36,384 research outputs found
Distributed Maximum Matching in Bounded Degree Graphs
We present deterministic distributed algorithms for computing approximate
maximum cardinality matchings and approximate maximum weight matchings. Our
algorithm for the unweighted case computes a matching whose size is at least
(1-\eps) times the optimal in \Delta^{O(1/\eps)} +
O\left(\frac{1}{\eps^2}\right) \cdot\log^*(n) rounds where is the number
of vertices in the graph and is the maximum degree. Our algorithm for
the edge-weighted case computes a matching whose weight is at least (1-\eps)
times the optimal in
\log(\min\{1/\wmin,n/\eps\})^{O(1/\eps)}\cdot(\Delta^{O(1/\eps)}+\log^*(n))
rounds for edge-weights in [\wmin,1].
The best previous algorithms for both the unweighted case and the weighted
case are by Lotker, Patt-Shamir, and Pettie~(SPAA 2008). For the unweighted
case they give a randomized (1-\eps)-approximation algorithm that runs in
O((\log(n)) /\eps^3) rounds. For the weighted case they give a randomized
(1/2-\eps)-approximation algorithm that runs in O(\log(\eps^{-1}) \cdot
\log(n)) rounds. Hence, our results improve on the previous ones when the
parameters , \eps and \wmin are constants (where we reduce the
number of runs from to ), and more generally when
, 1/\eps and 1/\wmin are sufficiently slowly increasing functions
of . Moreover, our algorithms are deterministic rather than randomized.Comment: arXiv admin note: substantial text overlap with arXiv:1402.379
Best of Two Local Models: Local Centralized and Local Distributed Algorithms
We consider two models of computation: centralized local algorithms and local
distributed algorithms. Algorithms in one model are adapted to the other model
to obtain improved algorithms.
Distributed vertex coloring is employed to design improved centralized local
algorithms for: maximal independent set, maximal matching, and an approximation
scheme for maximum (weighted) matching over bounded degree graphs. The
improvement is threefold: the algorithms are deterministic, stateless, and the
number of probes grows polynomially in , where is the number of
vertices of the input graph.
The recursive centralized local improvement technique by Nguyen and
Onak~\cite{onak2008} is employed to obtain an improved distributed
approximation scheme for maximum (weighted) matching. The improvement is
twofold: we reduce the number of rounds from to for a
wide range of instances and, our algorithms are deterministic rather than
randomized
Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching
We present a deterministic distributed algorithm that computes a
-edge-coloring, or even list-edge-coloring, in any -node graph
with maximum degree , in rounds. This answers
one of the long-standing open questions of \emph{distributed graph algorithms}
from the late 1980s, which asked for a polylogarithmic-time algorithm. See,
e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and
Elkin. The previous best round complexities were by
Panconesi and Srinivasan [STOC'92] and
by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our
deterministic list-edge-coloring also improves the randomized complexity of
-edge-coloring to poly rounds.
The key technical ingredient is a deterministic distributed algorithm for
\emph{hypergraph maximal matching}, which we believe will be of interest beyond
this result. In any hypergraph of rank --- where each hyperedge has at most
vertices --- with nodes and maximum degree , this algorithm
computes a maximal matching in rounds.
This hypergraph matching algorithm and its extensions lead to a number of
other results. In particular, a polylogarithmic-time deterministic distributed
maximal independent set algorithm for graphs with bounded neighborhood
independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a
-round deterministic
algorithm for -approximation of maximum matching, and a
quasi-polylogarithmic-time deterministic distributed algorithm for orienting
-arboricity graphs with out-degree at most ,
for any constant , hence partially answering Open Problem 10 of
Barenboim and Elkin's book
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
Distributed Approximate Maximum Matching in the CONGEST Model
We study distributed algorithms for the maximum matching problem in the CONGEST model, where each message must be bounded in size. We give new deterministic upper bounds, and a new lower bound on the problem.
We begin by giving a distributed algorithm that computes an exact maximum (unweighted) matching in bipartite graphs, in O(n log n) rounds. Next, we give a distributed algorithm that approximates the fractional weighted maximum matching problem in general graphs. In a graph with maximum degree at most Delta, the algorithm computes a (1-epsilon)-approximation for the problem in time O(log(Delta W)/epsilon^2), where W is a bound on the ratio between the largest and the smallest edge weight. Next, we show a slightly improved and generalized version of the deterministic rounding algorithm of Fischer [DISC \u2717]. Given a fractional weighted maximum matching solution of value f for a given graph G, we show that in time O((log^2(Delta)+log^*n)/epsilon), the fractional solution can be turned into an integer solution of value at least (1-epsilon)f for bipartite graphs and (1-epsilon) * (g-1)/g * f for general graphs, where g is the length of the shortest odd cycle of G. Together with the above fractional maximum matching algorithm, this implies a deterministic algorithm that computes a (1-epsilon)* (g-1)/g-approximation for the weighted maximum matching problem in time O(log(Delta W)/epsilon^2 + (log^2(Delta)+log^* n)/epsilon).
On the lower-bound front, we show that even for unweighted fractional maximum matching in bipartite graphs, computing an (1 - O(1/sqrt{n}))-approximate solution requires at least Omega~(D+sqrt{n}) rounds in CONGEST. This lower bound requires the introduction of a new 2-party communication problem, for which we prove a tight lower bound
An Estimator for Matching Size in Low Arboricity Graphs with Two Applications
In this paper, we present a new simple degree-based estimator for the size of
maximum matching in bounded arboricity graphs. When the arboricity of the graph
is bounded by , the estimator gives a factor approximation
of the matching size. For planar graphs, we show the estimator does better and
returns a approximation of the matching size.
Using this estimator, we get new results for approximating the matching size
of planar graphs in the streaming and distributed models of computation. In
particular, in the vertex-arrival streams, we get a randomized
space algorithm for approximating the
matching size within factor in a planar graph on vertices.
Similarly, we get a simultaneous protocol in the vertex-partition model for
approximating the matching size within factor using
communication from each player.
In comparison with the previous estimators, the estimator in this paper does
not need to know the arboricity of the input graph and improves the
approximation factor for the case of planar graphs
An optimal maximal independent setalgorithm for bounded-independence graphs
We present a novel distributed algorithm for the maximal independent set problem (This is an extended journal version of Schneider and Wattenhofer in Twenty-seventh annual ACM SIGACT-SIGOPS symposium on principles of distributed computing, 2008). On bounded-independence graphs our deterministic algorithm finishes in O(log* n) time, n being the number of nodes. In light of Linial's Ω(log* n) lower bound our algorithm is asymptotically optimal. Furthermore, it solves the connected dominating set problem for unit disk graphs in O(log* n) time, exponentially faster than the state-of-the-art algorithm. With a new extension our algorithm also computes a δ+1 coloring and a maximal matching in O(log* n) time, where δ is the maximum degree of the grap
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