9,990 research outputs found
Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics
Simple heuristics often show a remarkable performance in practice for
optimization problems. Worst-case analysis often falls short of explaining this
performance. Because of this, "beyond worst-case analysis" of algorithms has
recently gained a lot of attention, including probabilistic analysis of
algorithms.
The instances of many optimization problems are essentially a discrete metric
space. Probabilistic analysis for such metric optimization problems has
nevertheless mostly been conducted on instances drawn from Euclidean space,
which provides a structure that is usually heavily exploited in the analysis.
However, most instances from practice are not Euclidean. Little work has been
done on metric instances drawn from other, more realistic, distributions. Some
initial results have been obtained by Bringmann et al. (Algorithmica, 2013),
who have used random shortest path metrics on complete graphs to analyze
heuristics.
The goal of this paper is to generalize these findings to non-complete
graphs, especially Erd\H{o}s-R\'enyi random graphs. A random shortest path
metric is constructed by drawing independent random edge weights for each edge
in the graph and setting the distance between every pair of vertices to the
length of a shortest path between them with respect to the drawn weights. For
such instances, we prove that the greedy heuristic for the minimum distance
maximum matching problem, the nearest neighbor and insertion heuristics for the
traveling salesman problem, and a trivial heuristic for the -median problem
all achieve a constant expected approximation ratio. Additionally, we show a
polynomial upper bound for the expected number of iterations of the 2-opt
heuristic for the traveling salesman problem.Comment: An extended abstract appeared in the proceedings of WALCOM 201
Generalized Buneman pruning for inferring the most parsimonious multi-state phylogeny
Accurate reconstruction of phylogenies remains a key challenge in
evolutionary biology. Most biologically plausible formulations of the problem
are formally NP-hard, with no known efficient solution. The standard in
practice are fast heuristic methods that are empirically known to work very
well in general, but can yield results arbitrarily far from optimal. Practical
exact methods, which yield exponential worst-case running times but generally
much better times in practice, provide an important alternative. We report
progress in this direction by introducing a provably optimal method for the
weighted multi-state maximum parsimony phylogeny problem. The method is based
on generalizing the notion of the Buneman graph, a construction key to
efficient exact methods for binary sequences, so as to apply to sequences with
arbitrary finite numbers of states with arbitrary state transition weights. We
implement an integer linear programming (ILP) method for the multi-state
problem using this generalized Buneman graph and demonstrate that the resulting
method is able to solve data sets that are intractable by prior exact methods
in run times comparable with popular heuristics. Our work provides the first
method for provably optimal maximum parsimony phylogeny inference that is
practical for multi-state data sets of more than a few characters.Comment: 15 page
Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach
In this paper, we study the -forest problem in the model of resource
augmentation. In the -forest problem, given an edge-weighted graph ,
a parameter , and a set of demand pairs , the
objective is to construct a minimum-cost subgraph that connects at least
demands. The problem is hard to approximate---the best-known approximation
ratio is . Furthermore, -forest is as hard to
approximate as the notoriously-hard densest -subgraph problem.
While the -forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the -forest problem can be
viewed as to remove at most demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the -forest
problem that, for every , removes at most demands and has
cost no more than times the cost of an optimal algorithm
that removes at most demands
Rational Deployment of CSP Heuristics
Heuristics are crucial tools in decreasing search effort in varied fields of
AI. In order to be effective, a heuristic must be efficient to compute, as well
as provide useful information to the search algorithm. However, some well-known
heuristics which do well in reducing backtracking are so heavy that the gain of
deploying them in a search algorithm might be outweighed by their overhead.
We propose a rational metareasoning approach to decide when to deploy
heuristics, using CSP backtracking search as a case study. In particular, a
value of information approach is taken to adaptive deployment of solution-count
estimation heuristics for value ordering. Empirical results show that indeed
the proposed mechanism successfully balances the tradeoff between decreasing
backtracking and heuristic computational overhead, resulting in a significant
overall search time reduction.Comment: 7 pages, 2 figures, to appear in IJCAI-2011, http://www.ijcai.org
A Codebook Generation Algorithm for Document Image Compression
Pattern-matching-based document-compression systems (e.g. for faxing) rely on
finding a small set of patterns that can be used to represent all of the ink in
the document. Finding an optimal set of patterns is NP-hard; previous
compression schemes have resorted to heuristics. This paper describes an
extension of the cross-entropy approach, used previously for measuring pattern
similarity, to this problem. This approach reduces the problem to a k-medians
problem, for which the paper gives a new algorithm with a provably good
performance guarantee. In comparison to previous heuristics (First Fit, with
and without generalized Lloyd's/k-means postprocessing steps), the new
algorithm generates a better codebook, resulting in an overall improvement in
compression performance of almost 17%
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
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Locating Depots for Capacitated Vehicle Routing
We study a location-routing problem in the context of capacitated vehicle
routing. The input is a set of demand locations in a metric space and a fleet
of k vehicles each of capacity Q. The objective is to locate k depots, one for
each vehicle, and compute routes for the vehicles so that all demands are
satisfied and the total cost is minimized. Our main result is a constant-factor
approximation algorithm for this problem. To achieve this result, we reduce to
the k-median-forest problem, which generalizes both k-median and minimum
spanning tree, and which might be of independent interest. We give a
(3+c)-approximation algorithm for k-median-forest, which leads to a
(12+c)-approximation algorithm for the above location-routing problem, for any
constant c>0. The algorithm for k-median-forest is just t-swap local search,
and we prove that it has locality gap 3+2/t; this generalizes the corresponding
result known for k-median. Finally we consider the "non-uniform"
k-median-forest problem which has different cost functions for the MST and
k-median parts. We show that the locality gap for this problem is unbounded
even under multi-swaps, which contrasts with the uniform case. Nevertheless, we
obtain a constant-factor approximation algorithm, using an LP based approach.Comment: 12 pages, 1 figur
Insertion Heuristics for Central Cycle Problems
A central cycle problem requires a cycle that is
reasonably short and keeps a the maximum distance
from any node not on the cycle to its nearest
node on the cycle reasonably low. The objective
may be to minimise maximumdistance or cycle
length and the solution may have further constraints.
Most classes of central cycle problems
are NP-hard. This paper investigates insertion
heuristics for central cycle problems, drawing on
insertion heuristics for p-centres [7] and travelling
salesman tours [21]. It shows that a modified
farthest insertion heuristic has reasonable worstcase
bounds for a particular class of problem.
It then compares the performance of two farthest
insertion heuristics against each other and
against bounds (where available) obtained by integer
programming on a range of problems from
TSPLIB [20]. It shows that a simple farthest insertion
heuristic is fast, performs well in practice
and so is likely to be useful for a general problems
or as the basis for more complex heuristics
for specific problems
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