267 research outputs found
Improving Christofides' Algorithm for the s-t Path TSP
We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t
path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices
including two prespecified endpoints, the problem is to find a shortest
Hamiltonian path between the two endpoints; Hoogeveen showed that the natural
variant of Christofides' algorithm is a 5/3-approximation algorithm for this
problem, and this asymptotically tight bound in fact has been the best
approximation ratio known until now. We modify this algorithm so that it
chooses the initial spanning tree based on an optimal solution to the Held-Karp
relaxation rather than a minimum spanning tree; we prove this simple but
crucial modification leads to an improved approximation ratio, surpassing the
20-year-old barrier set by the natural Christofides' algorithm variant. Our
algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of
the path-variant Held-Karp relaxation. The techniques devised in this paper can
be applied to other optimization problems as well: these applications include
improved approximation algorithms and improved LP integrality gap upper bounds
for the prize-collecting s-t path problem and the unit-weight graphical metric
s-t path TSP.Comment: 31 pages, 5 figure
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"
A 4/3-Approximation Algorithm for Half-Integral Cycle Cut Instances of the TSP
A long-standing conjecture for the traveling salesman problem (TSP) states
that the integrality gap of the standard linear programming relaxation of the
TSP is at most 4/3. Despite significant efforts, the conjecture remains open.
We consider the half-integral case, in which the LP has solution values in
. Such instances have been conjectured to be the most difficult
instances for the overall four-thirds conjecture. Karlin, Klein, and Oveis
Gharan, in a breakthrough result, were able to show that in the half-integral
case, the integrality gap is at most 1.49993. This result led to the first
significant progress on the overall conjecture in decades; the same authors
showed the integrality gap is at most in the non-half-integral
case. For the half-integral case, the current best-known ratio is 1.4983, a
result by Gupta et al.
With the improvements on the 3/2 bound remaining very incremental even in the
half-integral case, we turn the question around and look for a large class of
half-integral instances for which we can prove that the 4/3 conjecture is
correct.
The previous works on the half-integral case perform induction on a hierarchy
of critical tight sets in the support graph of the LP solution, in which some
of the sets correspond to "cycle cuts" and the others to "degree cuts". We show
that if all the sets in the hierarchy correspond to cycle cuts, then we can
find a distribution of tours whose expected cost is at most 4/3 times the value
of the half-integral LP solution; sampling from the distribution gives us a
randomized 4/3-approximation algorithm. We note that the known bad cases for
the integrality gap have a gap of 4/3 and have a half-integral LP solution in
which all the critical tight sets in the hierarchy are cycle cuts; thus our
result is tight.Comment: Comments, questions, and suggestions are welcome
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