The edge-bandwidth minimization problem by M.C.E. van der Ven BSc. (286463) A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Operations Research and Management Science Supervised by Dr. R. Sotirov Department of Econometrics & Operations Research Tilburg School of Economics and Management Tilburg University January 10, 2012 Abstract In this thesis, we investigate the edge-bandwidth of some well known classes of graphs. Two known lower bounds for the graph bandwidth problem are tested for lattice graphs and triangular graphs. One of these is based on semidefinite programming (SDP) re- laxations and one is based on a reformulation-linearization technique for the quadratic assignment problem. The results of these relaxations are compared to the upper bounds constructed by a new heuristic exploiting the structure of lattice graphs. Furthermore, the edge-bandwidths of the hypercube graph Q3 and the Petersen graph are determined. Contents 1 Introduction 3 2 The bandwidth problem 5 2.1 The bandwidth minimization problem . 5 2.2 Edge-bandwidth . 6 2.3 Cyclic bandwidth . 7 2.4 Antibandwidth . 8 3 Known bandwidths and edge-bandwidths 9 3.1 Path graph . 9 3.2 Complete graph . 9 3.3 Complete bipartite graph . 10 3.4 Rectangular grid graph . 10 3.5 Hypercube graph . 10 3.6 Petersen graph . 13 3.7 Triangular grid graph . 13 3.8 Theta graph . 13 3.9 Hamming graph . 14 4 Known bounds 16 4.1 Relation bandwidth and edge-bandwidth . 16 4.2 Harper's boundary bound . 17 4.3 Chung's density bound . 18 5 Semidefinite programming 19 5.1 Linear algebra . 19 5.2 Semidefinite programs . 19 6 Relaxations 22 6.1 SDP for graph bandwidth . 22 6.2 Reformulation-linearization technique . 23 1 7 Heuristic 25 7.1 Assumptions . 25 7.2 Local search . 25 7.3 Improvements . 26 8 Numerical results 27 8.1 Theoretical lower bounds . 27 8.2 Computational lower bounds . 27 8.3 Heuristic results . 31 9 Conclusions and recommendations 33 9.1 Conclusions . 33 9.2 Recommendations for future research . 34 2 Chapter 1 Introduction In this thesis the bandwidth problem is investigated as presented by Harper [20] in 1985. We provide a short survey on the bandwidth and the edge-bandwidth. After this, we test the quality of several lower bounds and a new heuristic. In Chapter 2, the bandwidth minimization problem and its variations, i.e. the edge- bandwidth, the cyclic bandwidth and the anti-bandwidth, are introduced. Further, the bandwidth minimization problem is formulated as the quadratic assignment problem (QAP). In Chapter 3, different classes of graphs and their line graphs are summarized to- gether with their bandwidth. The expressions for the value of the bandwidth and the edge-bandwidth or the best known bounds of the corresponding graphs are given. Chapter 4 elaborates on known bounds for the bandwidth problem and the rela- tionship between the bandwidth and the edge-bandwidth of a graph. Chung's density bound and Harper's boundary bound are discussed. Chapter 5 gives a brief introduction to semidefinite programming (SDP). The basics from linear algebra are explained, such as semidefinite matrices, to construct semidefi- nite programs. In Chapter 6, two relaxations are presented for the QAP that are used to compute lower bounds for the edge-bandwidth. First, the SDP relaxations that are introduced by Zhao et al. in [54] and by de Klerk et al. in [37] are presented. Second, the relaxation-linearization technique introduced by [1] is presented. In Chapter 7, a heuristic is designed to construct an upper bound on the bandwidth. This heuristic uses the properties from the adjacency matrix of the lattice graph, but shows to work for other types of graphs too. The numerical results of the relaxations from Chapter 6 and the heuristic from Chapter 7 are discussed in Chapter 8. Finally, Chapter 9 gives a concise overview of the results found in this thesis. The chapter will conclude with a section on recommendations for future research in this field. Our main focus lies on the lattice graph, the line graph of the complete bipartite graph. The bandwidth of the lattice graph is unknown. However, to verify our approach, we carry out computations on triangular graphs, for which the bandwidth is known. We 3 find new results on the exact values of the bandwidth for small lattice graphs (up to size 12), the line graph of the hypercube graph Q3 and the line graph of the Petersen graph, using the techniques from Chapter 6 and 7. The edge-bandwidths of graphs are computed by taking into consideration that the edge-bandwidth of a graph is equal to the bandwidth of its line graph. 