14 research outputs found
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
Approximation algorithms for general cluster routing problem
Graph routing problems have been investigated extensively in operations
research, computer science and engineering due to their ubiquity and vast
applications. In this paper, we study constant approximation algorithms for
some variations of the general cluster routing problem. In this problem, we are
given an edge-weighted complete undirected graph whose vertex set
is partitioned into clusters We are also given a subset
of and a subset of The weight function satisfies the
triangle inequality. The goal is to find a minimum cost walk that visits
each vertex in only once, traverses every edge in at least once and
for every all vertices of are traversed consecutively.Comment: In COCOON 202
Improving on Best-of-Many-Christofides for -tours
The -tour problem is a natural generalization of TSP and Path TSP. Given a
graph , edge cost , and an even
cardinality set , we want to compute a minimum-cost -join
connecting all vertices of (and possibly containing parallel edges).
In this paper we give an -approximation for the -tour
problem and show that the integrality ratio of the standard LP relaxation is at
most . Despite much progress for the special case Path TSP, for
general -tours this is the first improvement on Seb\H{o}'s analysis of the
Best-of-Many-Christofides algorithm (Seb\H{o} [2013])