the first set V K is matched to the node j in the second set V L otherwise x ji = 0. The elements of the indicator vector x {1, 0} KL are ordered as follows: x = (x 11,, x L1, x 12,, x L2,, x 1K,, x LK ). (2) We illustrate such an indicator vector in figure 1 where a bipartite matching between two sets of nodes and the corresponding indicator vector are shown. The matrix Q within the objective function x Qx of the optimisation problem ( ) Fig. 1. The illustration of the 0, 1-indicator vector on the right side is a representation of the matching which is shown on the left side of this figure. (1) can be written in a short form using the Kronecker product: Definition 1. Relational Structure Matrix Q = N K N L + N K N L (3) Here N K and N L are the 0, 1-adjacency matrices of the two s. The matrices N K and N L represent the complementary adjacency matrices which are computed as follows: Definition 2. Complementary Adjacency Matrices N L = E LL N L I L NK = E KK N K I K

4 4 These complementary adjacency matrices can be interpreted as 0, 1-indicator matrices for non-adjacent nodes. They have the element ( N) ij = 1 if the corresponding nodes i and j are not directly connected in the. The adjacency matrix N K for a small along with its complementary adjacency matrix are shown in figure 2. In the following we show that a 0,1-solution vector x of N K = B A N K = B A Fig.2. An example and its adjacency matrix N K along with its complementary adjacency matrix N K. the optimisation problem (1) with an optimal objective value of zero represents a sub isomorphism. We first show that zero is the smallest possible value and than we show that every derivation of a sub isomorphism results in an objective value > 0. Proposition 1. The minimal value of the combinatorial optimisation problem (1) is zero. Proof. The elements of Q and x are all non-negative. In fact the elements are either zero or one. Therefore the lowest possible value of the quadratic cost term which can be rewritten as the following sum x Qx = x (N K N L + N K N L )x K,L K,L = [(N K ) ab ( N L ) rs + ( N K ) ab (N L ) rs ]x ra x sb (4) a,r b,s is zero. Proposition 2. A solution with the minimal value of zero of the quadratic optimisation problem (1) represents a sub- isomorphism. To proof this we consider the term within the sum and show it leads only to a cost > 0 if the considered matching violates the condition for a sub isomorphism. Proof. Only if the product x ra x sb is one the term within the sum (4) can be different from zero and the part [(N K ) ab ( N L ) rs + ( N K ) ab (N L ) rs ] must be considered. In the following we refer to this part of the term also as structure comparison term. There are two cases that lead to x ra x sb = 1:

5 5 Case A: The node a and node b in G K represent the same node (a = b). But as the diagonals of N K and N K are zero one obtains that (N K ) aa = 0 and ( N K ) aa = 0. In this case the term [(N K ) aa ( N L ) rr +( N K ) aa (N L ) rs ]x ra x ra is always zero and does not contribute to the sum. Case B: The nodes a and b in G K represent different nodes (a b) in G K and due to the bipartite matching constraint a value x ar x bs = 1 represents the situation x ar = 1 and x bs = 1 which means that the nodes a and b are mapped to two different nodes r and s in the second G L, respectively. Considering now the term [(N K ) ab ( N L ) rs + ( N K ) ab (N L ) rs ] we observe that all four possible structural cases between two pairs of nodes in the two s are valued with a cost of zero or one. All these sub-cases from case B that could lead to a non-zero value in the structure comparison term and therefore in the sum are listed in the table 1. In the following we summarise the meaning of the cases and we will see that costs are only added for every difference between the structure of G K and the considered sub of the second G L. case configuration (N K) ab ( N L) rs ( N K) ab (N L) rs cost I a,b adjacent; r,s adjacent II a,b adjacent; r,s not adjacent III a,b not adjacent; r,s adjacent IV a,b not adjacent; r,s not adjacent Table 1. List of all outcomes of the structure comparison term between two different nodes a and b of G K that are mapped to two different nodes r and s in the second G L. Only in case I and IV the structure is preserved and can lead to an isomorphism. No cost is added in this cases. The other cases (II and III) don t preserve the structure and lead to an total cost > 0. For details see the text. I: If the two nodes a and b in the first are neighbours, (N K ) ab = 1, then no cost is added in (4) if the nodes r and s in the scene are neighbours, too: ( N L ) rs = 0. II: Otherwise if a and b are neighbours in G K and the corresponding nodes r and s are no neighbours in the second, ( N L ) rs = 1, then a cost of 1 is added. The configurations I and II are visualised in figure 3. III: Analogously, the structure comparison term penalises assignments where pairs of nodes (a and b) in the G K become neighbours in the second G L which were not adjacent before. IV: Finally if a and b are not adjacent in the first G K and the nodes r and s in G L are also not adjacent, no cost is added.

