1,844 research outputs found

    Dynamic Connectivity in Disk Graphs

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    Let S ⊆ R2 be a set of n sites in the plane, so that every site s ∈ S has an associated radius rs > 0. Let D(S) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii rs , rt intersect. Our goal is to design data structures that maintain the connectivity structure of D(S) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix rs = 1, for all sites s ∈ S. For this case, we describe a data structure that has O(log2 n) amortized update time and O(log n/ log log n) query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a parameter Ψ that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time O(Ψ log4 n) and query time O(log n/ log log n). This improves the currently best update time by a factor of Ψ. In the incremental case, we achieve logarithmic dependency on Ψ, with a data structure that has O(α(n)) amortized query time and O(log Ψ log4 n) amortized expected update time, where α(n) denotes the inverse Ackermann function. For the decremental setting, we first develop an efficient decremental disk revealing data structure: given two sets R and B of disks in the plane, we can delete disks from B, and upon each deletion, we receive a list of all disks in R that no longer intersect the union of B. Using this data structure, we get decremental data structures with a query time of O(log n/ log log n) that supports deletions in O(n log Ψ log4 n) overall expected time for disk graphs with bounded radius ratio Ψ and O(n log5 n) overall expected time for disk graphs with arbitrary radii, assuming that the deletion sequence is oblivious of the internal random choices of the data structures

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    (Almost) Ruling Out SETH Lower Bounds for All-Pairs Max-Flow

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    The All-Pairs Max-Flow problem has gained significant popularity in the last two decades, and many results are known regarding its fine-grained complexity. Despite this, wide gaps remain in our understanding of the time complexity for several basic variants of the problem. In this paper, we aim to bridge this gap by providing algorithms, conditional lower bounds, and non-reducibility results. Our main result is that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out n4o(1)n^{4-o(1)} time algorithms under a hypothesis called NSETH. In particular, to obtain our result for the setting of undirected graphs with unit node-capacities, we design a new randomized O(m2+o(1))O(m^{2+o(1)}) time combinatorial algorithm, improving on the recent O(m11/5+o(1))O(m^{11/5+o(1)}) time algorithm [Huang et al., STOC 2023] and matching their m2o(1)m^{2-o(1)} lower bound (up to subpolynomial factors), thus essentially settling the time complexity for this setting of the problem. More generally, our main technical contribution is the insight that stst-cuts can be verified quickly, and that in most settings, stst-flows can be shipped succinctly (i.e., with respect to the flow support). This is a key idea in our non-reducibility results, and it may be of independent interest

    From coordinate subspaces over finite fields to ideal multipartite uniform clutters

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    Take a prime power qq, an integer n2n\geq 2, and a coordinate subspace SGF(q)nS\subseteq GF(q)^n over the Galois field GF(q)GF(q). One can associate with SS an nn-partite nn-uniform clutter C\mathcal{C}, where every part has size qq and there is a bijection between the vectors in SS and the members of C\mathcal{C}. In this paper, we determine when the clutter C\mathcal{C} is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether qq is 2,42,4, a higher power of 22, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of C\mathcal{C} depends solely on the underlying matroid of SS. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and τ=2\tau=2 Conjectures for this class of clutters.Comment: 32 pages, 6 figure

