183 research outputs found
Streaming algorithms for 2-coloring uniform hypergraphs
We consider the problem of two-coloring n-uniform hypergraphs. It is known that any such hypergraph with at most 1/10ân/ln n 2n hyperedges can be two-colored [7]. In fact, there is an efficient (requiring polynomial time in the size of the input) randomized algorithm that produces such a coloring. As stated [7], this algorithm requires random access to the hyperedge set of the input hypergraph. In this paper, we show that a variant of this algorithm can be implemented in the streaming model (with just one pass over the input), using space O(|V|B), where V is the vertex set of the hypergraph and each vertex is represented by B bits. (Note that the number of hyperedges in the hypergraph can be superpolynomial in |V|, and it is not feasible to store the entire hypergraph in memory.) We also consider the question of the minimum number of hyperedges in non-two-colorable n-uniform hypergraphs. Erdos showed that there exist non-2-colorable n-uniform hypegraphs with O(n2 2n) hyperedges and Î(n2) vertices. We show that the choice Î(n2) for the number of vertices in Erdös's construction is crucial: any hypergraph with at most 2n2/t vertices and 2nexp(t/8) hyperedges is 2-colorable. (We present a simple randomized streaming algorithm to construct the two-coloring.) Thus, for example, if the number of vertices is at most n1.5, then any non-2-colorable hypergraph must have at least 2n exp(ân/8) » n22n hyperedges. We observe that the exponential dependence on t in our result is optimal up to constant factors
Hypergraph-Based Analysis of Clustered Cooperative Beamforming with Application to Edge Caching
The evaluation of the performance of clustered cooperative beamforming in
cellular networks generally requires the solution of complex non-convex
optimization problems. In this letter, a framework based on a hypergraph
formalism is proposed that enables the derivation of a performance
characterization of clustered cooperative beamforming in terms of per-user
degrees of freedom (DoF) via the efficient solution of a coloring problem. An
emerging scenario in which clusters of cooperative base stations (BSs) arise is
given by cellular networks with edge caching. In fact, clusters of BSs that
share the same requested files can jointly beamform the corresponding encoded
signals. Based on this observation, the proposed framework is applied to obtain
quantitative insights into the optimal use of cache and backhaul resources in
cellular systems with edge caching. Numerical examples are provided to
illustrate the merits of the proposed framework.Comment: 10 pages, 5 figures, Submitte
Online Disjoint Set Cover Without Prior Knowledge
The disjoint set cover (DSC) problem is a fundamental combinatorial optimization problem concerned with partitioning the (hyper)edges of a hypergraph into (pairwise disjoint) clusters so that the number of clusters that cover all nodes is maximized. In its online version, the edges arrive one-by-one and should be assigned to clusters in an irrevocable fashion without knowing the future edges. This paper investigates the competitiveness of online DSC algorithms. Specifically, we develop the first (randomized) online DSC algorithm that guarantees a poly-logarithmic (O(log^{2} n)) competitive ratio without prior knowledge of the hypergraph\u27s minimum degree. On the negative side, we prove that the competitive ratio of any randomized online DSC algorithm must be at least Omega((log n)/(log log n)) (even if the online algorithm does know the minimum degree in advance), thus establishing the first lower bound on the competitive ratio of randomized online DSC algorithms
Counting Simplices in Hypergraph Streams
We consider the problem of space-efficiently estimating the number of
simplices in a hypergraph stream. This is the most natural hypergraph
generalization of the highly-studied problem of estimating the number of
triangles in a graph stream. Our input is a -uniform hypergraph with
vertices and hyperedges. A -simplex in is a subhypergraph on
vertices such that all possible hyperedges among exist in .
The goal is to process a stream of hyperedges of and compute a good
estimate of , the number of -simplices in .
