160 research outputs found
Approximation Algorithms for Multi-Criteria Traveling Salesman Problems
In multi-criteria optimization problems, several objective functions have to
be optimized. Since the different objective functions are usually in conflict
with each other, one cannot consider only one particular solution as the
optimal solution. Instead, the aim is to compute a so-called Pareto curve of
solutions. Since Pareto curves cannot be computed efficiently in general, we
have to be content with approximations to them.
We design a deterministic polynomial-time algorithm for multi-criteria
g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate
Pareto curves for all 1/2<=g<=1. In particular, we obtain a
(2+eps)-approximation for multi-criteria metric STSP. We also present two
randomized approximation algorithms for multi-criteria g-metric STSP that
achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1
+3g -4g^2) + eps, respectively.
Moreover, we present randomized approximation algorithms for multi-criteria
g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with
weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do
this, we design randomized approximation schemes for multi-criteria cycle cover
and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented
at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006
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
Minimum-weight Cycle Covers and Their Approximability
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L.
We investigate how well L-cycle covers of minimum weight can be approximated.
For undirected graphs, we devise a polynomial-time approximation algorithm that
achieves a constant approximation ratio for all sets L. On the other hand, we
prove that the problem cannot be approximated within a factor of 2-eps for
certain sets L.
For directed graphs, we present a polynomial-time approximation algorithm
that achieves an approximation ratio of O(n), where is the number of
vertices. This is asymptotically optimal: We show that the problem cannot be
approximated within a factor of o(n).
To contrast the results for cycle covers of minimum weight, we show that the
problem of computing L-cycle covers of maximum weight can, at least in
principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2007). Minor change
Deterministic algorithms for multi-criteria TSP
We present deterministic approximation algorithms for the multi-criteria traveling salesman problem (TSP). Our algorithms are faster and simpler than the existing randomized algorithms.\ud
First, we devise algorithms for the symmetric and asymmetric multi-criteria Max-TSP that achieve ratios of 1/2k − ε and 1/(4k − 2) − ε, respectively, where k is the number of objective functions. For two objective functions, we obtain ratios of 3/8 − ε and 1/4 − ε for the symmetric and asymmetric TSP, respectively. Our algorithms are self-contained and do not use existing approximation schemes as black boxes.\ud
Second, we adapt the generic cycle cover algorithm for Min-TSP. It achieves ratios of 3/2 + ε, , and for multi-criteria Min-ATSP with distances 1 and 2, Min-ATSP with -triangle inequality and Min-STSP with -triangle inequality, respectively
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.
On Approximating Restricted Cycle Covers
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L. The weight of a cycle cover of an edge-weighted
graph is the sum of the weights of its edges.
We come close to settling the complexity and approximability of computing
L-cycle covers. On the one hand, we show that for almost all L, computing
L-cycle covers of maximum weight in directed and undirected graphs is APX-hard
and NP-hard. Most of our hardness results hold even if the edge weights are
restricted to zero and one.
On the other hand, we show that the problem of computing L-cycle covers of
maximum weight can be approximated within a factor of 2 for undirected graphs
and within a factor of 8/3 in the case of directed graphs. This holds for
arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change
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