In some applications of matching, the structural or hierarchical properties of the two graphs being aligned must be maintained. The hierarchical properties are induced by the direction of the edges in the two directed graphs. These structural relationships defined by the hierarchy in the graphs act as a constraint on the alignment. In this paper, we formalize the above problem as the weighted alignment between two directed acyclic graphs. We prove that this problem is NP-complete, show several upper bounds for approximating the solution, and finally introduce polynomial time algorithms for sub-classes of directed acyclic graphs.
Both of these problems have many practical applications, in particular, graph isomorphism has received a lot of attention in the area of computer vision. Images or objects can be represented as a graph. A weighted graph can be used to formulate a structural description of an object @cite_24 . There have been two main approaches to solving graph isomorphism: state--space construction with searching and nonlinear optimization. The first method consists of building the state--space, which can then be searched. This method has an exponential running time in the worst case scenario, but by employing heuristics, the search can be reduced to a low--order polynomial for many types of graphs @cite_20 @cite_9 . With the second approach (nonlinear optimization), the most successful approaches have been relaxation labeling @cite_27 , neural networks @cite_18 , linear programming @cite_8 , eigendecomposition @cite_0 , genetic algorithms @cite_4 , and Lagrangian relaxation @cite_21 .
@cite_18 @cite_4 @cite_8 @cite_9 @cite_21 @cite_24 @cite_0 @cite_27 @cite_20
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A generalization of subgraph isomorphism for the fault-tolerant interpretation of disturbed line images has been achieved. Object recognition is effected by optimal matching of a reference graph to the graph of a distorted image. This optimization is based on the solution of linear and quadratic assignment problems. The efficiency of the procedures developed for this objective has been proved in practical applications. NP-complete problems such as subgraph recognition need exhaustive computation if exact (branch-and-bound) algorithms are used. In contrast to this, heuristics are very fast and sufficiently reliable for less complex relational structures of the kind investigated in the first part of this paper. Constrained continuous optimization techniques, such as relaxation labeling and neural network strategies, solve recognition problems within a reasonable time, even in rather complex relational structures where heuristics can fail. They are also well suited to parallelism. The second part of this paper is devoted exclusively to them. The isomorphism problem for graphs G 1 and G 2 is to determine if there exists a one-to-one mapping of the vertices of G 1 onto the vertices of G 2 such that two vertices of G 1 are adjacent if and only if their images in G 2 are adjacent. In addition to determining the existence of such an isomorphism, it is useful to be able to produce an isomorphism-inducing mapping in the case where one exists. The isomorphism problem for triconnected planar graphs is particularly simple since a triconnected planar graph has a unique embedding on a sphere [6]. Weinberg [5] exploited this fact in developing an algorithm for testing isomorphism of triconnected planar graphs in O(|V| 2 ) time where V is the set consisting of the vertices of both graphs. The result has been extended to arbitrary planar graphs and improved to O(|V|log|V|) steps by Hopcroft and Tarjan [2,3]. In this paper, the time bound for planar graph isomorphism is improved to O(|V|). In addition to determining the isomorphism of two planar graphs, the algorithm can be easily extended to partition a set of planar graphs into equivalence classes of isomorphic graphs in time linear in the total number of vertices in all graphs in the set. A random access model of computation (see Cook [1]) is assumed. Although the proposed algorithm has a linear asymptotic growth rate, at the present stage of development it appears to be inefficient on account of a rather large constant. This paper is intended only to establish the existence of a linear algorithm which subsequent work might make truly efficient. A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing. Our approach is in the same spirit as a Lagrangian decomposition approach in that the row and column constraints are satisfied separately with a Lagrange multiplier used to equate the two "solutions". Due to the unavoidable symmetries in graph isomorphism (resulting in multiple global minima), we add a symmetry-breaking self-amplification term in order to obtain a permutation matrix. With the application of a fixpoint preserving algebraic transformation to both the distance measure and self-amplification terms, we obtain a Lagrangian relaxation network. The network performs minimization with respect to the Lagrange parameters and maximization with respect to the permutation matrix variables. Simulation results are shown on 100 node random graphs and for a wide range of connectivities. We show that the Graph Isomorphism (GI) problem and the more general problems of String Isomorphism (SI) andCoset Intersection (CI) can be solved in quasipolynomial(exp((logn)O(1))) time. The best previous bound for GI was exp(O( √n log n)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp(O (√ n)), where n is the size of the permutation domain (Babai, 1983). Following the approach of Luks’s seminal 1980 82 paper, the problem we actually address is SI. This problem takes two strings of length n and a permutation group G of degree n (the “ambient group”) as input (G is given by a list of generators) and asks whether or not one of the strings can be transformed into the other by some element of G. Luks’s divide-and-conquer algorithm for SI proceeds by recursion on the ambient group. We build on Luks’s framework and attack the obstructions to efficient Luks recurrence via an interplay between local and global symmetry. We construct group theoretic “local certificates” to certify the presence or absence of local symmetry, aggregate the negative certificates to canonical k-ary relations where k = O(log n), and employ combinatorial canonical partitioning techniques to split the k-ary relational structure for efficient divide-and- conquer. We show that in a well–defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. The central element of the algorithm is the “local certificates” routine which is based on a new group theoretic result, the “Unaffected stabilizers lemma,” that allows us to construct global automorphisms out of local information.