109 research outputs found
A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs
We present a near-optimal polynomial-time approximation algorithm for the
asymmetric traveling salesman problem for graphs of bounded orientable or
non-orientable genus. Our algorithm achieves an approximation factor of O(f(g))
on graphs with genus g, where f(n) is the best approximation factor achievable
in polynomial time on arbitrary n-vertex graphs. In particular, the
O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et
al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation
algorithm for genus-g graphs. Our result improves the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA
2011], which applies only to graphs with orientable genus g; ours is the first
approximation algorithm for graphs with bounded non-orientable genus.
Moreover, using recent progress on approximating the genus of a graph, our
O(log(g) / loglog(g))-approximation can be implemented even without an
embedding when the input graph has bounded degree. In contrast, the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a
genus-g embedding as part of the input.
Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on
graphs of genus g, with running time 2^O(g)*n^O(1)
Approximating ATSP by Relaxing Connectivity
The standard LP relaxation of the asymmetric traveling salesman problem has
been conjectured to have a constant integrality gap in the metric case. We
prove this conjecture when restricted to shortest path metrics of node-weighted
digraphs. Our arguments are constructive and give a constant factor
approximation algorithm for these metrics. We remark that the considered case
is more general than the directed analog of the special case of the symmetric
traveling salesman problem for which there were recent improvements on
Christofides' algorithm.
The main idea of our approach is to first consider an easier problem obtained
by significantly relaxing the general connectivity requirements into local
connectivity conditions. For this relaxed problem, it is quite easy to give an
algorithm with a guarantee of 3 on node-weighted shortest path metrics. More
surprisingly, we then show that any algorithm (irrespective of the metric) for
the relaxed problem can be turned into an algorithm for the asymmetric
traveling salesman problem by only losing a small constant factor in the
performance guarantee. This leaves open the intriguing task of designing a
"good" algorithm for the relaxed problem on general metrics.Comment: 25 pages, 2 figures, fixed some typos in previous versio
Constant-factor approximations for asymmetric TSP on nearly-embeddable graphs
In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a
closed walk of minimum cost in a directed graph visiting every vertex. We
consider the approximability of ATSP on topologically restricted graphs. It has
been shown by [Oveis Gharan and Saberi 2011] that there exists polynomial-time
constant-factor approximations on planar graphs and more generally graphs of
constant orientable genus. This result was extended to non-orientable genus by
[Erickson and Sidiropoulos 2014].
We show that for any class of \emph{nearly-embeddable} graphs, ATSP admits a
polynomial-time constant-factor approximation. More precisely, we show that for
any fixed , there exist , such that ATSP on
-vertex -nearly-embeddable graphs admits a -approximation in time
. The class of -nearly-embeddable graphs contains graphs with at
most apices, vortices of width at most , and an underlying surface
of either orientable or non-orientable genus at most . Prior to our work,
even the case of graphs with a single apex was open. Our algorithm combines
tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs
of pathwidth (and hence on -nearly embeddable graphs) requires time
, assuming the Exponential-Time Hypothesis (ETH). This is
surprising in light of the fact that both TSP on undirected graphs and Minimum
Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth
Solving Dynamic Traveling Salesman Problem Using Dynamic Gaussian Process Regression
This paper solves the dynamic traveling salesman problem (DTSP) using dynamic Gaussian Process Regression (DGPR) method. The problem of varying correlation tour is alleviated by the nonstationary covariance function interleaved with DGPR to generate a predictive distribution for DTSP tour. This approach is conjoined with Nearest Neighbor (NN) method and the iterated local search to track dynamic optima. Experimental results were obtained on DTSP instances. The comparisons were performed with Genetic Algorithm and Simulated Annealing. The proposed approach demonstrates superiority in finding good traveling salesman problem (TSP) tour and less computational time in nonstationary conditions
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