Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on $n$ nodes and $m$ edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each $k$ there is a threshold graph on the same number of nodes and edges whose sum of the $k$ largest Laplacian eigenvalues exceeds that of the $k$ largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the $k$ largest Laplacian eigenvalues

A Preconditioned Iterative Interior Point Approach to the Conic Bundle Subproblem

The conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.Comment: 29+9 pages, 4 figure

Network Models with Convex Cost Structure like Bundle Methods

For three rather diverse applications (truck scheduling for inter warehouse logistics, university-course timetabling, operational train timetabling) that contain integer multi-commodity flow as a major modeling element we present a computational comparison between a bundle and a full linear programming (LP) approach for solving the basic relaxations. In all three cases computing the optimal solutions with LP standard solvers is computationally very time consuming if not impractical due to high memory consumption while bundle methods produce solutions of sufficient but low accuracy in acceptable time. The rounding heuristics generate comparable results for the exact and the approximate solutions, so this entails no loss in quality of the final solution. Furthermore, bundle methods facilitate the use of nonlinear convex cost functions. In practice this not only improves the quality of the relaxation but even seems to speed up convergence of the method

An interlacing property of the signless Laplacian of threshold graphs

We show that for threshold graphs, the eigenvalues of the signless Laplacian matrix interlace with the degrees of the vertices. As an application, we show that the signless Brouwer conjecture holds for threshold graphs, i.e., for threshold graphs the sum of the k largest eigenvalues is bounded by the number of edges plus k + 1 choose 2.Comment: 14 pages, 3 figure

An Interior Point Method for Semidefinite Programming and Max-Cut Bounds

The presentation is divided into two parts. In Part I we introduce a primal-dual path-following interior point method for positive semidefinite programming based on the linearization of ZX \Gamma¯I . This linearization leads to unsymmetric updates \DeltaX for X but by using the symmetric part of \DeltaX only, convergence can be guaranteed. The algorithm is well suited for the predictor-corrector approach proposed by Mehrotra [48] and is very efficient in practice. In Part II we employ the code from Part I for computing upper bounds for the max-cut problem. In particular we combine the eigenvalue upper bound based on the maximal eigenvalue of the Laplacian of the graph with the polyhedral upper bound delivered by triangle inequalities. We also investigate the effect of larger clique inequalities. The bounds show considerable improvements to previous results on complete graphs. For the traditional approach the relaxation over the triangle inequalities was computationally too expensi..

Cutting Aluminium Coils with High Length Variabilities

A case study of a cutting stock problem in an aluminium mill is presented. Orders have release dates, due dates, a total length and may be delivered in any number of coils within rather large length intervals. A variety of different cutting machines is available, hierarchical cuts may be necessary to produce small widths. The mill is capable of producing custom--made coils within certain bounds but there is a declared preference for standard widths. The task is to group the orders into coils which can be produced by the mill and slit by the machines. Waste should be minimized, the dates should be obeyed, the load of the machines should be balanced. In spite of the fact that column generation is not possible the problem is efficiently solved by a multi--pattern approach using linear programming