On the advice complexity of the online <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>L</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>-coloring problem on paths and cycles

Bianchi, Maria Paola; Böckenhauer, Hans-Joachim; Hromkovič, Juraj; Krug, Sacha; Steffen, Björn

doi:10.1016/j.tcs.2014.06.027

Cited by 12 publications

(5 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Instead, the nodes are revealed one by one and results have been obtained for paths [22], bipartite graphs [3], and 3-colorable graphs [34]. In [29], a coloring problem with restrictions going beyond the immediate neighbors is considered. Furthermore, there are interesting connections between advice and randomization and sometimes results on advice complexity can be used to obtain efficient random algorithms [5,27,6].…”

Section: Previous Resultsmentioning

confidence: 99%

Online Multi-Coloring with Advice

Christ

Favrholdt

Larsen

2015

Approximation and Online Algorithms

View full text Add to dashboard Cite

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.

show abstract

Section: Previous Resultsmentioning

confidence: 99%

Online Multi-Coloring with Advice

Christ

Favrholdt

Larsen

2015

Approximation and Online Algorithms

View full text Add to dashboard Cite

show abstract

“…It is easy to see that L(2, 1)-coloring on paths is a strictly ∨-repeatable problem. Thus, combining the lower bound for deterministic algorithms without advice from [19] with Theorem 15, we get that for every ε > 0 and every (possibly randomized) algorithm reading o(n) bits of advice, there exists an input σ such that E[ALG(σ)] ≥ (1 − ε)6. Since costs are integral, this reproves the lower bound of 3/2 on the strict competitive ratio of algorithms with o(n) bits of advice.…”

Section: Applicationsmentioning

confidence: 93%

“…Application: L(2, 1)-coloring on paths and cycles See [19] for a formal definition of the problem. The input is a graph of maximum degree two.…”

Section: Applicationsmentioning

confidence: 99%

“…In [19], the following results were obtained: It is shown that there exists a deterministic algorithm D without advice such that D(σ) ≤ 6 for every input σ. A matching lower bound is provided by showing that there exists a finite set of inputs I such that, for each deterministic algorithm ALG without advice, ALG(σ) ≥ 6 for some σ ∈ I.…”

Section: Applicationsmentioning

confidence: 99%

“…A matching lower bound is provided by showing that there exists a finite set of inputs I such that, for each deterministic algorithm ALG without advice, ALG(σ) ≥ 6 for some σ ∈ I. Finally, it is also shown in [19] that a deterministic algorithm ALG which can guarantee to solve the L(2, 1) problem on paths with a cost of at most 5 must read at least 3.9402 • 10 −10 n = Ω(n) bits of advice. Since costs are integral and OPT(σ) is always at most 4, this gives a (tight) lower bound of 3/2 on the strict competitive ratio of algorithms with sublinear advice.…”

Section: Applicationsmentioning

confidence: 99%

See 2 more Smart Citations

Randomization can be as helpful as a glimpse of the future in online computation

Mikkelsen

2015

Preprint

View full text Add to dashboard Cite

We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. Intuitively, our direct product theorems say that if b bits of advice are needed to ensure a cost of at most t for some problem, then r • b bits of advice are needed to ensure a total cost of at most r • t when solving r independent instances of the problem. Using our direct product theorems, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, we show that

show abstract

The Secretary Problem with Reservation Costs

Burjons

Gehnen

Lotze

et al. 2021

Lecture Notes in Computer Science

View full text Add to dashboard Cite

We study the Feedback Vertex Set and the Vertex Cover problem in a natural variant of the classical online model that allows for delayed decisions and reservations. Both problems can be characterized by an obstruction set of subgraphs that the online graph needs to avoid. In the case of the Vertex Cover problem, the obstruction set consists of an edge (i.e., the graph of two adjacent vertices), while for the Feedback Vertex Set problem, the obstruction set contains all cycles.In the delayed-decision model, an algorithm needs to maintain a valid partial solution after every request, thus allowing it to postpone decisions until the current partial solution is no longer valid for the current request.The reservation model grants an online algorithm the new and additional option to pay a so-called reservation cost for any given element in order to delay the decision of adding or rejecting it until the end of the instance.For the Feedback Vertex Set problem, we first analyze the variant with only delayed decisions, proving a lower bound of 4 and an upper bound of 5 on the competitive ratio. Then we look at the variant with both delayed decisions and reservation. We show that given bounds on the competitive ratio of a problem with delayed decisions impliy lower and upper bounds for the same problem when adding the option of reservations. This observation allows us to give a lower bound of min {1 + 3α, 4} and an upper bound of min {1 + 5α, 5} for the Feedback Vertex Set problem. Finally, we show that the online Vertex Cover problem, when both delayed decisions and reservations are allowed, is min {1 + 2α, 2}-competitive, where α ∈ R ≥0 is the reservation cost per reserved vertex.

show abstract

On the advice complexity of the online L(2,1)-coloring problem on paths and cycles

Cited by 12 publications

References 5 publications

Online Multi-Coloring with Advice

Online Multi-Coloring with Advice

Randomization can be as helpful as a glimpse of the future in online computation

The Secretary Problem with Reservation Costs

Contact Info

Product

Resources

About