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The random walk’ correlation matrix of multi-objective combinatorial optimization problems utilizes both local structure and general statistics of the objective functions. Reckoning time of correlation, or the random walk of lag 0, is quadratic in problem size L and number of objectives D. The computational complexity of the correlation coefficients of mNK is $$O(D^2 K^2 L)$$ , and of mQAP is $$O(D^2 L^2)$$ , where K is the number of interacting bits. To compute the random walk of a lag larger than 0, we employ a weighted graph Laplacian that associates a mutation operator with the difference in the objective function. We calculate the expected objective vector of a neighbourhood function and the eigenvalues of the corresponding transition matrix. The computational complexity of random walk’s correlation coefficients is polynomial with the problem size L and the number of objectives D. The computational effort of the random walks correlation coefficients of mNK is $$O(2^K L D^2)$$ , whereas of mQAP is $$O(L^6 D^2)$$ . Numerical examples demonstrate the utilization of these techniques.

Relaxation techniques play a great role in solving the quadratic assignment problem, among which the convex quadratic programming bound (QPB) is competitive with existing bounds in the trade-off between cost and quality. In this article, we propose two new lower bounds based on QPB. The first dominates QPB at a high computational cost, which is shown equivalent to the recent second-order cone programming bound. The second is strictly tighter than QPB in most cases, while it is solved as easily as QPB.

Cloud computing has emerged as a new paradigm for hosting and supplying services over the Internet. This technology has brought many benefits, such as eliminating the need for maintaining expensive computing hardware. With an increasing demand for cloud computing, providing performance guarantees for applications that run over cloud become important. Applications can be abstracted into a set of virtual machines with certain guarantees depicting the quality of service of the application. In this paper, we consider the placement of these virtual machines across multiple data centers (VMPlacement), meeting the quality of service requirements while minimizing the bandwidth cost of the data centers. This problem is a generalization of the NP-hard generalized quadratic assignment problem (GQAP). In this paper, we present a greedy randomized adaptive search procedure and a biased random-key genetic algorithm, both hybridized with a path-relinking strategy and a local search based on variable neighborhood descent for solving this problem. The hybrid heuristics are also tested on instances of the GQAP. We show that both algorithms are effective in quickly solving small and large instances of VMPlacement problem, especially when the path-relinking is used. For GQAP, the results outperform the previous state-of-the-art algorithms.

The efficiency of local search is proportional to the number and the distribution of basins of attraction. Often combinatorial optimisation problems have a large number of local optima, uncountable with available computational resources. Approximating the number of basins of attraction and the minimal number of samples for visiting all basins at least once are complex problems for which we assume specific distributions of basins of attraction. We define two types of basins of attraction of multi-objective combinatorial optimisation problems with complementary properties. Acknowledging that each local search run generates a Pareto front of solutions, either each Pareto local solution corresponds to a basin of attraction, or a Pareto basin matches an entire Pareto local front. Simulations compare parametric and non-parametric estimators on bi-objective quadratic assignment problem instances.

Analysing the characters of the elements of the efficiency matrix of the assignment problem, we find some properties of the optimal solution. Different from the traditional algorithm, Hungarian method, we give several principles to recognize some optimal points rapidly, thus we can reduce the dimension of the assignment problem.

The linear ordering problem (LOP) is an $\mathcal{NP}$ -hard combinatorial optimization problem with a wide range of applications in economics, archaeology, the social sciences, scheduling, and biology. It has, however, drawn little attention compared to other closely related problems such as the quadratic assignment problem and the traveling salesman problem. Due to its computational complexity, it is essential in practice to develop solution approaches to rapidly search for solution of high-quality. In this paper we propose a new algorithm based on a greedy randomized adaptive search procedure (GRASP) to efficiently solve the LOP. The algorithm is integrated with a Path-Relinking (PR) procedure and a new local search scheme. We tested our implementation on the set of 49 real-world instances of input-output tables (LOLIB instances) proposed in Reinelt (Linear ordering library (LOLIB) 2002). In addition, we tested a set of 30 large randomly-generated instances proposed in Mitchell (Computational experience with an interior point cutting plane algorithm, Tech. rep., Mathematical Sciences, Rensellaer Polytechnic Institute, Troy, NY 12180-3590, USA 1997). Most of the LOLIB instances were solved to optimality within 0.87 seconds on average. The average gap for the randomly-generated instances was 0.0173% with an average running time of 21.98 seconds. The results indicate the efficiency and high-quality of the proposed heuristic procedure.

Quadratic assignment problems (QAPs) is a NP-hard combinatorial optimization problem. QAPs are often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator that can be used for benchmarking for heuristic algorithms. Our QAP generator combines small size QAPs with known optimum solution into a larger size QAP instance. We call these instances composite QAPs (cQAPs), and we show that the cost function of composite QAPs is additively decomposable. We give mild conditions for which a composite QAP instance has known optimum solution. We generate cQAP instances using uniform distributions with different bounds for the component QAPs and for the rest of the cQAP elements. Numerical and analytical techniques that measure the difficulty of the composite QAP instances in comparison with other QAPs from the literature are introduced. These methods point out that some cQAP instances are difficult for local search with many local optimum of various values, low epistasis and non-trivial asymptotic behaviour.

