62,065 research outputs found
On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings
The problem of establishing the number of perfect matchings necessary to
cover the edge-set of a cubic bridgeless graph is strictly related to a famous
conjecture of Berge and Fulkerson. In this paper we prove that deciding whether
this number is at most 4 for a given cubic bridgeless graph is NP-complete. We
also construct an infinite family of snarks (cyclically
4-edge-connected cubic graphs of girth at least five and chromatic index four)
whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs
were known. It turns out that the family also has interesting
properties with respect to the shortest cycle cover problem. The shortest cycle
cover of any cubic bridgeless graph with edges has length at least
, and we show that this inequality is strict for graphs of .
We also construct the first known snark with no cycle cover of length less than
.Comment: 17 pages, 8 figure
Covering cubic graphs with matchings of large size
Let m be a positive integer and let G be a cubic graph of order 2n. We
consider the problem of covering the edge-set of G with the minimum number of
matchings of size m. This number is called excessive [m]-index of G in
literature. The case m=n, that is a covering with perfect matchings, is known
to be strictly related to an outstanding conjecture of Berge and Fulkerson. In
this paper we study in some details the case m=n-1. We show how this parameter
can be large for cubic graphs with low connectivity and we furnish some
evidence that each cyclically 4-connected cubic graph of order 2n has excessive
[n-1]-index at most 4. Finally, we discuss the relation between excessive
[n-1]-index and some other graph parameters as oddness and circumference.Comment: 11 pages, 5 figure
The Lazy Flipper: MAP Inference in Higher-Order Graphical Models by Depth-limited Exhaustive Search
This article presents a new search algorithm for the NP-hard problem of
optimizing functions of binary variables that decompose according to a
graphical model. It can be applied to models of any order and structure. The
main novelty is a technique to constrain the search space based on the topology
of the model. When pursued to the full search depth, the algorithm is
guaranteed to converge to a global optimum, passing through a series of
monotonously improving local optima that are guaranteed to be optimal within a
given and increasing Hamming distance. For a search depth of 1, it specializes
to Iterated Conditional Modes. Between these extremes, a useful tradeoff
between approximation quality and runtime is established. Experiments on models
derived from both illustrative and real problems show that approximations found
with limited search depth match or improve those obtained by state-of-the-art
methods based on message passing and linear programming.Comment: C++ Source Code available from
http://hci.iwr.uni-heidelberg.de/software.ph
Transforming Monitoring Structures with Resilient Encoders. Application to Repeated Games
An important feature of a dynamic game is its monitoring structure namely,
what the players effectively see from the played actions. We consider games
with arbitrary monitoring structures. One of the purposes of this paper is to
know to what extent an encoder, who perfectly observes the played actions and
sends a complementary public signal to the players, can establish perfect
monitoring for all the players. To reach this goal, the main technical problem
to be solved at the encoder is to design a source encoder which compresses the
action profile in the most concise manner possible. A special feature of this
encoder is that the multi-dimensional signal (namely, the action profiles) to
be encoded is assumed to comprise a component whose probability distribution is
not known to the encoder and the decoder has a side information (the private
signals received by the players when the encoder is off). This new framework
appears to be both of game-theoretical and information-theoretical interest. In
particular, it is useful for designing certain types of encoders that are
resilient to single deviations and provide an equilibrium utility region in the
proposed setting; it provides a new type of constraints to compress an
information source (i.e., a random variable). Regarding the first aspect, we
apply the derived result to the repeated prisoner's dilemma.Comment: Springer, Dynamic Games and Applications, 201
Covering a cubic graph with perfect matchings
Let G be a bridgeless cubic graph. A well-known conjecture of Berge and
Fulkerson can be stated as follows: there exist five perfect matchings of G
such that each edge of G is contained in at least one of them. Here, we prove
that in each bridgeless cubic graph there exist five perfect matchings covering
a portion of the edges at least equal to 215/231 . By a generalization of this
result, we decrease the best known upper bound, expressed in terms of the size
of the graph, for the number of perfect matchings needed to cover the edge-set
of G.Comment: accepted for the publication in Discrete Mathematic
A method for pricing American options using semi-infinite linear programming
We introduce a new approach for the numerical pricing of American options.
The main idea is to choose a finite number of suitable excessive functions
(randomly) and to find the smallest majorant of the gain function in the span
of these functions. The resulting problem is a linear semi-infinite programming
problem, that can be solved using standard algorithms. This leads to good upper
bounds for the original problem. For our algorithms no discretization of space
and time and no simulation is necessary. Furthermore it is applicable even for
high-dimensional problems. The algorithm provides an approximation of the value
not only for one starting point, but for the complete value function on the
continuation set, so that the optimal exercise region and e.g. the Greeks can
be calculated. We apply the algorithm to (one- and) multidimensional diffusions
and to L\'evy processes, and show it to be fast and accurate
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