Graph Coloring via Degeneracy in Streaming and Other Space-Conscious Models

We study the problem of coloring a given graph using a small number of colors in several well-established models of computation for big data. These include the data streaming model, the general graph query model, the massively parallel computation (MPC) model, and the CONGESTED-CLIQUE and the LOCAL models of distributed computation. On the one hand, we give algorithms with sublinear complexity, for the appropriate notion of complexity in each of these models. Our algorithms color a graph $G$ using about $\kappa(G)$ colors, where $\kappa(G)$ is the degeneracy of $G$: this parameter is closely related to the arboricity $\alpha(G)$. As a function of $\kappa(G)$ alone, our results are close to best possible, since the optimal number of colors is $\kappa(G)+1$. On the other hand, we establish certain lower bounds indicating that sublinear algorithms probably cannot go much further. In particular, we prove that any randomized coloring algorithm that uses $\kappa(G)+1$ many colors, would require $\Omega(n^2)$ storage in the one pass streaming model, and $\Omega(n^2)$ many queries in the general graph query model, where $n$ is the number of vertices in the graph. These lower bounds hold even when the value of $\kappa(G)$ is known in advance; at the same time, our upper bounds do not require $\kappa(G)$ to be given in advance.Comment: 26 page

Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm

Toward more localized local algorithms: removing assumptions concerning global knowledge

International audienceNumerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and (∆+1)-coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆ 2)-coloring algorithm by Linial [28]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, local algorithms generally use good estimations of one or more global parameters of the network, e.g., the maximum degree ∆ or the number of nodes n. This paper provides a method for transforming a non-uniform local algorithm into a uniform one. Furthermore , the resulting algorithm enjoys the same asymp-totic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest

Local Algorithms for Bounded Degree Sparsifiers in Sparse Graphs

In graph sparsification, the goal has almost always been of global nature: compress a graph into a smaller subgraph (sparsifier) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in many cases leads to improvements in the overall runtime and memory. This paper studies sparsifiers that have bounded (maximum) degree, and are thus locally sparse, aiming to improve local measures of runtime and memory. To improve those local measures, it is important to be able to compute such sparsifiers locally. We initiate the study of local algorithms for bounded degree sparsifiers in unweighted sparse graphs, focusing on the problems of vertex cover, matching, and independent set. Let eps > 0 be a slack parameter and alpha ge 1 be a density parameter. We devise local algorithms for computing: 1. A (1+eps)-vertex cover sparsifier of degree O(alpha / eps), for any graph of arboricity alpha.footnote{In a graph of arboricity alpha the average degree of any induced subgraph is at most 2alpha.} 2. A (1+eps)-maximum matching sparsifier and also a (1+eps)-maximal matching sparsifier of degree O(alpha / eps, for any graph of arboricity alpha. 3. A (1+eps)-independent set sparsifier of degree O(alpha^2 / eps), for any graph of average degree alpha. Our algorithms require only a single communication round in the standard message passing model of distributed computing, and moreover, they can be simulated locally in a trivial way. As an immediate application we can extend results from distributed computing and local computation algorithms that apply to graphs of degree bounded by d to graphs of arboricity O(d / eps) or average degree O(d^2 / eps), at the expense of increasing the approximation guarantee by a factor of (1+eps). In particular, we can extend the plethora of recent local computation algorithms for approximate maximum and maximal matching from bounded degree graphs to bounded arboricity graphs with a negligible loss in the approximation guarantee. The inherently local behavior of our algorithms can be used to amplify the approximation guarantee of any sparsifier in time roughly linear in its size, which has immediate applications in the area of dynamic graph algorithms. In particular, the state-of-the-art algorithm for maintaining (2-eps)-vertex cover (VC) is at least linear in the graph size, even in dynamic forests. We provide a reduction from the dynamic to the static case, showing that if a t-VC can be computed from scratch in time T(n) in any (sub)family of graphs with arboricity bounded by alpha, for an arbitrary t ge 1, then a (t+eps)-VC can be maintained with update time frac{T(n)}{O((n / alpha) cdot eps^2)}, for any eps > 0. For planar graphs this yields an algorithm for maintaining a (1+eps)-VC with constant update time for any constant eps > 0

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem