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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 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 Lovász Theta function (Lovász 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. [ABSTRACT FROM AUTHOR]

In this paper we propose a geometric approach to solve the Graph Isomorphism (GI in short) problem. Given two graphs G1,G2, 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. [ABSTRACT FROM AUTHOR]

We consider special cases of the quadratic assignment problem (QAP) that are linearizable in the sense of Bookhold. We provide combinatorial characterizations of the linearizable instances of the weighted feedback arc set QAP, and of the linearizable instances of the traveling salesman QAP. As a by-product, this yields a new well-solvable special case of the weighted feedback arc set problem. [ABSTRACT FROM AUTHOR]

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 cQAPs is additively decomposable. We give mild conditions for which a cQAP 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 cQAP 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388-399, ). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), ) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

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. For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting the special structure of the relaxation. See e.g., Vandenverghe and Boyd (1995) for a similar approach for solving SDP problems arising from control applications. Numerical results are presented which indicate that the described methods yield at least competitive lower bounds. [ABSTRACT FROM AUTHOR]