We design a suite of algorithms for this problem. Under a promise that
, our algorithms use at most four passes and together imply a
space bound of for each fixed , in order to
guarantee an estimate within with probability at least
. We also give a simpler -pass algorithm that achieves
space, where (respectively, ) denotes
the maximum number of -simplices that share a hyperedge (respectively, a
vertex). We complement these algorithmic results with space lower bounds of the
form , , and
for multi-pass algorithms and
for -pass algorithms, which show that some of the dependencies on parameters
in our upper bounds are nearly tight. Our techniques extend and generalize
several different ideas previously developed for triangle counting in graphs,
using appropriate innovations to handle the more complicated combinatorics of
hypergraphs
Triangles, Long Paths, and Covered Sets
In chapter 2, we consider a generalization of the well-known Maker-Breaker triangle game for uniform hypergraphs in which Maker tries to build a triangle by choosing one edge in each round and Breaker tries to prevent her from doing so by choosing q edges in each round. The main result is the analysis of a new Breaker strategy using potential functions, introduced by Glazik and Srivastav (2019). Both bounds are of the order Î(n3/2) so they are asymptotically optimal. The constant for the lower bound is 2-o(1) and for the upper bound it is 3â2. In chapter 3, we describe another Maker-Breaker game, namely the P3-game in which Maker tries to build a path of length 3. First, we show that the methods of chapter 2 are not applicable in this scenario and give an intuition why that might be the case. Then, we give a more simple counting argument to bound the threshold bias. In chapter 4, we consider the longest path problem which is a classic NP-hard problem that arises in many contexts. Our motivation to investigate this problem in a big-data context was the problem of genome-assembly, where a long path in a graph that is constructed of the reads of a genome potentially represents a long contiguous sequence of the genome. We give a semi-streaming algorithm. Our algorithm delivers results competitive to algorithms that do not have a restriction on the amount of memory. In chapter 5, we investigate the b-SetMultiCover problem, a classic combinatorial problem which generalizes the set cover problem. Using an LP-relaxation and analysis with the bounded differences inequality of C. McDiarmid (1989), we show that there is a strong concentration around the expectation
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
Recommended from our members
ALGORITHMS FOR MASSIVE, EXPENSIVE, OR OTHERWISE INCONVENIENT GRAPHS
A long-standing assumption common in algorithm design is that any part of the input is accessible at any time for unit cost. However, as we work with increasingly large data sets, or as we build smaller devices, we must revisit this assumption. In this thesis, I present some of my work on graph algorithms designed for circumstances where traditional assumptions about inputs do not apply. 1. Classical graph algorithms require direct access to the input graph and this is not feasible when the graph is too large to fit in memory. For computation on massive graphs we consider the dynamic streaming graph model. Given an input graph defined by as a stream of edge insertions and deletions, our goal is to approximate properties of this graph using space that is sublinear in the size of the stream. In this thesis, I present algorithms for approximating vertex connectivity, hypergraph edge connectivity, maximum coverage, unique coverage, and temporal connectivity in graph streams. 2. In certain applications the input graph is not explicitly represented, but its edges may be discovered via queries which require costly computation or measurement. I present two open-source systems which solve real-world problems via graph algorithms which may access their inputs only through costly edge queries. M ESH is a memory manager which compacts memory efficiently by finding an approximate graph matching subject to stringent time and edge query restrictions. PathCache is an efficiently scalable network measurement platform that outperforms the current state of the art
Local computation algorithms for hypergraph coloring â following Beckâs approach
We investigate local computation algorithms (LCA) for two-coloring of k-uniform hypergraphs. We
focus on hypergraph instances that satisfy strengthened assumption of the LovĂĄsz Local Lemma
of the form , where â is the bound on the maximum edge degree. The main
question which arises here is for how large α there exists an LCA that is able to properly color such
hypergraphs in polylogarithmic time per query. We describe briefly how upgrading the classical
sequential procedure of Beck from 1991 with Moser and Tardosâ Resample yields polylogarithmic
LCA that works for α up to 1/4. Then, we present an improved procedure that solves wider range
of instances by allowing α up to 1/3
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