We provide a complete description of important geometric invariants of the Laplacian lattice of a multigraph under the distance function induced by a regular simplex, namely Voronoi Diagram, Delaunay Triangulation, Delaunay Polytope and its combinatorial structure, Shortest Vectors, Covering and Packing Radius. We use this information to obtain the following results: i. Every multigraph defines a Delaunay triangulation of its Laplacian lattice and this Delaunay triangulation contains complete information of the multigraph up to isomorphism. ii. The number of multigraphs with a given Laplacian lattice is controlled, in particular upper bounded, by the number of different Delaunay triangulations. iii. We obtain formulas for the covering and packing densities of a Laplacian lattice and deduce that in the space of Laplacian lattices of undirected connected multigraphs, the Laplacian lattices of highly connected multigraphs such as Ramanujan multigraphs possess good covering and packing properties.
A substantial body of work is devoted to the study of lattices constructed from graphs, the most well studied ones being the lattice of integral cuts and the lattice of integral flows. This line of investigation was pioneered by the work of Bacher, Harpe and Nagnibeda @cite_3 where they provide a combinatorial interpretation of various parameters of the lattice of integral flows and the lattice of integral cuts in the Euclidean distance function, for example they show that the square norm of the shortest vector of the lattice of integral flows is equal to the girth of the graph. In contrast, the study of the Laplacian lattice under simplicial distance function gives rise to some new and interesting phenomenon, that are not seen in the Euclidean case and in some cases provides more refined information about the graph. Here we note such instances:
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Arora, Rao and Vazirani [2] showed that the standard semi-definite programming (SDP) relaxation of the Sparsest Cut problem with the triangle inequality constraints has an integrality gap of O(√log n). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the ARV-Conjecture) by constructing an Ω(log log n) integrality gap instance. Khot and Vishnoi [16] had earlier disproved the non-uniform version of the ARV-Conjecture.A simple "stretching" of the integrality gap instance for the Sparsest Cut problem serves as an Ω(log log n) integrality gap instance for the SDP relaxation of the Minimum Linear Arrangement problem. This SDP relaxation was considered in [6, 11], where it was shown that its integrality gap is bounded from above by O(√log n log log n). The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference oper- ator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient. The random walks we consider are defined as follows. Let K be a connected graph (i.e. a one dimensional simplicial complex). For a vertex x E K, let m(x) denote the number of edges emanating from x. The probability that a particle moves from x to another vertex y E K is l m(x) if x and y are connected by an edge and it is zero otherwise. As observed by Courant, Friedrichs and Lewy (CFL) for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the Laplacian For some positive constant , we give a (3 2- )-approximation algorithm for the following problem: given a graph G_0=(V,E_0), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G_0. The result improves on the 3 2-approximation algorithm due to Christofides [C76] for this special case. Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest. General random coding theorems for lattices are derived from the Minkowski-Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski-Hlawka theorem itself is obtained as the limit, for p spl rarr spl infin , of a simple lemma for linear codes over GF(p) used with p-level amplitude modulation. The relation between the combinatorial packing of solid bodies and the information-theoretic "soft packing" with arbitrarily small, but positive, overlap is illuminated. The "soft-packing" results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda-Poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 (P N).