6 6 a GK x ra=1 GK GL GL a r x ra=1 r b x sb=1 s b x s b =1 Good assignment (no costs) Bad assignment (costly) s Fig.3. Left: Adjacent nodes a and b in the G K are assigned to adjacent nodes r and s in the G L. Right. Adjacent nodes a and b are no longer adjacent in the G L after the assignment. The left assignment leads to no additional costs while the right undesired assignment adds 1 to the total cost. a GK x ra=1 r GL a GK x ra=1 r GL b x sb=1 Good assignment (no costs) s b x s b =1 Bad assignment (costly) s Fig.4. Left: Nodes a and b which are not adjacent in the object G K are assigned to nodes which are also not adjacent in the scene G L. Right: A pair of nodes a and b become neighbours r and s after assignment. The left assignment is associated with no additional costs in (4). The undesired assignment on the right side adds 1 to these costs. Figure 4 illustrates situation III and IV in detail. This shows that only mappings that lead to a change in the structure are penelised with a cost. Structure preserving mappings which are compatible with a sub isomorphism are without costs. Note that due to the symmetry of the adjacency matrices the quadratic cost term x Qx is symmetric too and every difference in the compared structures of the two s is considered twice resulting in a cost of 2 for every difference in the structure. Finally the sum (4) and therefore the objective function x Qx considers all possible combinations of node pairs a and b that are mapped to r and s, respectively. And only for matchings which lead to no difference in the mapped sub-structure and vice versa all the terms within the sum (4) are zero. In this case the bipartite matching represents a sub isomorphism. We wish to emphasise that the minimisation of (1) represents the search for a bipartite matching which has the smallest possible structural deviation between G K and the considered sub of G L. Therefore (1) can be seen as a edit distance with a cost of 2 for each addition or removal of an edge that is needed to turn the first into the considered sub of the other.

7 7 4 Convex Problem Relaxation The combinatorial isomorphism approach (1) can be relaxed to a (convex) semidefinite program (SDP) which has the following standard form: [ ] min Tr QX s.t. Tr[A i X] = c i for i = 1,..., m (5) X 0 The constraint X 0 means that X has to be positive semidefinite. This convex optimization problem can be solved with standard methods like interior point algorithms (see e.g. [13]). Note that the solution of the relaxation (5) provides a lower bound to (1). Below, we describe how we derive such a semidefinite program from (1). For more information on semidefinite programming we refer to [14]. 5 Convex Relaxation The convex relaxation in this section follows the relaxation explained in detail in [15]. In order to obtain an appropriate SDP relaxation for the combinatorial sub matching problem, we start with the reformulation of the objective function of (1) f(x) = x Qx = Tr [ x Qx ] = Tr [ Qxx ] [ ] = Tr QX, (6) We take into account the following summarised constraints of the form Tr[A i X] = c i which intend to include the original bipartite matching constrainst in a suitable way. In particular we describe the constraint marices A i. The equality constraint L j=1 x ij = 1, i = 1,...,K, which are part of the bipartite matching constraints represent the constraint that each node of the smaller is mapped to exactly one node of the scene. We define K constraint matrices sum A j IR (KL+1) (KL+1), j = 1,...,K which ensure (taking the order of the diagonal elements into account) that the sum of the appropriate portion of the diagonal elements of X is 1. As we deal with the diagonal elements of X we exploit also the fact that x i = x 2 i holds true for 0/1-variables. The matrix elements for the j-th constraint matrix sum A j can be expressed as follows: sum A j kl = jl+1 i=(j 1)L+1 δ ik δ il for k, l = 1,...,KL + 1 For these constraints the constants c j are: c j, j = 1,...,K. As all integer solutions X = xx IR KL KL, where x represents a bipartite matching, have zerovalues at those matrix positions where I K (E LL I L ) and (E KK I K ) I L have non-zero elements we want to force the corresponding elements in X IR KL KL

8 8 to be zero. The matrices E LL IR L L and E KK IR R R are matrices where all elements are 1. The matrices I nn IR n n represent the unit matrices. This can be achieved with the constraint matrices A ars, Aŝâˆb IR KL KL which are determined by the indices a, r,s and ŝ, â,ˆb. They have the following matrix elements A ars kl =δ k,(al+r) δ l,(al+s) + δ k,(al+s) δ l,(al+r), (7) Aŝâˆb kl =δ k,(ŝk+ˆb) δ l,(ŝk+â) + δ k,(ŝk+â) δ l,(ŝk+ˆb), (8) where k, l = 1,...,KL. The indices a, r,s and ŝ, â,ˆb attain all valid combinations of the following triples where s > r and ˆb > â: (a, r, s) : a = 1,...,K; r = 1,...,L; s = (r + 1),...,L (ŝ, â,ˆb) : ŝ = 1,...,L; â = 1,...,K; ˆb = (â + 1),...,K For this constraints the constant c has to be zero. With this we define (LL L)K/2 + (KK K)L/2 additional constraints that ensure zero-values at the corresponding matrix positions of X. 6 Early Results to the Non-Isomorphism Bound For the early results presented in this section we used our implementation described in [15] where we had to set the similarity vector to a zero vector. Furthermore we introduced a parameter α > 0 which is just a scaling parameter for the objective function and should not have a influence on the solution other than a scaling. An illustrative example for a sub isomorphism problem is depicted in figure 5. For this example we compute a lower bound > 0 using the Fig. 5. Example for a randomly created sub problem. Is there a sub isomorphism? For the shown problem instance we can compute a lower bound > 0 for (1) which proves that no sub isomorphism is present. SDP relaxation (5), which proves that a sub isomorphism does not exist in this problem instance. Note that we did not eliminate mappings that could

