1,404 research outputs found
Relaxing the Irrevocability Requirement for Online Graph Algorithms
Online graph problems are considered in models where the irrevocability
requirement is relaxed. Motivated by practical examples where, for example,
there is a cost associated with building a facility and no extra cost
associated with doing it later, we consider the Late Accept model, where a
request can be accepted at a later point, but any acceptance is irrevocable.
Similarly, we also consider a Late Reject model, where an accepted request can
later be rejected, but any rejection is irrevocable (this is sometimes called
preemption). Finally, we consider the Late Accept/Reject model, where late
accepts and rejects are both allowed, but any late reject is irrevocable. For
Independent Set, the Late Accept/Reject model is necessary to obtain a constant
competitive ratio, but for Vertex Cover the Late Accept model is sufficient and
for Minimum Spanning Forest the Late Reject model is sufficient. The Matching
problem has a competitive ratio of 2, but in the Late Accept/Reject model, its
competitive ratio is 3/2
Optimal Online Edge Coloring of Planar Graphs with Advice
Using the framework of advice complexity, we study the amount of knowledge
about the future that an online algorithm needs to color the edges of a graph
optimally, i.e., using as few colors as possible. For graphs of maximum degree
, it follows from Vizing's Theorem that bits of
advice suffice to achieve optimality, where is the number of edges. We show
that for graphs of bounded degeneracy (a class of graphs including e.g. trees
and planar graphs), only bits of advice are needed to compute an optimal
solution online, independently of how large is. On the other hand, we
show that bits of advice are necessary just to achieve a
competitive ratio better than that of the best deterministic online algorithm
without advice. Furthermore, we consider algorithms which use a fixed number of
advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to
this class of algorithms). We show that for bipartite graphs, any such
algorithm must use at least bits of advice to achieve
optimality.Comment: CIAC 201
The Advice Complexity of a Class of Hard Online Problems
The advice complexity of an online problem is a measure of how much knowledge
of the future an online algorithm needs in order to achieve a certain
competitive ratio. Using advice complexity, we define the first online
complexity class, AOC. The class includes independent set, vertex cover,
dominating set, and several others as complete problems. AOC-complete problems
are hard, since a single wrong answer by the online algorithm can have
devastating consequences. For each of these problems, we show that
bits of advice are
necessary and sufficient (up to an additive term of ) to achieve a
competitive ratio of .
The results are obtained by introducing a new string guessing problem related
to those of Emek et al. (TCS 2011) and B\"ockenhauer et al. (TCS 2014). It
turns out that this gives a powerful but easy-to-use method for providing both
upper and lower bounds on the advice complexity of an entire class of online
problems, the AOC-complete problems.
Previous results of Halld\'orsson et al. (TCS 2002) on online independent
set, in a related model, imply that the advice complexity of the problem is
. Our results improve on this by providing an exact formula for
the higher-order term. For online disjoint path allocation, B\"ockenhauer et
al. (ISAAC 2009) gave a lower bound of and an upper bound of
on the advice complexity. We improve on the upper bound by a
factor of . For the remaining problems, no bounds on their advice
complexity were previously known.Comment: Full paper to appear in Theory of Computing Systems. A preliminary
version appeared in STACS 201
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Exact bounds for distributed graph colouring
We prove exact bounds on the time complexity of distributed graph colouring.
If we are given a directed path that is properly coloured with colours, by
prior work it is known that we can find a proper 3-colouring in communication rounds. We close the gap between upper and
lower bounds: we show that for infinitely many the time complexity is
precisely communication rounds.Comment: 16 pages, 3 figure
Algorithms and complexity analyses for some combinational optimization problems
The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design.
In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios.
In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs.
Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
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