8,040 research outputs found
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
Given a simple graph , a subset of is called a triangle cover if
it intersects each triangle of . Let and denote the
maximum number of pairwise edge-disjoint triangles in and the minimum
cardinality of a triangle cover of , respectively. Tuza conjectured in 1981
that holds for every graph . In this paper, using a
hypergraph approach, we design polynomial-time combinatorial algorithms for
finding small triangle covers. These algorithms imply new sufficient conditions
for Tuza's conjecture on covering and packing triangles. More precisely,
suppose that the set of triangles covers all edges in . We
show that a triangle cover of with cardinality at most can be
found in polynomial time if one of the following conditions is satisfied: (i)
, (ii) , (iii)
.
Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs,
Combinatorial algorithm
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
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