4,118 research outputs found
Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP
2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on âreal worldâ Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every pâN , a family of L p instances on which 2-Opt can take an exponential number of steps.
Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by O~(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1] d , for an arbitrary dâ„2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density Ï of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3â
Ï8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3â1/dâ
Ï8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4â1/dâ
Ï) . In addition, we prove an upper bound of O(Ïâd) on the expected approximation factor with respect to all L p metrics.
Let us remark that our probabilistic analysis covers as special cases the uniform input model with Ï=1 and a smoothed analysis with Gaussian perturbations of standard deviation Ï with ÏâŒ1/Ï d
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
Algebraic Methods in the Congested Clique
In this work, we use algebraic methods for studying distance computation and
subgraph detection tasks in the congested clique model. Specifically, we adapt
parallel matrix multiplication implementations to the congested clique,
obtaining an round matrix multiplication algorithm, where
is the exponent of matrix multiplication. In conjunction
with known techniques from centralised algorithmics, this gives significant
improvements over previous best upper bounds in the congested clique model. The
highlight results include:
-- triangle and 4-cycle counting in rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in rounds, which is the first
non-trivial solution in this model.
In addition, we present a novel constant-round combinatorial algorithm for
detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
An Efficient Data Structure for Dynamic Two-Dimensional Reconfiguration
In the presence of dynamic insertions and deletions into a partially
reconfigurable FPGA, fragmentation is unavoidable. This poses the challenge of
developing efficient approaches to dynamic defragmentation and reallocation.
One key aspect is to develop efficient algorithms and data structures that
exploit the two-dimensional geometry of a chip, instead of just one. We propose
a new method for this task, based on the fractal structure of a quadtree, which
allows dynamic segmentation of the chip area, along with dynamically adjusting
the necessary communication infrastructure. We describe a number of algorithmic
aspects, and present different solutions. We also provide a number of basic
simulations that indicate that the theoretical worst-case bound may be
pessimistic.Comment: 11 pages, 12 figures; full version of extended abstract that appeared
in ARCS 201
A -Vertex Kernel for Maximum Internal Spanning Tree
We consider the parameterized version of the maximum internal spanning tree
problem, which, given an -vertex graph and a parameter , asks for a
spanning tree with at least internal vertices. Fomin et al. [J. Comput.
System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a
simple application of this rule is sufficient to yield a -vertex kernel.
Here we propose a novel way to use the same reduction rule, resulting in an
improved -vertex kernel. Our algorithm applies first a greedy procedure
consisting of a sequence of local exchange operations, which ends with a
local-optimal spanning tree, and then uses this special tree to find a
reducible structure. As a corollary of our kernel, we obtain a deterministic
algorithm for the problem running in time
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