216 research outputs found
Asymmetric Traveling Salesman Path and Directed Latency Problems
We study integrality gaps and approximability of two closely related problems
on directed graphs. Given a set V of n nodes in an underlying asymmetric metric
and two specified nodes s and t, both problems ask to find an s-t path visiting
all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the
objective is to minimize the total cost of this path. In the directed latency
problem, the objective is to minimize the sum of distances on this path from s
to each node. Both of these problems are NP-hard. The best known approximation
algorithms for ATSPP had ratio O(log n) until the very recent result that
improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the
integrality gap of its linear programming relaxation has been known. For
directed latency, the best previously known approximation algorithm has a
guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new
algorithm for the ATSPP problem that has an approximation ratio of O(log n),
but whose analysis also bounds the integrality gap of the standard LP
relaxation of ATSPP by the same factor. This solves an open problem posed by
Chekuri and Pal [2007]. We then pursue a deeper study of this linear program
and its variations, which leads to an algorithm for the k-person ATSPP (where k
s-t paths of minimum total length are sought) and an O(log n)-approximation for
the directed latency problem
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
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
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
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