770 research outputs found
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
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
Approximation Algorithms for Path TSP, ATSP, and TAP via Relaxations
Linear programming (LP) relaxations provide a powerful technique to design approximation algorithms for combinatorial optimization problems. In the first part of the thesis, we study the metric s-t path Traveling Salesman Problem (TSP) via LP relaxations.
We first consider the s-t path graph-TSP, a critical special case of the metric s-t path TSP. We design a new simple LP-based algorithm for the s-t path graph-TSP that achieves the best known approximation factor of 1.5. Then, we turn our attention to the general metric s-t path TSP. [An, Kleinberg, and Shmoys, STOC 2012] improved on the long standing 5/3-approximation factor and presented an algorithm that achieves an approximation factor of (1+\sqrt{5})/2 \approx 1.61803. Later, [Sebo, IPCO 2013] further improved the approximation factor to 8/5. We present a simple, self-contained analysis that unifies both results. Additionally, we compare two different LP relaxations of the s-t path TSP, namely the path version of the Held-Karp LP relaxation for TSP and a weaker LP relaxation, and we show that both LPs have the same (fractional) optimal value. Also, we show that the minimum cost of integral solutions of the two LPs are within a factor of 3/2 of each other. Furthermore, we prove that a half-integral solution of the stronger LP relaxation of cost c can be rounded to an integral solution of cost at most 3c/2. Finally, we give an instance that presents obstructions to two natural methods that aim for an approximation factor of 3/2.
The Sherali-Adams (SA) system and the Lasserre (Las) system are two popular Lift-and-Project systems that tighten a given LP relaxation in a systematic way. In the second part of the thesis, we study the Asymmetric Traveling Salesman Problem (ATSP) and unweighted Tree Augmentation Problem, respectively, in the framework of the SA system and the Las system.
For ATSP, our focus is on negative results. For any fixed integer t>=0 and small \epsilon, 0<\epsilon<<1, we prove that the integrality ratio for level t of the SA system starting with the standard LP relaxation of ATSP is at least 1+(1-\epsilon)/(2t+3). For a further relaxation of ATSP called the balanced LP relaxation, we obtain an integrality ratio lower bound of 1+(1-\epsilon)/(t+1) for level t of the SA system. Also, our results for the standard LP relaxation extend to the path version of ATSP.
For the unweighted Tree Augmentation Problem, our focus is on positive results. We study this problem via the Las system. We prove an upper bound of (1.5+\epsilon) on the integrality ratio of a semidefinite programming (SDP) relaxation, where \epsilon>0 can be any small constant, by analyzing a combinatorial algorithm. This SDP relaxation is derived by applying the Las system to an initial LP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Las system via the decomposition result of [Karlin, Mathieu, and Nguyen, IPCO 2011]
Approximation Algorithms for the Asymmetric Traveling Salesman Problem : Describing two recent methods
The paper provides a description of the two recent approximation algorithms
for the Asymmetric Traveling Salesman Problem, giving the intuitive description
of the works of Feige-Singh[1] and Asadpour et.al\ [2].\newline [1] improves
the previous approximation algorithm, by improving the constant
from 0.84 to 0.66 and modifying the work of Kaplan et. al\ [3] and also shows
an efficient reduction from ATSPP to ATSP. Combining both the results, they
finally establish an approximation ratio of for ATSPP,\ considering a small ,\ improving the
work of Chekuri and Pal.[4]\newline Asadpour et.al, in their seminal work\ [2],
gives an randomized algorithm for
the ATSP, by symmetrizing and modifying the solution of the Held-Karp
relaxation problem and then proving an exponential family distribution for
probabilistically constructing a maximum entropy spanning tree from a spanning
tree polytope and then finally defining the thin-ness property and transforming
a thin spanning tree into an Eulerian walk.\ The optimization methods used in\
[2] are quite elegant and the approximation ratio could further be improved, by
manipulating the thin-ness of the cuts.Comment: 12 page
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