4 Chapter 2 The bandwidth problem 2.1 The bandwidth minimization problem Let G = (V; E) be a simple graph with jV j = n and jEj = m. A simple graph is an unweighed, undirected graph that has no loops. A layout or labeling f of the vertices of G is a bijection from V to f1; :::; ng. The bandwidth of a labeling f is B(f; G) = maxfjf(u) − f(v)j : uv 2 E(G)g: The bandwidth of a graph is the minimum bandwidth over all labelings, so B(G) is given by B(G) = minfB(f; G)g: f This notion was first introduced by Harper in 1985 [20]. The bandwidth problem can be described in the following way. Let A be the adja- cency matrix of G. The problem is finding a permutation of the rows and columns of A that brings all the non-zero elements of A in a band as close as possible to the diagonal. In other words, minimize the bandwidth of XAX−1 where X 2 Rn×n is a permutation matrix. Define the matrix B = (bij) as follows: { 1 for ji − jj > k b := (1) ij 0 otherwise. We can now formulate the bandwidth minimization problem. Define µ∗ as µ∗ := min tr(AXBXT ) (2) s.t. X 2 Πn; where tr(X) is the trace of matrix X and Πn is the set of permutation matrices n×n T Πn := fX 2 R : X · en = en;X · en = en; xij 2 f0; 1gg ∗ where en is the all ones vector of length n. Then B(G) > maxfk : µ > 0g. 5 Alternatively, the bandwidth minimization problem can be formulated as a maxi- mization problem, where B = (bij): { 0 for ji − jj > k b = ij 1 otherwise. This is the opposite of the matrix B in the minimization problem, where the zeros are replaced by ones and the ones by zeros. The maximization problem is as follows: γ∗ := max tr(AXBXT ) (3) s.t. X 2 Πn: Then B(G) = maxfk : γ∗ < 2jEjg, see [37]. Optimization problems of the form (1) and (2) are known as the quadratic assign- ment problem (QAP). The QAP was introduced by Koopmans and Beckmann in 1957 in the context of analysis of the location of economic activity. It is a model for various real life problems, such as hospital lay-out or the assignment of letters to a typewriter. The Dutch scientist Koopmans was awarded the Nobel Prize in economic science in 1975 for his contributions to the theory of optimum allocation of resources. The formulation of the QAP is as follows. Assign n facilities to n locations such that the cost of transportation between facilities is minimized. Take the n × n symmetric matrix A = (aij) where aij is the cost of transporting one unit from location i to location j. Take the n × n symmetric matrix B = (bpq) where (bpq) is the number of units to be transported between facility p and facility q. Finally, the matrix n × n matrix C = (cip) is the matrix that gives the construction costs of building facility p on location i. The assignment of the facilities to locations is given by a permutation matrix Πn. The Koopmans-Beckmann problem minimizes the total costs. This can be written as: µ∗ = min tr(AXBXT + CXT ) (2) s.t. X 2 Πn: The bandwidth problem is proven to be NP-hard [39]. The bandwidth problem was originally used for modeling the problem of re-ordering the rows and columns of a sparse matrix so that the non-zero entries of the matrix form a band along the diagonal with a minimum width. Computations on such a sparse matrix can be carried out more efficiently. Studies on this problem date back to the 1950s. Sparse matrix computations are used in solving linear equations and differential equations. The bandwidth problem has more applications, for example VLSI (very-large-scale integration) layout, intercon- nection networks and the constraint satisfaction problem. Besides this, the bandwidth problem has applications in the field of physics and biology. For an extensive survey on the applications of the bandwidth, see [30]. There are a few variations on the bandwidth problem which will be discussed in the rest of this chapter. 2.2 Edge-bandwidth The edge-bandwidth problem, introduced by Hwang and Lagarias, labels the edges instead of the vertices [27]. A labeling h of the edges is a bijection from E to f1; :::; mg: 6 The edge-bandwidth of a labeling is the maximum difference between the labels of a pair of incident edges. The bandwidth of an edge-labeling h of E is given by: B0(h; G) = maxfjh(uv) − h(vw)j : uv; vw 2 E(G)g: The edge-bandwidth of a graph G is B0(G) = minfB0(h; G)g: h The edge-bandwidth of a graph is related to the bandwidth of its line graph.