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Abstract—Ant colony optimization is a metaheuristic approach belonging to the class of model-based search algorithms. In this paper, we propose a new framework for implementing ant colony optimization algorithms called the hyper-cube framework for ant colony optimization. In contrast to the usual way of implementing ant colony optimization algorithms, this framework limits the pheromone values to the interval [0,1]. This is obtained by introducing changes in the pheromone value update rule. These changes can in general be applied to any pheromone value update rule used in ant colony optimization. We discuss the benefits coming with this new framework. The benefits are twofold. On the theoretical side, the new framework allows us to prove that in Ant System, the ancestor of all ant colony optimization algorithms, the average quality of the solutions produced increases in expectation over time when applied to unconstrained problems. On the practical side, the new framework automatically handles the scaling of the objective function values. We experimentally show that this leads on average to a more robust behavior of ant colony optimization algorithms. Index Terms—Ant colony optimization (ACO), metaheuristics. I.

We present a new energy-minimization framework for the graph isomorphism problem that is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear one-to-one correspondence exists between the solutions of the quadratic program and those in the original, combinatorial problem. To solve the program we use the so-called replicator equations—a class of straightforward continuous- and discrete-time dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results that are competitive with those obtained using more elaborate mean-field annealing heuristics.

We present some new results which definitively explain the behavior of the classical, heuristic nonlinear relaxation labeling algorithm of Rosenfeld, Hummel, and Zucker in terms of the Hummel-Zucker consistency theory and dynamical systems theory. In particular, it is shown that, when a certain symmetry condition is met, the algorithm possesses a Liapunov function which turns out to be (the negative of) a well-known consistency measure. This follows almost immediately from a powerful result of Baum and Eagon developed in the context of Markov chain theory. Moreover, it is seen that most of the essential dynamical properties of the algorithm are retained when the symmetry restriction is relaxed. These properties are also shown to naturally generalize to higher-order relaxation schemes. Some applications and implications of the presented results are finally outlined.

Given an undirected graph with 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. This is a generalization of the classical problem of finding the maximum cardinality clique of an unweighted graph, which arises as a special case of the MWCP when all the weights associated to the vertices are equal. 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. Although there has recently been much interest around neural network algorithms for the unweighted maximum clique problem, no effort has been directed so far towards its weighted counterpart. In this paper, we present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles developed and studied in various branches of mathematical biology. The proposed framework centers aroun...

In this paper, a new heuristic for approximating the maximum clique problem is proposed, based on a detailed analysis of a class of continuous optimization models which yield a complete solution to this NP-hard combinatorial problem. The idea is to alter a regularization parameter iteratively in such a way that an iterative procedure with the updated parameter value would avoid unwanted, inefficient local solutions, i.e., maximal cliques which contain less than the maximum possible number of vertices. The local search procedure is performed with the help of the replicator dynamics, and the regularization parameter is chosen deliberately as to render dynamical instability of the (formerly) stable solutions which we want to discard in order to get an improvement. In this respect, the proposed procedure differs from usual simulated annealing approaches which mostly use a "black-box" cooling schedule. To demonstrate the validity of this approach, we report on the performance applied to sel...

INTRODUCTION Let G = (V; E) be an undirected graph, where V = f1; \Delta \Delta \Delta ; ng is the set of vertices, and E ` V \Theta V is the set of edges. Vertices i and j are called adjacent if they are connected by an edge. A clique of G is a subset of V in which every pair of vertices is adjacent. A clique C is called maximal if no strict superset of C is a clique. The highest-cardinality maximal clique is called a maximum clique. The maximum clique problem is to find a maximum clique in a given graph G. The problem is NP-hard [1], even to approximate well [2]. 0 The authors thank J. Shawe-Taylor for

A classical way of matching relational structures consists of finding a maximum clique in a derived "association graph." However, it is not clear how to apply this approach to problems where the graphs are hierarchically organized, i.e. are trees, since maximum cliques are not constrained to preserve the partial order. We have recently provided a solution to this problem by constructing the association graph using the graph-theoretic concept of connectivity. In this paper, we extend the approach to the problem of matching attributed trees. Specifically, we show how to derive a "weighted" association graph, and prove that the attributed tree matching problem is equivalent to finding a maximum weight clique in it. We then formulate the maximum weight clique problem in terms of a continuous optimization problem, which we solve using "replicator" dynamical systems developed in theoretical biology. This formulation is attractive because it can motivate analog and biological implementations....

We present a new (continuous) quadratic programming approach for graph- and tree-isomorphism problems which is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear one-to-one correspondence exists between the solutions of the quadratic programs and those in the original, combinatorial problems. To approximately solve the program we use the so-called "replicator" equations, a class of straightforward continuous- and discrete-time dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results which are competitive with those obtained using more sophisticated mean-fiel...

Evolutionary game-theoretic models and, in particular, the so-called replicator equations have recently proven to be remarkably effective at approximately solving the maximum clique and related problems. The approach is centered around a classic result from graph theory that formulates the maximum clique problem as a standard (continuous) quadratic program and exploits the dynamical properties of these models, which, under a certain symmetry assumption, possess a Lyapunov function. In this letter, we generalize previous work along these lines in several respects. We introduce a wide family of game-dynamic equations known as payoffmonotonic dynamics, of which replicator dynamics are a special instance, and show that they enjoy precisely the same dynamical properties as standard replicator equations. These properties make any member of this family a potential heuristic for solving standard quadratic programs and, in particular, the maximum clique problem. Extensive simulations, performed on random as well as DIMACS benchmark graphs, show that