11,194 research outputs found
Improving bounds on large instances of graph coloring
This thesis explores new methods, using both vertex cover and exact graph coloring algorithms in addition to our implementation of the state of the art, to develop a hybrid algorithm that on most instances is able to tighten the bounds or determine the optimal number of colors outright
Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints
We propose two strategies for Presenter in the on-line interval graph
coloring games. Specifically, we consider a setting in which each interval is
associated with a -dimensional vector of weights and the coloring needs to
satisfy the -dimensional bandwidth constraint, and the -cardinality
constraint. Such a variant was first introduced by Epstein and Levy and it is a
natural model for resource-aware task scheduling with different shared
resources where at most tasks can be scheduled simultaneously on a single
machine.
The first strategy forces any on-line interval coloring algorithm to use at
least different colors on an -colorable set of intervals. The second strategy forces any
on-line interval coloring algorithm to use at least
different colors on an
-colorable set of unit intervals
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
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
Algorithms and almost tight results for 3-colorability of small diameter graphs.
The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring cannot be solved in time 2o(n) on graphs with n vertices and diameter at most 4. In spite of extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been an open problem. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity 2O(nlogn√). Furthermore we extend the notion of an articulation vertex to that of an articulation neighborhood, and we provide a polynomial algorithm for 3-coloring on graphs with diameter 2 that have at least one articulation neighborhood. For graphs with diameter at most 3, we establish the complexity of 3-coloring by proving for every ε∈[0,1) that 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree δ=Θ(nε). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every ε∈[0,1) subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree δ=Θ(nε). Finally, we provide a 3-coloring algorithm with running time 2O(min{δΔ, nδlogδ}) for arbitrary graphs with diameter 3, where n is the number of vertices and δ (resp. Δ) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this is the first subexponential algorithm for graphs with δ=ω(1) and for graphs with δ=O(1) and Δ=o(n). Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree of the input graph is δ=Θ(nε), where ε∈[12,1), as its time complexity is 2O(nδlogδ)=2O(n1−εlogn) and the corresponding lower bound states that there is no 2o(n1−ε)-time algorithm
Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time
The described multicoloring problem has direct applications in the context of
wireless ad hoc and sensor networks. In order to coordinate the access to the
shared wireless medium, the nodes of such a network need to employ some medium
access control (MAC) protocol. Typical MAC protocols control the access to the
shared channel by time (TDMA), frequency (FDMA), or code division multiple
access (CDMA) schemes. Many channel access schemes assign a fixed set of time
slots, frequencies, or (orthogonal) codes to the nodes of a network such that
nodes that interfere with each other receive disjoint sets of time slots,
frequencies, or code sets. Finding a valid assignment of time slots,
frequencies, or codes hence directly corresponds to computing a multicoloring
of a graph . The scarcity of bandwidth, energy, and computing resources in
ad hoc and sensor networks, as well as the often highly dynamic nature of these
networks require that the multicoloring can be computed based on as little and
as local information as possible
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