    Improved Approximation Bounds for Minimum Weight Cycle in the CONGEST Model

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    Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of minimum weight in a graph G=(V,E)G=(V,E). This is a fundamental graph problem with classical sequential algorithms that run in O~(n3)\tilde{O}(n^3) and O~(mn)\tilde{O}(mn) time where n=Vn=|V| and m=Em=|E|. In recent years this problem has received significant attention in the context of hardness through fine grained sequential complexity as well as in design of faster sequential approximation algorithms. For computing minimum weight cycle in the distributed CONGEST model, near-linear in nn lower and upper bounds on round complexity are known for directed graphs (weighted and unweighted), and for undirected weighted graphs; these lower bounds also apply to any (2ϵ)(2-\epsilon)-approximation algorithm. This paper focuses on round complexity bounds for approximating MWC in the CONGEST model: For coarse approximations we show that for any constant α>1\alpha >1, computing an α\alpha-approximation of MWC requires Ω(nlogn)\Omega (\frac{\sqrt n}{\log n}) rounds on weighted undirected graphs and on directed graphs, even if unweighted. We complement these lower bounds with sublinear O~(n2/3+D)\tilde{O}(n^{2/3}+D)-round algorithms for approximating MWC close to a factor of 2 in these classes of graphs. A key ingredient of our approximation algorithms is an efficient algorithm for computing (1+ϵ)(1+\epsilon)-approximate shortest paths from kk sources in directed and weighted graphs, which may be of independent interest for other CONGEST problems. We present an algorithm that runs in O~(nk+D)\tilde{O}(\sqrt{nk} + D) rounds if kn1/3k \ge n^{1/3} and O~(nk+k2/5n2/5+o(1)D2/5+D)\tilde{O}(\sqrt{nk} + k^{2/5}n^{2/5+o(1)}D^{2/5} + D) rounds if k<n1/3k<n^{1/3}, and this round complexity smoothly interpolates between the best known upper bounds for approximate (or exact) SSSP when k=1k=1 and APSP when k=nk=n

    Algorithms for Geometric Facility Location: Centers in a Polygon and Dispersion on a Line

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    We study three geometric facility location problems in this thesis. First, we consider the dispersion problem in one dimension. We are given an ordered list of (possibly overlapping) intervals on a line. We wish to choose exactly one point from each interval such that their left to right ordering on the line matches the input order. The aim is to choose the points so that the distance between the closest pair of points is maximized, i.e., they must be socially distanced while respecting the order. We give a new linear-time algorithm for this problem that produces a lexicographically optimal solution. We also consider some generalizations of this problem. For the next two problems, the domain of interest is a simple polygon with n vertices. The second problem concerns the visibility center. The convention is to think of a polygon as the top view of a building (or art gallery) where the polygon boundary represents opaque walls. Two points in the domain are visible to each other if the line segment joining them does not intersect the polygon exterior. The distance to visibility from a source point to a target point is the minimum geodesic distance from the source to a point in the polygon visible to the target. The question is: Where should a single guard be located within the polygon to minimize the maximum distance to visibility? For m point sites in the polygon, we give an O((m + n) log (m + n)) time algorithm to determine their visibility center. Finally, we address the problem of locating the geodesic edge center of a simple polygon—a point in the polygon that minimizes the maximum geodesic distance to any edge. For a triangle, this point coincides with its incenter. The geodesic edge center is a generalization of the well-studied geodesic center (a point that minimizes the maximum distance to any vertex). Center problems are closely related to farthest Voronoi diagrams, which are well- studied for point sites in the plane, and less well-studied for line segment sites in the plane. When the domain is a polygon rather than the whole plane, only the case of point sites has been addressed—surprisingly, more general sites (with line segments being the simplest example) have been largely ignored. En route to our solution, we revisit, correct, and generalize (sometimes in a non-trivial manner) existing algorithms and structures tailored to work specifically for point sites. We give an optimal linear-time algorithm for finding the geodesic edge center of a simple polygon

    Two-sets cut-uncut on planar graphs

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    We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein, one is given an undirected planar graph GG and two sets of vertices SS and TT. The question is, what is the minimum number of edges to remove from GG, such that we separate all of SS from all of TT, while maintaining that every vertex in SS, and respectively in TT, stays in the same connected component. We show that this problem can be solved in time 2S+TnO(1)2^{|S|+|T|} n^{O(1)} with a one-sided error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut remains fixed-parameter tractable even when parameterized by the number rr of faces in the plane graph covering the terminals STS \cup T, by providing an algorithm of running time 4r+O(r)nO(1)4^{r + O(\sqrt r)} n^{O(1)}.Comment: 22 pages, 5 figure
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