317 research outputs found
Even-cycle decompositions of graphs with no odd--minor
An even-cycle decomposition of a graph G is a partition of E(G) into cycles
of even length. Evidently, every Eulerian bipartite graph has an even-cycle
decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian
planar graph with an even number of edges also admits an even-cycle
decomposition. Later, Zhang (1994) generalized this to graphs with no
-minor.
Our main theorem gives sufficient conditions for the existence of even-cycle
decompositions of graphs in the absence of odd minors. Namely, we prove that
every 2-connected loopless Eulerian odd--minor-free graph with an even
number of edges has an even-cycle decomposition.
This is best possible in the sense that `odd--minor-free' cannot be
replaced with `odd--minor-free.' The main technical ingredient is a
structural characterization of the class of odd--minor-free graphs, which
is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio
P_4-Decomposability in Regular Graphs and Multigraphs
The main objective of this thesis is to review and expand the study of graph decomposability. An H-decomposition of a graph G=(V,E) is a partitioning of the edge set, , into edge-disjoint isomorphic copies of a subgraph H. In particular we focus on the decompositions of graphs into paths. We prove that a 2,4 mutligraph with maximum multiplicity 2 admits a C_2,C_3-free Euler tour (and thus, a decomposition into paths of length 3 if it has size a multiple of 3) if and only if it avoids a set of 15 forbidden structures. We also prove that a 4-regular multigraph with maximum multiplicity 2 admits a decomposition into paths of length three if and only if it has size a multiple of 3 and no three vertices induce more than 4 edges. We go on to outline drafted work reflecting further research into path decomposition problems
Eight-Fifth Approximation for TSP Paths
We prove the approximation ratio 8/5 for the metric -path-TSP
problem, and more generally for shortest connected -joins.
The algorithm that achieves this ratio is the simple "Best of Many" version
of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys
(2012), which consists in determining the best Christofides -tour out
of those constructed from a family \Fscr_{>0} of trees having a convex
combination dominated by an optimal solution of the fractional
relaxation. They give the approximation guarantee for
such an -tour, which is the first improvement after the 5/3 guarantee
of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao
(2012) extended this result to a 13/8-approximation of shortest connected
-joins, for .
The ratio 8/5 is proved by simplifying and improving the approach of An,
Kleinberg and Shmoys that consists in completing in order to dominate
the cost of "parity correction" for spanning trees. We partition the edge-set
of each spanning tree in \Fscr_{>0} into an -path (or more
generally, into a -join) and its complement, which induces a decomposition
of . This decomposition can be refined and then efficiently used to
complete without using linear programming or particular properties of
, but by adding to each cut deficient for an individually tailored
explicitly given vector, inherent in .
A simple example shows that the Best of Many Christofides algorithm may not
find a shorter -tour than 3/2 times the incidentally common optima of
the problem and of its fractional relaxation.Comment: 15 pages, corrected typos in citations, minor change
The traveling salesman problem on cubic and subcubic graphs
We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on TeX vertices a tour of length TeX exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented an algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Mömke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs
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