121 research outputs found
Towards explaining the speed of -means
The -means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the -means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences
Minimum-weight Cycle Covers and Their Approximability
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L.
We investigate how well L-cycle covers of minimum weight can be approximated.
For undirected graphs, we devise a polynomial-time approximation algorithm that
achieves a constant approximation ratio for all sets L. On the other hand, we
prove that the problem cannot be approximated within a factor of 2-eps for
certain sets L.
For directed graphs, we present a polynomial-time approximation algorithm
that achieves an approximation ratio of O(n), where is the number of
vertices. This is asymptotically optimal: We show that the problem cannot be
approximated within a factor of o(n).
To contrast the results for cycle covers of minimum weight, we show that the
problem of computing L-cycle covers of maximum weight can, at least in
principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2007). Minor change
On Approximating Multi-Criteria TSP
We present approximation algorithms for almost all variants of the
multi-criteria traveling salesman problem (TSP).
First, we devise randomized approximation algorithms for multi-criteria
maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP,
where the edge weights have to be symmetric, we devise an algorithm with an
approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge
weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps.
Our algorithms work for any fixed number k of objectives. Furthermore, we
present a deterministic algorithm for bi-criteria Max-STSP that achieves an
approximation ratio of 7/27.
Finally, we present a randomized approximation algorithm for the asymmetric
multi-criteria minimum TSP with triangle inequality Min-ATSP. This algorithm
achieves a ratio of log n + eps.Comment: Preliminary version at STACS 2009. This paper is a revised full
version, where some proofs are simplifie
Improved Smoothed Analysis of the k-Means Method
The k-means method is a widely used clustering algorithm. One of its
distinguished features is its speed in practice. Its worst-case running-time,
however, is exponential, leaving a gap between practical and theoretical
performance. Arthur and Vassilvitskii (FOCS 2006) aimed at closing this gap,
and they proved a bound of \poly(n^k, \sigma^{-1}) on the smoothed
running-time of the k-means method, where n is the number of data points and
is the standard deviation of the Gaussian perturbation. This bound,
though better than the worst-case bound, is still much larger than the
running-time observed in practice.
We improve the smoothed analysis of the k-means method by showing two upper
bounds on the expected running-time of k-means. First, we prove that the
expected running-time is bounded by a polynomial in and
. Second, we prove an upper bound of k^{kd} \cdot \poly(n,
\sigma^{-1}), where d is the dimension of the data space. The polynomial is
independent of k and d, and we obtain a polynomial bound for the expected
running-time for .
Finally, we show that k-means runs in smoothed polynomial time for
one-dimensional instances.Comment: To be presented at the 20th ACM-SIAM Symposium on Discrete Algorithms
(SODA 2009
Bisimplicial edges in bipartite graphs
Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to nd such edges in bipartite graphs. The expected time complexity of our new algorithm is on random bipartite graphs in which each edge is present with a fixed probability p, a polynomial improvement over the fastest algorithm found in the existing literature
Approximation Algorithms for Multi-Criteria Traveling Salesman Problems
In multi-criteria optimization problems, several objective functions have to
be optimized. Since the different objective functions are usually in conflict
with each other, one cannot consider only one particular solution as the
optimal solution. Instead, the aim is to compute a so-called Pareto curve of
solutions. Since Pareto curves cannot be computed efficiently in general, we
have to be content with approximations to them.
We design a deterministic polynomial-time algorithm for multi-criteria
g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate
Pareto curves for all 1/2<=g<=1. In particular, we obtain a
(2+eps)-approximation for multi-criteria metric STSP. We also present two
randomized approximation algorithms for multi-criteria g-metric STSP that
achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1
+3g -4g^2) + eps, respectively.
Moreover, we present randomized approximation algorithms for multi-criteria
g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with
weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do
this, we design randomized approximation schemes for multi-criteria cycle cover
and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented
at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006
Deterministic algorithms for multi-criteria TSP
We present deterministic approximation algorithms for the multi-criteria traveling salesman problem (TSP). Our algorithms are faster and simpler than the existing randomized algorithms.\ud
First, we devise algorithms for the symmetric and asymmetric multi-criteria Max-TSP that achieve ratios of 1/2k − ε and 1/(4k − 2) − ε, respectively, where k is the number of objective functions. For two objective functions, we obtain ratios of 3/8 − ε and 1/4 − ε for the symmetric and asymmetric TSP, respectively. Our algorithms are self-contained and do not use existing approximation schemes as black boxes.\ud
Second, we adapt the generic cycle cover algorithm for Min-TSP. It achieves ratios of 3/2 + ε, , and for multi-criteria Min-ATSP with distances 1 and 2, Min-ATSP with -triangle inequality and Min-STSP with -triangle inequality, respectively
Multi-criteria TSP:Min and max combined
We present randomized approximation algorithms for multi-criteria traveling salesman problems (TSP), where some objective functions should be minimized while others should be maximized. For the symmetric multi-criteria TSP (STSP), we present an algorithm that computes (2/3,)-approximate Pareto curves. Here, the first parameter is the approximation ratio for the objectives that should be maximized, and the second parameter is the ratio for the objectives that should be minimized. For the asymmetric multi-criteria TSP (ATSP), we obtain an approximation performance of (1/2, )
Note on VCG vs. Price Raising for Matching Markets
In \cite{EK10} the use of VCG in matching markets is motivated by saying that
in order to compute market clearing prices in a matching market, the auctioneer
needs to know the true valuations of the bidders. Hence VCG and corresponding
personalized prices are proposed as an incentive compatible mechanism. The same
line of argument pops up in several lecture sheets and other documents related
to courses based on Easley and Kleinberg's book, seeming to suggest that
computing market clearing prices and corresponding assignments were \emph{not}
incentive compatible. Main purpose of our note is to observe that, in contrast,
assignments based on buyer optimal market clearing prices are indeed incentive
compatible
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