In this paper, we present a simple algorithm to obtain mechanically SDP relaxations for any quadratic or linear program with bivalent variables, starting from an existing linear relaxation of the considered combinatorial problem. A significant advantage of our approach is that we obtain an improvement on the linear relaxation we start from. Moreover, we can take into account all the existing theoretical and practical experience accumulated in the linear approach. After presenting the rules to treat each type of constraint, we describe our algorithm, and then apply it to obtain semidefinite relaxations for three classical combinatorial problems: the K-CLUSTER problem, the Quadratic Assignment Problem, and the Constrained-Memory Allocation Problem. We show that we obtain better SDP relaxations than the previous ones, and we report computational experiments for the three problems.

One of the most notorious network design problems is the Quadratic Assignment Problem (QAP). We develop an heuristic algorithm for QAPs along with an M/G/C/C state dependent queueing model for capturing congestion in the traffic system interconnecting the nodes in the network. Computational results are also presented.

Quadratic bottleneck assignment problems (QBAP) are obtained by replacing the addition of cost terms in the objective function of a quadratic (sum) assignment problem by taking their maximum. Since the QBAP is an $\mathcal{NP}$ -hard problem, polynimially solvable special cases of the QBAP are of interest. In this paper we specify conditions on the cost matrices of QBAP leading to special cases which can be solved to optimality in polynomial time. In particular, the following three cases are discussed: (i) any permutation is optimal (constant QBAP), (ii) a certain specified permutation is optimal (constant permutation QBAP) and (iii) the solution can be found algorithmically by a polynomial algorithm. Moreover, the max-cone of bottleneck Monge matrices is investigated, its generating matrices are identified and it is used as a tool in proving polynomiality results.

Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e., the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods.

Topology control is an important technology of wireless ad hoc networks to achieve energy efficiency and fault tolerance. In this paper, we study the dual power assignment problem for 2-edge connectivity and 2-vertex connectivity in the symmetric graphical model which is a combinatorial optimization problem from topology control technology.

Given a (combinatorial) optimization problem and a feasible solution to it, the corresponding inverse optimization problem is to find a minimal adjustment of the cost function such that the given solution becomes optimum. Several such problems have been studied in the last twelve years. After formalizing the notion of an inverse problem and its variants, we present various methods for solving them. Then we discuss the problems considered in the literature and the results that have been obtained. Finally, we formulate some open problems.

In this paper we consider some inverse combinatorial optimization problems, that is, for a given set of feasible solutions of an optimization problem, we wish to know: under what weight vectors (or capacity vectors) will these feasible solutions become optimal solutions? We analysed shortest path problem, minimum cut problem, minimum spanning tree problem and maximum-weight matching problem, and found that in each of these cases, the solution set of the inverse problem is charaterized by solving another combinatorial optimization problem. The main tool in our approach is Fulkerson's theory of blocking and anti-blocking polyhedra with some necessary revisions.

Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) show that two graphs $$G_1,G_2$$ , on n vertices each, are isomorphic if and only if the feasible region of a certain linear program, LP-GI, intersects with the Quadratic Assignment Problem (QAP)-polytope in $${\mathbb {R}}^{(n^4+n^2)/2}$$ . The linear program LP-GI in Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) is obtained by relaxing an integer linear program whose feasible points correspond to the isomorphisms between $$G_1,G_2$$ . In this paper we take an analogous approach with the linear programs replaced with conic programs. A completely positive description of the QAP-polytope was obtained in Povh and Rendl (Discrete Optim 6(3):231–241, 2009). By adding the graph conditions to this description we get a completely positive formulation of the graph isomorphism problem. However, analogous to integer linear programs, it is NP-hard to optimize over the cone of completely positive matrices. So we relax this formulation by replacing the cone of completely positive matrices with the cone of positive semidefinite matrices. We observe that the resulting SDP is the Lovasz Theta function (Lovasz in IEEE Trans Inf Theory 25(1):1–7, 1979) of a graph product of $$G_1,G_2$$ and can be efficiently computed. We provide a natural heuristic that uses the SDP to solve the graph isomorphism problem. We run our heuristic on several pairs of non-isomorphic strongly regular graphs and find the results to be encouraging. Further, by adding the non-negativity constraints to the SDP, we obtain a doubly non-negative formulation, DNN-GI. We show that if the set of optimal points in DNN-GI contains a point of rank at most 3, then the given pair of graphs must be isomorphic.

In this paper we propose a geometric approach to solve the Graph Isomorphism (GI in short) problem. Given two graphs $$G_1, G_2$$ , the GI problem is to decide if the given graphs are isomorphic i.e., there exists an edge preserving bijection between the vertices of the two graphs. We propose an Integer Linear Program (ILP) that has a non-empty solution if and only if the given graphs are isomorphic. The convex hull of all possible solutions of the ILP has been studied in literature as the Quadratic Assignment Problem (QAP) polytope. We study the feasible region of the linear programming relaxation of the ILP and show that the given graphs are isomorphic if and only if this region intersects with the QAP-polytope. As a consequence, if the graphs are not isomorphic, the feasible region must lie entirely outside the QAP-polytope. We study the facial structure of the QAP-polytope with the intention of using the facet defining inequalities to eliminate the feasible region outside the polytope. We determine two new families of facet defining inequalities of the QAP-polytope and show that all the known facet defining inequalities are special instances of a general inequality. Further we define a partial ordering on each exponential sized family of facet defining inequalities and show that if there exists a common minimal violated inequality for all points in the feasible region outside the QAP-polytope, then we can solve the GI problem in polynomial time. We also study the general case when there are k such inequalities and give an algorithm for the GI problem that runs in time exponential in k.