2,153 research outputs found
Packing Odd Walks and Trails in Multiterminal Networks
Let G be an undirected network with a distinguished set of terminals T ? V(G) and edge capacities cap: E(G) ? ?_+. By an odd T-walk we mean a walk in G (with possible vertex and edge self-intersections) connecting two distinct terminals and consisting of an odd number of edges. Inspired by the work of Schrijver and Seymour on odd path packing for two terminals, we consider packings of odd T-walks subject to capacities cap.
First, we present a strongly polynomial time algorithm for constructing a maximum fractional packing of odd T-walks. For even integer capacities, our algorithm constructs a packing that is half-integer. Additionally, if cap(?(v)) is divisible by 4 for any v ? V(G)-T, our algorithm constructs an integer packing.
Second, we establish and prove the corresponding min-max relation.
Third, if G is inner Eulerian (i.e. degrees of all nodes in V(G)-T are even) and cap(e) = 2 for all e ? E, we show that there exists an integer packing of odd T-trails (i.e. odd T-walks with no repeated edges) of the same value as in case of odd T-walks, and this packing can be found in polynomial time.
To achieve the above goals, we establish a connection between packings of odd T-walks and T-trails and certain multiflow problems in undirected and bidirected graphs
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Subdivision into i-packings and S-packing chromatic number of some lattices
An -packing in a graph is a set of vertices at pairwise distance
greater than . For a nondecreasing sequence of integers
, the -packing chromatic number of a graph is
the least integer such that there exists a coloring of into colors
where each set of vertices colored , , is an -packing.
This paper describes various subdivisions of an -packing into -packings
(j\textgreater{}i) for the hexagonal, square and triangular lattices. These
results allow us to bound the -packing chromatic number for these graphs,
with more precise bounds and exact values for sequences ,
S-Packing Colorings of Cubic Graphs
Given a non-decreasing sequence of positive
integers, an {\em -packing coloring} of a graph is a mapping from
to such that any two vertices with color
are at mutual distance greater than , . This paper
studies -packing colorings of (sub)cubic graphs. We prove that subcubic
graphs are -packing colorable and -packing
colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we
provide an example of a cubic graph of order which is not
-packing colorable
How many matchings cover the nodes of a graph?
Given an undirected graph, are there matchings whose union covers all of
its nodes, that is, a matching--cover? A first, easy polynomial solution
from matroid union is possible, as already observed by Wang, Song and Yuan
(Mathematical Programming, 2014). However, it was not satisfactory neither from
the algorithmic viewpoint nor for proving graphic theorems, since the
corresponding matroid ignores the edges of the graph.
We prove here, simply and algorithmically: all nodes of a graph can be
covered with matchings if and only if for every stable set we have
. When , an exception occurs: this condition is not
enough to guarantee the existence of a matching--cover, that is, the
existence of a perfect matching, in this case Tutte's famous matching theorem
(J. London Math. Soc., 1947) provides the right `good' characterization. The
condition above then guarantees only that a perfect -matching exists, as
known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953).
Some results are then deduced as consequences with surprisingly simple
proofs, using only the level of difficulty of bipartite matchings. We give some
generalizations, as well as a solution for minimization if the edge-weights are
non-negative, while the edge-cardinality maximization of matching--covers
turns out to be already NP-hard.
We have arrived at this problem as the line graph special case of a model
arising for manufacturing integrated circuits with the technology called
`Directed Self Assembly'.Comment: 10 page
Interleaving schemes for multidimensional cluster errors
We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice
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