24,945 research outputs found
Packing tight Hamilton cycles in 3-uniform hypergraphs
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C
\subset H is a collection of N edges for which there is an ordering of the
vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i,
v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop
new techniques which enable us to prove that under certain natural
pseudo-random conditions, almost all edges of H can be covered by edge-disjoint
tight Hamilton cycles, for N divisible by 4. Consequently, we derive the
corollary that random 3-uniform hypergraphs can be almost completely packed
with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small.
Along the way, we develop a similar result for packing Hamilton cycles in
pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur
Maximum weight cycle packing in directed graphs, with application to kidney exchange programs
Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration
On Approximating the Sum-Rate for Multiple-Unicasts
We study upper bounds on the sum-rate of multiple-unicasts. We approximate
the Generalized Network Sharing Bound (GNS cut) of the multiple-unicasts
network coding problem with independent sources. Our approximation
algorithm runs in polynomial time and yields an upper bound on the joint source
entropy rate, which is within an factor from the GNS cut. It
further yields a vector-linear network code that achieves joint source entropy
rate within an factor from the GNS cut, but \emph{not} with
independent sources: the code induces a correlation pattern among the sources.
Our second contribution is establishing a separation result for vector-linear
network codes: for any given field there exist networks for which
the optimum sum-rate supported by vector-linear codes over for
independent sources can be multiplicatively separated by a factor of
, for any constant , from the optimum joint entropy
rate supported by a code that allows correlation between sources. Finally, we
establish a similar separation result for the asymmetric optimum vector-linear
sum-rates achieved over two distinct fields and
for independent sources, revealing that the choice of field
can heavily impact the performance of a linear network code.Comment: 10 pages; Shorter version appeared at ISIT (International Symposium
on Information Theory) 2015; some typos correcte
Parameterization Above a Multiplicative Guarantee
Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ? g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ?>0, multiplicative parameterization above g(I)^(1+?) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth
Optimal Recombination in Genetic Algorithms
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results
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