20 research outputs found
Online coloring problem with a randomized adversary and infinite advice
Online problems are those in which the instance is not given as a whole but by parts named requests. They arrise naturaly in computer science. Several examples are given such as ski rental problem, the server problem and the coloring problem. The performance of the online algorithms is analized in terms of the ratio between the cost of the algorithm and the cost of the optimal offline. This ratio is called the competitive ratio. Several models of online algorithms are described. They are deterministic algorithms, randomized algorithms and algorithms with advice. We present several upper and lower bounds for the competitive ratio in a particular case of the k-server problem. We review the known bounds for the coloring problem in the diferent models. We present a new model, the randomized adversary. For this model we present an upper bound and a restricted lower bound. Finally we conjecture an unrestricted lower bound and we present several approaches to the result
Online coloring problem with a randomized adversary and infinite advice
Online problems are those in which the instance is not given as a whole but by parts named requests. They arrise naturaly in computer science. Several examples are given such as ski rental problem, the server problem and the coloring problem. The performance of the online algorithms is analized in terms of the ratio between the cost of the algorithm and the cost of the optimal offline. This ratio is called the competitive ratio. Several models of online algorithms are described. They are deterministic algorithms, randomized algorithms and algorithms with advice. We present several upper and lower bounds for the competitive ratio in a particular case of the k-server problem. We review the known bounds for the coloring problem in the diferent models. We present a new model, the randomized adversary. For this model we present an upper bound and a restricted lower bound. Finally we conjecture an unrestricted lower bound and we present several approaches to the result
The Online Simple Knapsack Problem with Reservation and Removability
In the online simple knapsack problem, a knapsack of unit size 1 is given and an algorithm is tasked to fill it using a set of items that are revealed one after another. Each item must be accepted or rejected at the time they are presented, and these decisions are irrevocable. No prior knowledge about the set and sequence of items is given. The goal is then to maximize the sum of the sizes of all packed items compared to an optimal packing of all items of the sequence.
In this paper, we combine two existing variants of the problem that each extend the range of possible actions for a newly presented item by a new option. The first is removability, in which an item that was previously packed into the knapsack may be finally discarded at any point. The second is reservations, which allows the algorithm to delay the decision on accepting or rejecting a new item indefinitely for a proportional fee relative to the size of the given item.
If both removability and reservations are permitted, we show that the competitive ratio of the online simple knapsack problem rises depending on the relative reservation costs. As soon as any nonzero fee has to be paid for a reservation, no online algorithm can be better than 1.5-competitive. With rising reservation costs, this competitive ratio increases up to the golden ratio (? ? 1.618) that is reached for relative reservation costs of 1-?5/3 ? 0.254. We provide a matching upper and lower bound for relative reservation costs up to this value. From this point onward, the tight bound by Iwama and Taketomi for the removable knapsack problem is the best possible competitive ratio, not using any reservations
Finding Optimal Solutions With Neighborly Help
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory\u27s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted).
We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level.
Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class
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
Delaying Decisions and Reservation Costs
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 and an upper bound of 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 and an upper bound of
for the Feedback Vertex Set problem. Finally, we show
that the online Vertex Cover problem, when both delayed decisions and
reservations are allowed, is -competitive, where
is the reservation cost per reserved vertex.Comment: 14 Pages, submitte
The Impact of Additional Information on Online and Parameterized Problems
There are many open problems in the field of complexity. This means that, when
analyzing the complexity of a specific computer science problem, we are often not
satisfied with a standard complexity analysis of the problem. There are several
known models that help us analyze the complexity of a problem in a more realistic
setting or in a way that provides us more information than can be given by the
classical complexity-theoretic tools.
In this thesis, we take a look at several methods to help provide a more precise
or adapted complexity analysis for some problems. The goal is always to find ways
to measure the information content of the problem. We do this by adapting known
models in several ways.
First, we take a look at online problems, beginning with the k-server problem. In
the advice complexity model, given an online instance for a problem, the algorithm
is allowed access to an advice tape constructed by an all-knowing oracle. For the
k-server problem, we give several results that provide a tradeoff between the advice
complexity and the competitive ratio of this problem in several metric spaces. In
particular, we focus on the Euclidean space and on the sphere, both in two dimensions
and then generalizing the results to several dimensions.
Then, we present the model of a randomized adversary, an alternative model to
that of advice complexity, where instead of giving more power to the algorithm by
giving it access to an advice tape, we restrict the power of the adversary by making
it construct a set of instances, one of which will be chosen at random. In this model
it is easier to find implementable upper bounds than in the advice complexity model,
which is inherently nondeterministic but enables us to prove stronger lower bounds.
We analyze the graph coloring problem in this model.
The final online problem we analyze in this thesis is the weighted disjoint path
allocation problem. For this problem, we take a look at several parameters, that
might be reasonable for this problem, and we analyze the competitiveness of the
problem both with and without advice with respect to these parameters. The results
obtained show that there is a tradeoff; the more restricted the value of a parameter
is, the better the algorithm behaves. This tradeoff, not unexpected, serves to show
that, for particular problems, finding an appropriate parameter that is restricted in
most usual instances can help us get a better understanding of the complexity of the
problem itself.
Finally, we take a look at parameterized complexity. In particular, we give a
model of reoptimization of parameterized problems and analyze several problems
under this model. Reoptimization gives us essentially the solution to a similar
instance and the goal is to adapt this solution to the instance we want to solve. Here,
we see how much information the neighbouring solution is providing. The answer is
that it depends on the problem and the measure of similarity. We find polynomial
kernels under this setting, for some problems, that do not have polynomial kernels
under some standard complexity-theoretic assumptions in the classical model. We
also provide several lower bound results that show that some problems are just as
hard under reoptimization as they are in their classical form