3,889 research outputs found
Online graph coloring against a randomized adversary
Electronic version of an article published as
Online graph coloring against a randomized adversary. "International journal of foundations of computer science", 1 Juny 2018, vol. 29, nĂșm. 4, p. 551-569. DOI:10.1142/S0129054118410058 © 2018 copyright World Scientific Publishing Company. https://www.worldscientific.com/doi/abs/10.1142/S0129054118410058We consider an online model where an adversary constructs a set of 2s instances S instead of one single instance. The algorithm knows S and the adversary will choose one instance from S at random to present to the algorithm. We further focus on adversaries that construct sets of k-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in s and matching upper and lower bound are given for a specific set of algorithms that we call âminimalistic online algorithmsâ.Peer ReviewedPostprint (author's final draft
Tight Bounds for Online Coloring of Basic Graph Classes
We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is Theta(log n), where n is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size O(n/log n) or access to a reordering buffer of size n^(1-epsilon), for any 0 < epsilon <= 1. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural First Fit coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized
online algorithms
Lower Bounds for Structuring Unreliable Radio Networks
In this paper, we study lower bounds for randomized solutions to the maximal
independent set (MIS) and connected dominating set (CDS) problems in the dual
graph model of radio networks---a generalization of the standard graph-based
model that now includes unreliable links controlled by an adversary. We begin
by proving that a natural geographic constraint on the network topology is
required to solve these problems efficiently (i.e., in time polylogarthmic in
the network size). We then prove the importance of the assumption that nodes
are provided advance knowledge of their reliable neighbors (i.e, neighbors
connected by reliable links). Combined, these results answer an open question
by proving that the efficient MIS and CDS algorithms from [Censor-Hillel, PODC
2011] are optimal with respect to their dual graph model assumptions. They also
provide insight into what properties of an unreliable network enable efficient
local computation.Comment: An extended abstract of this work appears in the 2014 proceedings of
the International Symposium on Distributed Computing (DISC
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
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
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