Introduction
Some lower and upper bounds
PTAS
PTAS for the k-Tour Cover Problem on the
Euclidean Plane for Moderately Large Values of k
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
DIMAP, University of Warwick, UK
Lund University, Sweden
ISAAC 18.12.2009
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
k-tour problem in R 2
input: a set of n points P and a special
point O called the origin (in R 2 )
O
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
k-tour problem in R 2
input: a set of n points P and a special
point O called the origin (in R 2 )
O
k-tour cover: a set of tours (cycles), each
including the origin and at most k points
from P, which covers all points from P
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
k-tour problem in R 2
input: a set of n points P and a special
point O called the origin (in R 2 )
O
k-tour cover: a set of tours (cycles), each
including the origin and at most k points
from P, which covers all points from P
goal: find a k-tour cover with minimum
total length
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
k-tour problem in R 2
input: a set of n points P and a special
point O called the origin (in R 2 )
O
k-tour cover: a set of tours (cycles), each
including the origin and at most k points
from P, which covers all points from P
goal: find a k-tour cover with minimum
total length
example: depot and clients
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Results
In the metric space:
APX-hard for all k ≥ 3
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Results
In the metric space:
APX-hard for all k ≥ 3
In the Euclidean plane:
PTAS for k = O(log n/ log log n) [Asano et al., 1997]
PTAS for k = Ω(n) [Asano et al., 1997]
quasi-polynomial time approximation scheme (QPTAS);
O(1/ε)
n [Das and Mathieu, 2008]
running time: nlog
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Results
In the metric space:
APX-hard for all k ≥ 3
In the Euclidean plane:
PTAS for k = O(log n/ log log n) [Asano et al., 1997]
PTAS for k = Ω(n) [Asano et al., 1997]
quasi-polynomial time approximation scheme (QPTAS);
O(1/ε)
n [Das and Mathieu, 2008]
running time: nlog
Our result:
δ
PTAS for k ≤ 2log n , where δ = δ(ε)
in particular for k = polylog(n)
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Lemma
TSP ≤ OPT
From a solution to the k-tour cover problem we can easily get a
solution to the TSP problem using only shortcutting.
O
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
⇒
O
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Lemma
2 P
p∈P d(p, O) ≤ OPT
k ·
Let T be a tour from a k-tour cover.
c(T ) — cost of the tour T
p
For any p on T : c(T ) ≥ 2 · d(p, O).
Tour T visits at mostP
k points,
therefore c(T ) ≥ k2 · p∈T d(p, O).
O
Now it is enough to sum up the results for all tours.
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Lemma
OPT ≤ TSP +
2
k
·
P
p∈P
d(p, O)
Suppose that k|n. Let T be the optimal TSP tour for our points.
Remove point O from T .
We can divide the tour into paths, such that each path visits
exactly k consecutive points from P. We add edges to the origin
to transform paths into tours.
P
Average cost of such a tour cover — TSP + k2 · p∈P d(p, O)
O
O
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
O
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
We have shown the following inequalities:
max{TSP,
2 X
2 X
·
d(p, O)} ≤ OPT ≤ TSP + ·
d(p, O)
k
k
p∈P
p∈P
Using a simple 2-approximation algorithm for the TSP we can
construct in polynomial
time a k-tour cover with cost at most
2 P
2 · TSP + k · p∈P d(p, O).
This gives us a 3-approximation algorithm.
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Removing close points
L - maximum distance from a point in P to the origin
We can cover all points at a distance at most Lε/n from the origin
by 1-tours.
L
Lε/n
O
The additional cost generated is at most n · 2Lε/n = 2Lε ≤ εOPT .
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Locations
Idea:
OPT ≥
2
k
·
P
p∈P
d(p, O)
moving each point p ∈ P by a distance at most
can change the total cost by at most ε · OPT
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
ε
k
· d(p, O)
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Locations
Idea:
OPT ≥
2
k
·
P
p∈P
d(p, O)
moving each point p ∈ P by a distance at most
can change the total cost by at most ε · OPT
ε
k
· d(p, O)
Circles and rays:
smallest radius: Lε/n
radius growing by a factor (1 + kε )
distance between rays ≈ radius
O
location: intersection of a circle
and a ray
T = Θ(k 2 ε−2 log(n/ε)) locations
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Moving points to nearest location
O
O
Such an operation changes the cost by at most ε · OPT .
Advantage: all points are in T = Θ(k 2 ε−2 log(n/ε)) locations.
Remark: for the rest of the talk we will consider the modified
problem (all points are in locations).
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Trivial and nontrivial tours
We call a tour trivial if it visits only a single location; a tour is
nontrivial otherwise.
Theorem
There is an optimal solution in which each location is visited by at
most T nontrivial tours (T — number of locations).
Idea:
we take the optimal k-tour cover
as long as we have a pair of nontrivial tours which have two
locations in common we will perform an operation which
eliminates such a pair without increasing the cost
if each pair of nontrivial tours has at most one location in
common — each location is visited by at most T nontrivial
tours
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Trivial and nontrivial tours
Suppose that we have tours T1 and T2 which visit locations p and
q.
p
q
We will swap the points from locations
p and q between the tours. We can do
it in such a way that after our operation
one of the tours will visit only one of
the locations p, q.
This operation does not increase the
cost of the solution.
O
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Decreasing the number of points
We already know that there is an optimal solution in which:
each location is visited by at most T nontrivial tours
in each location there are at most T · k points visited by
nontrivial tours
Corollary
Using trivial tours we can reduce the problem to one with at most
T 2 · k = (k/ε · log n)O(1) points.
Remark: we can improve this result and reduce the problem to a
collection of independent problems, each with (k/ε)O(1) points.
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Polynomial time approximation scheme
There is a quasi-polynomial time approximation scheme (QPTAS)
for k-tour cover problem on the Euclidean plane. Its running time
O(1/ε)
n.
is nlog
We can run this algorithm on the modified problem instance with
(k/ε · log n)O(1) points. We will achieve (1 + ε)-approximation.
δ
Running time will be polynomial for k ≤ 2log n , where δ = δ(ε).
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...
Introduction
Some lower and upper bounds
PTAS
Summary
Our approximation algorithm:
Removing close points
Moving points to nearest location
Decreasing the number of points by using trivial tours
Applying QPTAS
Open problem: Does there exist a polynomial time approximation
scheme for arbitrary k?
Anna Adamaszek, Artur Czumaj, Andrzej Lingas
PTAS for the k-Tour Cover Problem on the Euclidean Plane ...