7,540 research outputs found
Max-Min Greedy Matching
A bipartite graph G(U,V;E) that admits a perfect matching is given. One player imposes a permutation pi over V, the other player imposes a permutation sigma over U. In the greedy matching algorithm, vertices of U arrive in order sigma and each vertex is matched to the highest (under pi) yet unmatched neighbor in V (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals pi, and the second (min) player responds with the worst possible sigma for pi, does there exist a permutation pi ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time?
The main result of this paper is an affirmative answer for these questions: we show that there exists a polytime algorithm to compute pi for which for every sigma at least rho > 0.51 fraction of the vertices of V are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian graphs. Our solution solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations
The -matching problem on bipartite graphs
The -matching problem on bipartite graphs is studied with a local
algorithm. A -matching () on a bipartite graph is a set of matched
edges, in which each vertex of one type is adjacent to at most matched edge
and each vertex of the other type is adjacent to at most matched edges. The
-matching problem on a given bipartite graph concerns finding -matchings
with the maximum size. Our approach to this combinatorial optimization are of
two folds. From an algorithmic perspective, we adopt a local algorithm as a
linear approximate solver to find -matchings on general bipartite graphs,
whose basic component is a generalized version of the greedy leaf removal
procedure in graph theory. From an analytical perspective, in the case of
random bipartite graphs with the same size of two types of vertices, we develop
a mean-field theory for the percolation phenomenon underlying the local
algorithm, leading to a theoretical estimation of -matching sizes on
coreless graphs. We hope that our results can shed light on further study on
algorithms and computational complexity of the optimization problem.Comment: 15 pages, 3 figure
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms
In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for
random sampling of combinatorial objects. The running time of such an algorithm
depends on the total mixing time of the underlying Markov chain and is unknown
in general. For some Markov chains, upper bounds on this total mixing time
exist but are too large to be applicable in practice. We try to answer the
question, whether the total mixing time is close to its upper bounds, or if
there is a significant gap between them. In doing so, we present the software
library marathon which is designed to support the analysis of MCMC based
sampling algorithms. The main application of this library is to compute
properties of so-called state graphs which represent the structure of Markov
chains. We use marathon to investigate the quality of several bounding methods
on four well-known Markov chains for sampling perfect matchings and bipartite
graph realizations. In a set of experiments, we compute the total mixing time
and several of its bounds for a large number of input instances. We find that
the upper bound gained by the famous canonical path method is several
magnitudes larger than the total mixing time and deteriorates with growing
input size. In contrast, the spectral bound is found to be a precise
approximation of the total mixing time
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