Journal: Leibniz International Proceedings in Informatics (LIPIcs)

Abbreviation

Leibniz Int. Proc. Inform.

Publisher

Schloss Dagstuhl – Leibniz-Zentrum für Informatik

Journal Volumes

ISSN

1868-8969

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Publications 1 - 10 of 292
  • Distributed Arboricity-Dependent Graph Coloring via All-to-All Communication
    Item type: Conference Paper
    Ghaffari, Mohsen; Sayyadi, Ali (2019)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
    We present a constant-time randomized distributed algorithms in the congested clique model that computes an O(alpha)-vertex-coloring, with high probability. Here, alpha denotes the arboricity of the graph, which is, roughly speaking, the edge-density of the densest subgraph. Congested clique is a well-studied model of synchronous message passing for distributed computing with all-to-all communication: per round each node can send one O(log n)-bit message algorithm to each other node. Our O(1)-round algorithm settles the randomized round complexity of the O(alpha)-coloring problem. We also explain that a similar method can provide a constant-time randomized algorithm for decomposing the graph into O(alpha) edge-disjoint forests, so long as alpha <= n^{1-o(1)}.
  • The Maximum Label Propagation Algorithm onSparse Random Graphs
    Item type: Conference Paper
    Knierim, Charlotte; Lengler, Johannes; Pfister, Pascal; et al. (2019)
    Leibniz International Proceedings in Informatics (LIPIcs)
    In the Maximum Label Propagation Algorithm (Max-LPA), each vertex draws a distinct random label. In each subsequent round, each vertex updates its label to the label that is most frequent among its neighbours (including its own label), breaking ties towards the larger label. It is known that this algorithm can detect communities in random graphs with planted communities if the graphs are very dense, by converging to a different consensus for each community. In [Kothapalli et al., 2013] it was also conjectured that the same result still holds for sparse graphs if the degrees are at least C log n. We disprove this conjecture by showing that even for degrees n^epsilon, for some epsilon>0, the algorithm converges without reaching consensus. In fact, we show that the algorithm does not even reach almost consensus, but converges prematurely resulting in orders of magnitude more communities.
  • Bootstrap Percolation on Geometric Inhomogeneous Random Graphs
    Item type: Conference Paper
    Koch, Christoph; Lengler, Johannes (2016)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
    Geometric inhomogeneous random graphs (GIRGs) are a model for scale-free networks with underlying geometry. We study bootstrap percolation on these graphs, which is a process modelling the spread of an infection of vertices starting within a (small) local region. We show that the process exhibits a phase transition in terms of the initial infection rate in this region. We determine the speed of the process in the supercritical case, up to lower order terms, and show that its evolution is fundamentally influenced by the underlying geometry. For vertices with given position and expected degree, we determine the infection time up to lower order terms. Finally, we show how this knowledge can be used to contain the infection locally by removing relatively few edges from the graph. This is the first time that the role of geometry on bootstrap percolation is analysed mathematically for geometric scale-free networks.
  • The Niceness of Unique Sink Orientations
    Item type: Conference Paper
    Gärtner, Bernd; Thomas, Antonis (2016)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)
    Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of reachmaps and niceness of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least n^{Omega(2^n)} many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness.
  • Random sampling with removal
    Item type: Conference Paper
    Gärtner, Bernd; Lengler, Johannes; Szedlák, May (2016)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ 32nd International Symposium on Computational Geometry (SoCG 2016)
    Random sampling is a classical tool in constrained optimization. Under favorable conditions, the optimal solution subject to a small subset of randomly chosen constraints violates only a small subset of the remaining constraints. Here we study the following variant that we call random sampling with removal: suppose that after sampling the subset, we remove a fixed number of constraints from the sample, according to an arbitrary rule. Is it still true that the optimal solution of the reduced sample violates only a small subset of the constraints? The question naturally comes up in situations where the solution subject to the sampled constraints is used as an approximate solution to the original problem. In this case, it makes sense to improve cost and volatility of the sample solution by removing some of the constraints that appear most restricting. At the same time, the approximation quality (measured in terms of violated constraints) should remain high. We study random sampling with removal in a generalized, completely abstract setting where we assign to each subset R of the constraints an arbitrary set V(R) of constraints disjoint from R; in applications, V(R) corresponds to the constraints violated by the optimal solution subject to only the constraints in R. Furthermore, our results are parametrized by the dimension d, i.e., we assume that every set R has a subset B of size at most d with the same set of violated constraints. This is the first time this generalized setting is studied. In this setting, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of k elements. For a large range of values of k, the new upper bounds improve the previously best bounds for LP-type problems, which moreover had only been known in special cases. We show that this bound on special LP-type problems, can be derived in the much more general setting of violator spaces, and with very elementary proofs.
  • Convex hulls in polygonal domains
    Item type: Conference Paper
    Barba, Luis; Hoffmann, Michael; Korman, Matias; et al. (2018)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)
    We study generalizations of convex hulls to polygonal domains with holes. Convexity in Euclidean space is based on the notion of shortest paths, which are straight-line segments. In a polygonal domain, shortest paths are polygonal paths called geodesics. One possible generalization of convex hulls is based on the "rubber band" conception of the convex hull boundary as a shortest curve that encloses a given set of sites. However, it is NP-hard to compute such a curve in a general polygonal domain. Hence, we focus on a different, more direct generalization of convexity, where a set X is geodesically convex if it contains all geodesics between every pair of points x,y in X. The corresponding geodesic convex hull presents a few surprises, and turns out to behave quite differently compared to the classic Euclidean setting or to the geodesic hull inside a simple polygon. We describe a class of geometric objects that suffice to represent geodesic convex hulls of sets of sites, and characterize which such domains are geodesically convex. Using such a representation we present an algorithm to construct the geodesic convex hull of a set of O(n) sites in a polygonal domain with a total of n vertices and h holes in O(n^3h^{3+epsilon}) time, for any constant epsilon > 0.
  • Arc Diagrams, Flip Distances, and Hamiltonian Triangulations
    Item type: Conference Paper
    Cardinal, Jean; Hoffmann, Michael; Kusters, Vincent; et al. (2015)
    Leibniz International Proceedings in Informatics (LIPIcs)
    We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.
  • Drawings of Complete Multipartite Graphs up to Triangle Flips
    Item type: Conference Paper
    Aichholzer, Oswin; Chiu, Man-Kwun; Hoang, Hung P.; et al. (2023)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ 39th International Symposium on Computational Geometry (SoCG 2023)
    For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph Kₙ with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (also known as Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n¹⁶). The latter proof uses a Carath & eacute;odory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph Kₘ,ₙ minus two edges and Kₘ,ₙ plus one edge for any m, n ≥ 4, as well as Kₙ minus a 4-cycle for any n ≥ 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges.
  • The crossing tverberg theorem
    Item type: Conference Paper
    Fulek, Radoslav; Gärtner, Bernd; Kupavskii, Andrey; et al. (2019)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ 35th International Symposium on Computational Geometry (SoCG 2019)
    The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.
  • Sampling geometric inhomogeneous random graphs in linear time
    Item type: Conference Paper
    Bringmann, Karl; Keusch, Ralph; Lengler, Johannes (2017)
    Leibniz International Proceedings in Informatics (LIPIcs) ~ 25th Annual European Symposium on Algorithms (ESA 2017)
    Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs. With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we introduce a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies. We prove the following fundamental structural and algorithmic results on GIRGs. (1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in Omega(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits.
Publications 1 - 10 of 292