214 research outputs found
Shorter tours and longer detours: Uniform covers and a bit beyond
Motivated by the well known four-thirds conjecture for the traveling salesman
problem (TSP), we study the problem of {\em uniform covers}. A graph
has an -uniform cover for TSP (2EC, respectively) if the everywhere
vector (i.e. ) dominates a convex combination of
incidence vectors of tours (2-edge-connected spanning multigraphs,
respectively). The polyhedral analysis of Christofides' algorithm directly
implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP.
Seb\H{o} asked if such graphs have -uniform covers for TSP for
some . Indeed, the four-thirds conjecture implies that such
graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform
covers for TSP. We also study uniform covers for 2EC and show that the
everywhere 15/17 vector can be efficiently written as a convex combination of
2-edge-connected spanning multigraphs.
For a weighted, 3-edge-connected, cubic graph, our results show that if the
everywhere 2/3 vector is an optimal solution for the subtour linear programming
relaxation, then a tour with weight at most 27/19 times that of an optimal tour
can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall
into this category. In this special case, we can apply our tools to obtain an
even better approximation guarantee.
To extend our approach to input graphs that are 2-edge-connected, we present
a procedure to decompose an optimal solution for the subtour relaxation for TSP
into spanning, connected multigraphs that cover each 2-edge cut an even number
of times. Using this decomposition, we obtain a 17/12-approximation algorithm
for minimum weight 2-edge-connected spanning subgraphs on subcubic,
node-weighted graphs
On the minimum leaf number of cubic graphs
The \emph{minimum leaf number} of a connected graph is
defined as the minimum number of leaves of the spanning trees of . We
present new results concerning the minimum leaf number of cubic graphs: we show
that if is a connected cubic graph of order , then , improving on the best known result in [Inf. Process.
Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph
Theory and Applications 5 (2017) 207-211]. We further prove that if is also
2-connected, then , improving on the best
known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new
conjectures concerning the minimum leaf number of several types of cubic graphs
and examples showing that the bounds of the conjectures are best possible.Comment: 17 page
Graphs with large palette index
Given an edge-coloring of a graph, the palette of a vertex is defined as the
set of colors of the edges which are incident with it. We define the palette
index of a graph as the minimum number of distinct palettes, taken over all
edge-colorings, occurring among the vertices of the graph. Several results
about the palette index of some specific classes of graphs are known. In this
paper we propose a different approach that leads to new and more general
results on the palette index. Our main theorem gives a sufficient condition for
a graph to have palette index larger than its minimum degree. In the second
part of the paper, by using such a result, we answer to two open problems on
this topic. First, for every odd, we construct a family of -regular
graphs with palette index reaching the maximum admissible value. After that, we
construct the first known family of simple graphs whose palette index grows
quadratically with respect to their maximum degree.Comment: 7 pages, 1 figur
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
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