MADAM: a parallel exact solver for max-cut based on semidefinite programming and ADMM

Computational Optimization and Applications https://doi.org/10.1007/s10589-021-00310-6 MADAM: a parallel exact solver for max‑cut based on semidefinite programming and ADMM Timotej Hrga1 · Janez Povh1,2 Received: 11 September 2020 / Accepted: 18 August 2021 © The Author(s) 2021 Abstract We present MADAM, a parallel semidefinite-based exact solver for Max-Cut, a problem of finding the cut with the maximum weight in a given graph. The algorithm uses the branch and bound paradigm that applies the alternating direction method of multipliers as the bounding routine to solve the basic semidefinite relaxation strengthened by a subset of hypermetric inequalities. The benefit of the new approach is a less computationally expensive update rule for the dual variable with respect to the inequality constraints. We provide a theoretical convergence of the algorithm as well as extensive computational experiments with this method, to show that our algorithm outperforms state-of-the-art approaches. Furthermore, by combining algorithmic ingredients from the serial algorithm, we develop an efficient distributed parallel solver based on MPI. Keywords Semidefinite programming · Alternating direction method of multipliers · Maximum cut problem · Parallel computing 1 Introduction 1.1 Motivation The Max-Cut problem is a classical NP-hard optimization problem [1, 2] on graphs with the quadratic objective function and unconstrained binary variables. During the last decades, it has attracted the interest of many researchers, from its theoretical and algorithmic perspective to its applicability in different fields, for instance * Timotej Hrga Janez Povh 1 Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia 2 Institute of Mathematics, Physics and Mechanics Ljubljana, Ljubljana, Slovenia 13 Vol.:(0123456789) T. Hrga, J. Povh mathematics, physics, and computer science [3–5]. Several exact algorithmic approaches have been proposed in the literature, including the BiqMac [5] and BiqCrunch solvers [6], which are among the best solvers and are based on semidefinite programming. Lassere [7] has proved that the Max-Cut problem can be considered as a canonical model of linearly constrained linear and quadratic 0/1 programs. The transformation is based on the exact penalty approach and was further explored and advanced by Gusmeroli and Wiegele [8]. Even for instances of moderate size, it is considered a computational challenge to solve the Max-Cut to optimality. In practice, we typically solve such problems only approximately by using a heuristic or an approximation algorithm [9, 10]. However, to compare these algorithms and evaluate their performance, we still require the optimum solutions. Considering all this, solving increasingly large instances of Max-Cut to optimality on parallel computers is highly needed in scientific computing. 1.2 Problem formulation and notations The central problem that we consider is the Max-Cut problem, which can be defined as follows. For a given undirected graph G = (V, E) on n = |V| vertices and with edge weights we for e ∈ E , the Max-Cut problem asks to find a bipartition of the vertices such that the sum of the weights of the edges across the bipartition is maximized. Let A = (aij ) denote the weighted adjacency matrix with aij = aji = we for edge e = {i, j} ∈ E and zero otherwise. Encoding the partitions by vectors z = {0, 1}n, we obtain the following unconstrained binary quadratic optimization problem formulation for Max-Cut: maximize n ∑ aij zi (1 − zj ) = zT L0 z subject to z ∈ {0, 1}n , i,j (1) where L0 is the Laplacian matrix of the graph defined by L0 = diag(Ae) − A. By e we denote the vector of all ones. Note that due to the symmetry of the problem, we can fix the last element of vector z to zero and thus remove the last row and column of L0 to obtain the matrix L̂0 ∈ ℝ(n−1)×(n−1). The inner product on the space of symmetric matrices is given by ∑ X ∙ Y = ⟨X, Y⟩√= tr(XY) =� i,j Xij Yij and the associated Frobenious norm is defined ∑ 2 by ‖X‖F = ⟨X, X⟩ = Using the property of inner product i,j Xij . zT L0 z = ⟨L0 , zzT ⟩, we can reformulate problem (1) as: [ ] [ T ] zz z L̂0 0 maximize ∙ T z 1 0 0 subject to z ∈ {0, 1} n−1 (2) . We increase the dimension of the problem by one, since the last column of the solution matrix of a semidefinite relaxation will be used for determining the next branching variable. 13 MADAM: a parallel exact solver for max‑cut based on semidefinite… In the following, we denote the diagonal matrix, which has v on its diagonal, by Diag (v), and the vector obtained by extracting the main diagonal from the matrix X is denoted by diag (X). For the given symmetric matrices Ai , i = 1, … , m, let A ∶ Sn → ℝm denote the linear operator mapping n × n symmetric matrices to ℝm ∑ with A(X)i = ⟨Ai , X⟩. Its adjoint is well known to be AT (y) = i yi Ai . For some real number a, we denote its nonnegative part by a(+ =)max{a, ( )0}. This definition is extended to vectors as follows: if x ∈ ℝn, then x+ i = xi +, for i = 1, … , n. We denote the projection of some symmetric matrix X onto the positive semidefinite cone by X+ and its projection onto the negative semidefinite cone by X−. More specifically, if the eigendecomposition of X is given by X = S Diag (𝜆)ST with the eigenvalues 𝜆 ∈ ℝn and orthogonal matrix S ∈ ℝn×n, then we have X+ = S Diag (𝜆+ )ST , X− = S Diag (𝜆− )ST . 1.3 Related work and our contribution Over the decades, many methods have been proposed for finding exact solutions of Max-Cut. Some of them are linear programming-based methods [4, 11], which work particularly well when the underlying graph is sparse. Other algorithms combine semidefinite programming with the polyhedral approach [12] to strengthen the basic SDP relaxation with cutting planes. Two of them, the BiqMac [5] and BiqCrunch [6] solvers, respectively, have turned out to be the best performing solvers for MaxCut in the last decade, also when compared to commercial solvers. Both solvers utilize the branch and bound (B&B) paradigm. However, the distinction is in using different algorithms to solve the underlying SDP relaxation. Furthermore, they do not use any parallelization. For other recent computational approaches, the reader is referred to [13]. Recently, we have developed a BiqBin solver [14] with other authors for the class of binary quadratic problems with linear constraints, which includes the Max-Cut problem. This solver uses the exact penalty approach to reformulate every instance of the binary quadratic problem into an instance of the Max-Cut problem and then solves it using an enhanced version of BiqMac where the underlying SDP relaxations are tight due to the inclusion of hypermetric inequalities. Additionally, the B&B part of BiqBin has been improved compared to BiqMac, and the solver has been fully parallelized. Extensive numerical evidence shows that BiqBin outperforms BiqMac, BiqCrunch, GUROBI, a (...truncated)