2,203 research outputs found
The Salesman's Improved Tours for Fundamental Classes
Finding the exact integrality gap for the LP relaxation of the
metric Travelling Salesman Problem (TSP) has been an open problem for over
thirty years, with little progress made. It is known that , and a famous conjecture states . For this problem,
essentially two "fundamental" classes of instances have been proposed. This
fundamental property means that in order to show that the integrality gap is at
most for all instances of metric TSP, it is sufficient to show it only
for the instances in the fundamental class. However, despite the importance and
the simplicity of such classes, no apparent effort has been deployed for
improving the integrality gap bounds for them. In this paper we take a natural
first step in this endeavour, and consider the -integer points of one such
class. We successfully improve the upper bound for the integrality gap from
to for a superclass of these points, as well as prove a lower
bound of for the superclass. Our methods involve innovative applications
of tools from combinatorial optimization which have the potential to be more
broadly applied
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
A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case
Given a connected undirected graph on vertices, and
non-negative edge costs , the 2ECM problem is that of finding a
-edge~connected spanning multisubgraph of of minimum cost. The
natural linear program (LP) for 2ECM, which coincides with the subtour LP for
the Traveling Salesman Problem on the metric closure of , gives a
lower bound on the optimal cost. For instances where this LP is optimized by a
half-integral solution , Carr and Ravi (1998) showed that the integrality
gap is at most : they show that the vector dominates a
convex combination of incidence vectors of -edge connected spanning
multisubgraphs of .
We present a simpler proof of the result due to Carr and Ravi by applying an
extension of Lov\'{a}sz's splitting-off theorem. Our proof naturally leads to a
-approximation algorithm for half-integral instances. Given a
half-integral solution to the LP for 2ECM, we give an -time
algorithm to obtain a -edge connected spanning multisubgraph of
whose cost is at most
Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours
We present a new approach for gluing tours over certain tight, 3-edge cuts.
Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles
in special graph classes and in proving bounds for 2-edge-connected subgraph
problem, but not much was known in this direction for gluing connected
multigraphs. We apply this approach to the traveling salesman problem (TSP) in
the case when the objective function of the subtour elimination relaxation is
minimized by a -cyclic point: ,
where the support graph is subcubic and each vertex is incident to at least one
edge with -value 1. Such points are sufficient to resolve TSP in general.
For these points, we construct a convex combination of tours in which we can
reduce the usage of edges with -value 1 from the of
Christofides algorithm to while keeping the
usage of edges with fractional -value the same as Christofides algorithm. A
direct consequence of this result is for the Uniform Cover Problem for TSP: In
the case when the objective function of the subtour elimination relaxation is
minimized by a -uniform point: , we
give a -approximation algorithm for TSP. For such points, this
lands us halfway between the approximation ratios of of
Christofides algorithm and implied by the famous "four-thirds
conjecture"
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