Cut Tree Construction from Massive Graphs

The construction of cut trees (also known as Gomory-Hu trees) for a given graph enables the minimum-cut size of the original graph to be obtained for any pair of vertices. Cut trees are a powerful back-end for graph management and mining, as they support various procedures related to the minimum cut, maximum flow, and connectivity. However, the crucial drawback with cut trees is the computational cost of their construction. In theory, a cut tree is built by applying a maximum flow algorithm for $n$ times, where $n$ is the number of vertices. Therefore, naive implementations of this approach result in cubic time complexity, which is obviously too slow for today's large-scale graphs. To address this issue, in the present study, we propose a new cut-tree construction algorithm tailored to real-world networks. Using a series of experiments, we demonstrate that the proposed algorithm is several orders of magnitude faster than previous algorithms and it can construct cut trees for billion-scale graphs.Comment: Short version will appear at ICDM'1

Completing Partial Packings of Bipartite Graphs

Given a bipartite graph $H$ and an integer $n$, let $f(n;H)$ be the smallest integer such that, any set of edge disjoint copies of $H$ on $n$ vertices, can be extended to an $H$-design on at most $n+f(n;H)$ vertices. We establish tight bounds for the growth of $f(n;H)$ as $n \rightarrow \infty$. In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that $f(n;H) = o(n)$. This settles a long-standing open problem

Decomposition of multiple packings with subquadratic union complexity

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function $f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine at most $f(n)$ holes, which means that the complement of their union has at most $f(n)$ bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that $\mathcal{X}$ can be decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant depending only on $k$ and $f$.Comment: Small generalization of the main result, improvements in the proofs, minor correction

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Contact graphs of ball packings

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $\mathbb{R}^3$ is not greater than $13.955$. We also find new upper bounds for the average degree of contact graphs in $\mathbb{R}^4$ and $\mathbb{R}^5$

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201