Pattern Recognition 32 (1999) 1255}1271
Deterministic search for relational graph matching Mark L. Williams , Richard C. Wilson, Edwin R. Hancock * Defence Research Agency, St. Andrews Road, Malvern, Worcestershire, WR14 3PS, UK Department of Computer Science, University of York, York, Y01 5DD, UK Received 13 April 1998; accepted 16 October 1998
Abstract This paper describes a comparative study of various deterministic discrete search-strategies for graph-matching. The framework for our study is provided by the Bayesian consistency measure recently reported by Wilson and Hancock (IEEE PAMI 19 (1997) 634}648; Pattern Recognition 17 (1996) 263}276) and Wilson et al. (Comput. Vision Image Understanding 72 (1998) 20}38') We investigate two classes of update process. The "rst of these aims to exploit discrete gradient ascent methods. We investigate the e!ect of searching in the direction of both the local and global gradient maximum. An experimental study demonstrates that although more computationally intensive, the global gradient method o!ers signi"cant performance advantages in terms of accuracy of match. Our second search strategy is based on tabu search. In order to develop this method we introduce memory into the search procedure by de"ning contextdependant search paths. We illustrate that although it is more e$cient than the global gradient method, tabu search delivers almost comparable performance. 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Tabu search; Graph-matching; Natural gradient; Consistent labelling; Discrete relaxation; Heuristic search
1. Introduction Relational graph matching is a process that is central to symbolic interpretation problems in arti"cial intelligence and pattern recognition [1, 2]. The formal aspects of the problem have been studied for over 30 years [3]. In particular, topics such as subgraph isomorphism [4], maximal clique "nding [5] and graph partitioning [6]have attracted considerable interest in the "elds of discrete mathematics and the theory of algorithms. Each of these problems is known to be NP-complete, and the
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quest for e$cient algorithms of polynomial complexity still raises many important theoretical issues. However, from the perspective of practical problemsolving the issue of theoretical complexity is of less importance than the ability to "nd useful, though suboptimal, solutions using "nite computing resources. In any case, since the graph-structures under study are likely to be inexact due to the presence of noise and segmentation errors, approximate solutions may be the best that is achievable [7}9]. Indeed, the idea of posing graph-theoretic problems as optimisation tasks has recently attracted considerable interest in the literature [10}13]. Mean-"eld networks have been applied to a variety of graph-theoretic topics including graph-partitioning [14], the travelling salesman problem [15] and graph-matching [10,12]. Although these examples can all
0031-3203/99/$20.00 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 1 - 3 2 0 3 ( 9 8 ) 0 0 1 5 2 - 6
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be regarded as instances of continuation methods, disdcrete con"gurational optimisation methods have also been used to great e!ect. For instance, Cross, Wilson and Hancock have used genetic sesarch for graph-matching [16]. Heuristic search techniques have also proved to be highly e!ective. In the classical pattern analysis literature Haralick and Elliot [2] have used forward-checking backtracking to search for consistent labellings. More recently, Messmer and Bunke [17] have addressed the issue of exponential complexity by demonstrating how subgraph isomorphisms can be located in polynomialtime if a form of structural-hashing is used to prune the search space. In fact, the idea of using domain-speci"c heuristics to improve the e$ciency of search was central to much of the pioneering work on arti"cial intelligence of the 1970s [18]. Perhaps the most popular of these is the A* algorithm which has been widely exploited in applications such as planning and feature extraction. With the advent of more generally applicable global con"gurational optimisation strategies such as simulated annealing [19], mean-"eld theory [14,15,20] and most recently evolutionary optimisation (genetic search), heuristic search has been largely neglected in the literature. However, as recently observed by Glover and his co-workers [6,21}23], the abandonment of the method may be somewhat premature, since the use of domain-speci"c knowledge can prove highly e!ective in rapidly locating useful though suboptimal solutions. It is this observation that has lead to the development of a new class of heuristic optimisation techniques known as tabu search. Tabu-search [6,21}23] exploits constraints known to apply in a speci"c domain to take maximum advantage of the available computational resources. Rather than adopting a computationally demanding exploration of the state space using stochastic (e.g. simulated annealing and genetic search) or continuation methods (e.g. mean"eld annealing), tabu search deploys the resources to preferentially search potentially pro"table areas. In essence, the search procedure possesses memory. This memory can be both short- and long-term. In the long term, regions of unpro"table search are deemed tabu and are not revisited. Short-term memory can be invoked to intensify search in certain regions. If intensi"cation fails to yield a useful solution, then diversi"cation strategies can be invoked over a longer time scale. In the broadest sense, this process can be viewed as planning the deployment of computational resources to gain maximum yield in terms of solution quality. This paper aims to investigate the use of di!erent deterministic search strategies for graph matching. In a recent series of papers we have developed a Bayesian framework which allows the consistency of graph matching to be gauged using probability distributions [24}26]. These distributions are de"ned over the Hamming distances between partially consistent subgraphs and a set
of model sub-graphs residing in a dictionary. We have explored the optimisation of this global consistency measure using a number of stochastic [18,27] and continuation methods [28}30]. In this paper we aim to compare the use of steepest gradient methods with a heuristic search method inspired by tabu search.
2. Relational graphs We abstract the matching process in terms of purely symbolic relational graphs [9,31}33]. We use the notation G"(
