A Complementary Pivoting Approach to Graph Matching

Massaro, Alessio; Pelillo, Marcello

doi:10.1007/3-540-44745-8_31

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“…The remarkable effectiveness of our approach and the empirical immunity of Lemke's method to saddle points seems to indicate that pivoting-based methods offer a promising new way to tackle this and related combinatorial problems. The algorithm has already been applied with success to graph matching problems arising in computer vision and pattern recognition [21]. Note also that our algorithm is completely devoid of working parameters, a valuable feature which distinguishes it from other heuristics proposed in the literature (see, e.g., [4]).…”

Section: Discussionmentioning

confidence: 99%

A Complementary Pivoting Approach to the Maximum Weight Clique Problem

Massaro¹,

Pelillo²,

Bomze³

2002

SIAM J. Optim.

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Abstract. Given an undirected graph with positive weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having largest total weight. The problem is known to be NP -hard, even to approximate. Motivated by a recent quadratic programming formulation, which generalizes an earlier remarkable result of Motzkin and Straus, in this paper we propose a new framework for the MWCP based on the corresponding linear complementarity problem (LCP). We show that, generically, all stationary points of the MWCP quadratic program exhibit strict complementarity. Despite this regularity result, however, the LCP turns out to be inherently degenerate, and we find that Lemke's well-known pivoting method, equipped with standard degeneracy resolution strategies, yields unsatisfactory experimental results. We exploit the degeneracy inherent in the problem to develop a variant of Lemke's algorithm which incorporates a new and effective "look-ahead" pivot rule. The resulting algorithm is tested extensively on various instances of random as well as DIMACS benchmark graphs, and the results obtained show the effectiveness of our method.Key words. maximum weight clique, linear complementarity, pivoting methods, quadratic programming, combinatorial optimization, heuristics AMS subject classifications. 90C27, 90C20, 90C33, 90C49, 90C59, 05C69PII. S1052623400381413 1. Introduction. Given an undirected graph, the maximum clique problem (MCP) consists of finding a subset of pairwise adjacent vertices (i.e., a clique) having largest cardinality. The problem is known to be NP -hard for arbitrary graphs and, according to recent theoretical results, so is the problem of approximating it within a constant factor. An important generalization of the MCP arises when positive weights are associated to the vertices of the graph. In this case the problem is known as the maximum weight clique problem (MWCP) and consists of finding a clique in the graph which has largest total weight. (Note that the maximum weight clique does not necessarily have largest cardinality.) It is clear that the classical unweighted version is a special case in which the weights assigned to the vertices are all equal. As an obvious corollary, the MWCP has at least the same computational complexity as its unweighted counterpart. The MWCP has important applications in such fields as computer vision, pattern recognition, and robotics, where weighted graphs are employed as a convenient means of representing high-level pictorial information (see, e.g., [17,28]). We refer to [4] for a recent review concerning algorithms, applications, and complexity issues of this important problem.Inspired by a classical result in graph theory contributed by Motzkin and Straus [24], Gibbons et al. [13] have recently formulated the MWCP in terms of a standard quadratic optimization problem (StQP), which consists of minimizing a quadratic form

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Section: Discussionmentioning

confidence: 99%