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1. Constructing near spanning trees with few local inspections

   Levi, Reut ; Moshkovitz, Guy ; Ron, Dana ; Rubinfeld, Ronitt ; Shapira, Asaf ( March 2017 , Random structures & algorithms)

   Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph math formula consisting of math formula edges (for a prespecified constant math formula), where the decision for different edges should be consistent with the same subgraph math formula. Can this task be performed by inspecting only a constant number of edgesmore »in G? Our main results are: We show that if every t-vertex subgraph of G has expansion math formula then one can (deterministically) construct a sparse spanning subgraph math formula of G using few inspections. To this end we analyze a “local” version of a famous minimum-weight spanning tree algorithm. We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion math formula. We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.« less
2. Local Computation Algorithms for Spanners

   Parter, Merav ; Rubinfeld, Ronitt ; Vakilian, Ali ; Yodpinyanee, Anak ( January 2019 , Innovations in Theoretical Computer Science (ITCS))

   A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a singlemore »processor to store. To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge (u,v)∈E belongs to the output spanner. Such LCAs give the user the `illusion' that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present the following results: -For general n-vertex graphs and r∈{2,3}, there exists an LCA for (2r−1)-spanners with O˜(n1+1/r) edges and sublinear probe complexity of O˜(n1−1/2r). These size/stretch tradeoffs are best possible (up to polylogarithmic factors). -For every k≥1 and n-vertex graph with maximum degree Δ, there exists an LCA for O(k2) spanners with O˜(n1+1/k) edges, probe complexity of O˜(Δ4n2/3), and random seed of size polylog(n). This improves upon, and extends the work of [Lenzen-Levi, 2018]. We also complement our results by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with o(m) edges.« less
3. Sampling Multiple Edges Efficiently

   Eden, T. ; Mossel, S. ; Rubinfeld, R. ( January 2021 , Random)

   We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ϵ-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ∗(n/ √ m) for sampling a single edge in generalmore »graphs (where O ∗(·) suppresses poly(1/ϵ) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O∗(√ q · (n/ √ m)), which is strictly preferable to O∗(q · (n/ √ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal.« less
4. On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

   Cohen-Addad, Vincent ; Filtser, Arnold ; Klein, Philip N ; Le, Hung ( October 2020 , Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science)

   Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a “small-complexity” graph that approximately preserves distances between pairs of points of the metric. We show the two following structural results for minor-free metrics: 1) Construction of a light subset spanner. Given a subset of vertices called terminals, and ϵ, in polynomial time wemore »construct a sub graph that preserves all pairwise distances between terminals up to a multiplicative 1+ϵ factor, of total weight at most Oϵ(1) times the weight of the minimal Steiner tree spanning the terminals. 2) Construction of a stochastic metric embedding into low treewidth graphs with expected additive distortion ϵD. Namely, given a minor-free graph G=(V,E,w) of diameter D, and parameter ϵ, we construct a distribution D over dominating metric embeddings into treewidth-Oϵ(log n) graphs such that ∀u,v∈V, Ef∼D[dH(f(u),f(v))]≤dG(u,v)+ϵD. Our results have the following algorithmic consequences: (1) the first efficient approximation scheme for subset TSP in minor-free metrics; (2) the first approximation scheme for bounded-capacity vehicle routing in minor-free metrics; (3) the first efficient approximation scheme for bounded-capacity vehicle routing on bounded genus metrics. En route to the latter result, we design the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth).« less
5. Universal Geometric Graphs

   https://doi.org/10.1007/978-3-030-60440-0_14  

   Frati, Fabrizio ; Hoffmann, Michael ; Toth, Csaba D. ( October 2020 , Graph-Theoretic Concepts in Computer Science (WG 2020))

   We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is universal for a class H of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in H . Our main result is that there exists a geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an n-vertexmore »graph with O(nlogn) edges that contains every n-vertex forest as a subgraph. Our O(nlogn) bound on the number of edges is asymptotically optimal. We also prove that, for every h>0 , every n-vertex convex geometric graph that is universal for the class of the n-vertex outerplanar graphs has Ωh(n2−1/h) edges; this almost matches the trivial O(n2) upper bound given by the n-vertex complete convex geometric graph. Finally, we prove that there is an n-vertex convex geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex caterpillars.« less