Solving the maximum edge weight clique problem via unconstrained quadratic programming

The unconstrained quadratic binary program (UQP) is proving to be a successful modeling and solution framework for a variety of combinatorial optimization problems. Experience reported in the literature with several problem classes has demonstrated that this approach works surprisingly well in terms of solution quality and computational times, often rivaling and sometimes surpassing more traditional methods. In this paper we report on the application of UQP to the maximum edge-weighted clique problem. Computational experience is reported illustrating the attractiveness of the approach.     

A new trust region technique for the maximum weight clique problem

A new simple generalization of the Motzkin-Straus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a Motzkin-Straus optimum coincides with such a point. The developed method has complexity O(n³), where n is the number of vertices of the graph. It was implemented in a publicly available software package QUALEX-MS.Computational experiments indicate that the algorithm is exact on small graphs and very efficient on the DIMACS benchmark graphs and various random maximum weight clique problem instances.     

Multi-neighborhood tabu search for the maximum weight clique problem

Given an undirected graph G=(V,E) with vertex set V={1,??,n} and edge set E?V×V. Let w:V??Z ⁺ be a weighting function that assigns to each vertex i??V a positive integer. The maximum weight clique problem (MWCP) is to determine a clique of maximum weight. This paper introduces a tabu search heuristic whose key features include a combined neighborhood and a dedicated tabu mechanism using a randomized restart strategy for diversification. The proposed algorithm is evaluated on a total of 136 benchmark instances from different sources (DIMACS, BHOSLIB and set packing). Computational results disclose that our new tabu search algorithm outperforms the leading algorithm for the maximum weight clique problem, and in addition rivals the performance of the best methods for the unweighted version of the problem without being specialized to exploit this problem class.     

DC approach to weakly convex optimization and nonconvex quadratic optimization problems

Motivated by weakly convex optimization and quadratic optimization problems, we first show that there is no duality gap between a difference of convex (DC) program over DC constraints and its associated dual problem. We then provide certificates of global optimality for a class of nonconvex optimization problems. As an application, we derive characterizations of robust solutions for uncertain general nonconvex quadratic optimization problems over nonconvex quadratic constraints.     

The maximum clique problem

In this paper we present a survey of results concerning algorithms, complexity, and applications of the maximum clique problem. We discuss enumerative and exact algorithms, heuristics, and a variety of other proposed methods. An up to date bibliography on the maximum clique and related problems is also provided.     

On a continuous approach for the maximum weighted clique problem

This paper is focused on computational study of continuous approach for the maximum weighted clique problem. The problem is formulated as a continuous optimization problem with a nonconvex quadratic constraint given by the difference of two convex functions (d.c. function). The proposed approach consists of two main ingredients: a local search algorithm, which provides us with crucial points; and a procedure which is based on global optimality condition and which allows us to escape from such points. The efficiency of the proposed algorithm is illustrated by computational results.     

Solving the maximum clique problem using a tabu search approach

We describe two variants of a tabu search heuristic, a deterministic one and a probabilistic one, for the maximum clique problem. This heuristic may be viewed as a natural alternative implementation of tabu search for this problem when compared to existing ones. We also present a new random graph generator, the -generator, which produces graphs with larger clique sizes than comparable ones obtained by classical random graph generating techniques. Computational results on a large set of test problems randomly generated with this new generator are reported and compared with those of other approximate methods.The authors are grateful to the Quebec Government (Fonds F.C.A.R.) and to the Canadian Natural Sciences and Engineering Research Council (grant 0GP0038816) for financial support.     

A fast algorithm for the maximum clique problem

Given a graph, in the maximum clique problem, one desires to find the largest number of vertices, any two of which are adjacent. A branch-and-bound algorithm for the maximum clique problem—which is computationally equivalent to the maximum independent (stable) set problem—is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy. The algorithm performs successfully for many instances when applied to random graphs and DIMACS benchmark graphs.     

A hybrid heuristic for the maximum clique problem

In this paper we present a heuristic based steady-state genetic algorithm for the maximum clique problem. The steady-state genetic algorithm generates cliques, which are then extended into maximal cliques by the heuristic. We compare our algorithm with three best evolutionary approaches and the overall best approach, which is non-evolutionary, for the maximum clique problem and find that our algorithm outperforms all the three evolutionary approaches in terms of best and average clique sizes found on majority of DIMACS benchmark instances. However, the obtained results are much inferior to those obtained with the best approach for the maximum clique problem.     

On solving the maximum clique problem

The Maximum Clique Problem (MCP) is regarded here as the maximization of an indefinite quadratic form over the canonical simplex. For solving MCP an algorithm based upon Global Optimality Conditions (GOC) is applied. Furthermore, each step of the algorithm is analytically investigated and tested. The computational results for the proposed algorithm are compared with other Global Search approaches.     

The maximum ratio clique problem

Computational Management Science - This paper introduces a fractional version of the classical maximum weight clique problem, the maximum ratio clique problem, which is to find a maximal clique...     