9 9 not lead to an sub isomorphism. The possible objective values of (1) are restricted to discrete values as the quadratic term αx Qx can only reach values which are multiples of 2α. The discrete distribution of the objective values for the sub isomorphism problem shown in figure 5 is depicted in figure 6 where we have set α = 0.3. For Discrete Probability Distribution Objective Function Values of a Sub Isorphism Problem Graph: K=7,L=15, α= Probability Density f * bound = f * max = 12.0 f * opt = Objective Value Fig. 6. The distribution of the objective values for the sub isomorphism problem which is shown in figure 5. The objective values are restricted to discrete values, as the quadratic term αx Qx can only attain values which are multiples of 2α. Here we have set α arbitrarily to 0.3. The optimal objective value is 0.6 and the obtained lower bound is > 0.0, which is a non-isomorphism proof for this problem instance. a first preliminary investigation of this bound we created 1000 small sub matching problem instances for which we have chosen the size of the two s G K and G L to be K = 7 and L = 15, respectively. The edge probability of the G K was set to 0.5 and the probability for an edge in the second was set to 0.2. The results for this experiment series reveal that for various problem instances it is indeed possible to conclude that no sub isomorphism exist. We have obtained 388 problem instances with a lower bound > 0.0 which proves that no sub isomorphism can occur in this problem instances. The other 612 problem instances have a lower bound 0.0. For 436 ( 71%) of these problem instances the combinatorial optimum is > 0.0 indicating that the relaxation is not tight enough to detect that no sub isomorphism can occur. 7 Discussion We proposed a bound to the sub isomorphism problem and showed that the bound is not only of theoretical interest but also applies to several instances of sub matching problems. It would be interesting to investigate which criteria a sub matching problem has to fullfill to result in a tight relaxation. Such

10 10 insights could be usefull in the process of creating or obtaining object s from images for object recognition tasks. The tightness and therefore the lower bound can be improved by reducing the dimension of the problem size. For example one can eliminate a mapping i j if the degree (The number of incident edges.) of an node i is larger than the degree of node j in the second. Such a mapping cannot lead to a sub isomorphism. An other improvement could be expected when also inequalities are included in the SDP relaxation. None of these improvements are used for the presented results. However, for increasing problem instances the relaxation will probably get less tight and a lower bound 0.0 becomes more likely. But note that even less tight solutions still lead to good integer solutions (see e.g. [15]). References 1. H. Bunke. Error correcting matching: On the influence of the underlying cost function. IEEE Trans. Pattern Analysis and Machine Intelligence, 21(9): , A. Sanfeliu and K. S. Fu. A distance measure between attributed relational s for pattern recognition. IEEE Transaction on Systems, Man and Cybernetics, 13(3): , B.T. Messmer and H. Bunke. A new algorithm for error-tolerant sub isomorphism detection. IEEE Trans. Patt. Anal. Mach. Intell., 20(5): , Yuan-Kai Wang, Kuo-Chin Fan, and Jorng-Tzong Horng. Genetic-based search for error-correcting isomorphism. IEEETSMC: IEEE Transactions on Systems, Man, and Cybernetics, 27, M. R. Garey and D. S. Johnson. Computers and Intractability, a Guide to the Theory of NP-Completeness. W. H. Freeman and Company, P. M. Pardalos and S. A. Vavasis. Quadratic programming with one negative eigenvalue is np-hard. J. Global Optim., 1:15 22, J.R. Ullmann. An algorithm for sub isomorphism. Journal of the ACM, 23(1):31 42, H. G. Barrow and R. M. Burstall. Sub isomorphism, matching relational structures and maximal cliques. Information Processing Letters, 4(4):83 84, David Eppstein. Sub isomorphism in planar s and related problems. Journal of Graph Algorithms and Applications, 3(3):1 27, I. Bomze, M. Budinich, P. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, M. Pelillo. Replicator equations, maximal cliques, and isomorphism. Neural Computation, 11(8): , A. Graham. Kronecker Products and Matrix Calculus with Applications. Ellis Horwood Limited and John Wiley and Sons, B. Borchers. CSDP: A C library for semidefinite programming. Optimization Methods and Software, 11(1): , H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidefinite Programming, Boston, Kluwer Acad. Publ. 15. Christian Schellewald and Christoph Schnörr. Probabilistic sub matching based on convex relaxation. In Proc. of EMMCVPR, volume 3757 of LNCS, pages Springer, 2005.