185 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
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
New Inapproximability Bounds for TSP
In this paper, we study the approximability of the metric Traveling Salesman
Problem (TSP) and prove new explicit inapproximability bounds for that problem.
The best up to now known hardness of approximation bounds were 185/184 for the
symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to
Papadimitriou and Vempala). We construct here two new bounded occurrence CSP
reductions which improve these bounds to 123/122 and 75/74, respectively. The
latter bound is the first improvement in more than a decade for the case of the
asymmetric TSP. One of our main tools, which may be of independent interest, is
a new construction of a bounded degree wheel amplifier used in the proof of our
results
On Approximating Multi-Criteria TSP
We present approximation algorithms for almost all variants of the
multi-criteria traveling salesman problem (TSP).
First, we devise randomized approximation algorithms for multi-criteria
maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP,
where the edge weights have to be symmetric, we devise an algorithm with an
approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge
weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps.
Our algorithms work for any fixed number k of objectives. Furthermore, we
present a deterministic algorithm for bi-criteria Max-STSP that achieves an
approximation ratio of 7/27.
Finally, we present a randomized approximation algorithm for the asymmetric
multi-criteria minimum TSP with triangle inequality Min-ATSP. This algorithm
achieves a ratio of log n + eps.Comment: Preliminary version at STACS 2009. This paper is a revised full
version, where some proofs are simplifie
Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP
Given a metric space on n points, an {\alpha}-approximate universal algorithm
for the Steiner tree problem outputs a distribution over rooted spanning trees
such that for any subset X of vertices containing the root, the expected cost
of the induced subtree is within an {\alpha} factor of the optimal Steiner tree
cost for X. An {\alpha}-approximate differentially private algorithm for the
Steiner tree problem takes as input a subset X of vertices, and outputs a tree
distribution that induces a solution within an {\alpha} factor of the optimal
as before, and satisfies the additional property that for any set X' that
differs in a single vertex from X, the tree distributions for X and X' are
"close" to each other. Universal and differentially private algorithms for TSP
are defined similarly. An {\alpha}-approximate universal algorithm for the
Steiner tree problem or TSP is also an {\alpha}-approximate differentially
private algorithm. It is known that both problems admit O(logn)-approximate
universal algorithms, and hence O(log n)-approximate differentially private
algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation
ratio achievable for the universal Steiner tree problem and the universal TSP,
matching the known upper bounds. Our lower bound for the Steiner tree problem
holds even when the algorithm is allowed to output a more general solution of a
distribution on paths to the root.Comment: 14 page
A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time
We give the first constant-factor approximation for the Directed Latency
problem in quasi-polynomial time. Here, the goal is to visit all nodes in an
asymmetric metric with a single vehicle starting at a depot to minimize the
average time a node waits to be visited by the vehicle. The approximation
guarantee is an improvement over the polynomial-time -approximation
[Friggstad, Salavatipour, Svitkina, 2013] and no better quasi-polynomial time
approximation algorithm was known.
To obtain this, we must extend a recent result showing the integrality gap of
the Asymmetric TSP-Path LP relaxation is bounded by a constant [K\"{o}hne,
Traub, and Vygen, 2019], which itself builds on the breakthrough result that
the integrality gap for standard Asymmetric TSP is also a constant [Svensson,
Tarnawsi, and Vegh, 2018]. We show the standard Asymmetric TSP-Path integrality
gap is bounded by a constant even if the cut requirements of the LP relaxation
are relaxed from to
for some constant . We also give a better approximation
guarantee in the special case of Directed Latency in regret metrics where the
goal is to find a path minimize the average time a node waits in excess
of , i.e.
Combinatorial Optimization
Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry
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