An LPCC approach to nonconvex quadratic programs

Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divided into two LPCCs: the first confirms whether the QP is bounded below on the feasible set and, if not, computes a feasible ray on which the QP is unbounded; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can be accomplished by a parameter-free, mixed integer-programming based, finitely terminating algorithm developed recently by the authors, which can be enhanced in this context by a new kind of valid cut derived from the second-order conditions of the QP and by exploiting the special structure of the LPCCs. Throughout, our treatment makes no boundedness assumption of the QP; this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption. The general theory is illustrated by 3 classes of indefinite problems: QPs with simple upper and lower bounds (existence of optimal solutions is guaranteed); same QPs with an additional inequality constraint (extending the case of simple bound constraints); and nonnegatively constrained copositive QPs (no guarantee of the existence of an optimal solution). We also present numerical results to support the special cuts obtained due to the QP connection.     

A Lagrangian relaxation approach to the edge-weighted clique problem

The b-clique polytope CPⁿ_b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QPⁿ=CPⁿ_n as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over CPⁿ_n. The problem of optimizing linear functions over CPⁿ_b has so far been approached via heuristic combinatorial algorithms and cutting-plane methods.We study the structure of CPⁿ_b in further detail and present a new computational approach to the linear optimization problem based on the idea of integrating cutting planes into a Lagrangian relaxation of an integer programming problem that Balas and Christofides had suggested for the traveling salesman problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.     

A branch and bound algorithm for the maximum clique problem

We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive lower and upper bounds for the clique number.

---

Zusammenfassung Wir stellen einen Branch and Bound Algorithmus für das Maximum Clique Problem in einem beliebigen Graphen vor. Das Hauptaugenmerk richtet sich dabei auf die Bestimmung oberer Schranken mit Hilfe von Färbungen von Graphen. Es wird eine Modifikation einer bekannten Färbungsmethode, genannt DSATUR, verwendet, mit der sich gleichzeitig obere und untere Schranken für die Cliquezahl erstellen lassen.

     

Cooperating local search for the maximum clique problem

The advent of desktop multi-core computers has dramatically improved the usability of parallel algorithms which, in the past, have required specialised hardware. This paper introduces cooperating local search (CLS), a parallelised hyper-heuristic for the maximum clique problem. CLS utilises cooperating low level heuristics which alternate between sequences of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, where vertices of the current clique are swapped with vertices not in the current clique. These low level heuristics differ primarily in their vertex selection techniques and their approach to dealing with plateaus. To improve the performance of CLS, guidance information is passed between low level heuristics directing them to particular areas of the search domain. In addition, CLS dynamically reconfigures the allocation of low level heuristics to cores, based on information obtained during a trial, to ensure that the mix of low level heuristics is appropriate for the instance being optimised. CLS has no problem instance dependent parameters, improves the state-of-the-art performance for the maximum clique problem over all the BHOSLIB benchmark instances and attains unprecedented consistency over the state-of-the-art on the DIMACS benchmark instances.     

Hidden conic quadratic representation of some nonconvex quadratic optimization problems

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.     

A study of ACO capabilities for solving the maximum clique problem

This paper investigates the capabilities of the Ant Colony Optimization (ACO) meta-heuristic for solving the maximum clique problem, the goal of which is to find a largest set of pairwise adjacent vertices in a graph. We propose and compare two different instantiations of a generic ACO algorithm for this problem. Basically, the generic ACO algorithm successively generates maximal cliques through the repeated addition of vertices into partial cliques, and uses “pheromone trails” as a greedy heuristic to choose, at each step, the next vertex to enter the clique. The two instantiations differ in the way pheromone trails are laid and exploited, i.e., on edges or on vertices of the graph. We illustrate the behavior of the two ACO instantiations on a representative benchmark instance and we study the impact of pheromone on the solution process. We consider two measures—the re-sampling and the dispersion ratio—for providing an insight into the performance at run time. We also study the benefit of integrating a local search procedure within the proposed ACO algorithm, and we show that this improves the solution process. Finally, we compare ACO performance with that of three other representative heuristic approaches, showing that the former obtains competitive results.     

A reformulation-convexification approach for solving nonconvex quadratic programming problems

In this paper, we consider the class of linearly constrained nonconvex quadratic programming problems, and present a new approach based on a novel Reformulation-Linearization/Convexification Technique. In this approach, a tight linear (or convex) programming relaxation, or outer-approximation to the convex envelope of the objective function over the constrained region, is constructed for the problem by generating new constraints through the process of employing suitable products of constraints and using variable redefinitions. Various such relaxations are considered and analyzed, including ones that retain some useful nonlinear relationships. Efficient solution techniques are then explored for solving these relaxations in order to derive lower and upper bounds on the problem, and appropriate branching/partitioning strategies are used in concert with these bounding techniques to derive a convergent algorithm. Computational results are presented on a set of test problems from the literature to demonstrate the efficiency of the approach. (One of these test problems had not previously been solved to optimality). It is shown that for many problems, the initial relaxation itself produces an optimal solution.