An Improved Approximation for k-median, and Positive Correlation in Budgeted Optimization

29 Pages·2015·0.42 MB·English

An Improved Approximation for k-median,
and Positive Correlation in Budgeted
Optimization
Jaroslaw Byrka 1 Thomas Pensyl 2 Bartosz Rybicki 1
Aravind Srinivasan 2 Khoa Trinh 2
1UniversityofWroclaw,Poland
2UniversityofMaryland,USA
ISMP 2015
k-median
(cid:73) Input:
(cid:73) setoffacilities
F
(cid:73) setofclients
C
(cid:73) symmetricdistancemetricd on
C∪F
(cid:73) anintegerk >0
(cid:73) Goal: pick k facilities which minimize total connection cost
of clients.
(cid:73) Example: k = 2, open facilities A,C and pay cost
2+2+3 = 7.
A B C
2
2 1 3
4 3
Uncapcitated Facility Location
(cid:73) Input:
(cid:73) setoffacilities
F
(cid:73) setofclients
C
(cid:73) symmetricdistancemetricd on
C∪F
(cid:73) eachfacilityhasanopeningcost
(cid:73) Goal: pick some facilities which minimize total opening
cost and connection cost.
(cid:73) Example: open facilities A,C and pay cost
2+2+3+3+1 = 11.
3 4 1
A B C
2
2 1 3
4 3
Bi-point solution as middle step
Bi-point solution: a feasible solution which is a convex
combination of two integral solutions.
(cid:73) Step I: Construct a bi-point solution.
(cid:73) UsesomespecialUFLalgorithmstogeta“cheap”bi-point
solution.
(cid:73) Step II: Round bi-point solution to integral one.
Previous Work
Year% Authors% Bi.point% Bi.point% k.median%
Construc3on% Rounding% Approxima3on%
‘99# Charikar#et.#al.# (LP#Rounding)# 6.667#
‘99# Jain#&#Vazirani# 3#+#ε# 2# 6#+#ε#
‘02# Jain,Mahdian#&#Saberi# 2#+#ε# 2# 4#+#ε#
‘01# Arya#et.#al.# (Local#Search)# 3#+#ε#
‘12# Li#&#Svensson# 2#+#ε# 1.366#+#ε# 2.732#+#ε#
‘15% Our%work% 2#+#ε# 1.337%+%ε% 2.674%+%ε%
Bi-point Rounding
(cid:73) Again, bi-point solution is convex combination of two
integral solutions, and .
1 2
F F
(cid:73) a +b = k, where (a,b > 0,a+b = 1)
1 2
|F | |F |
22
FF
11
FF
Stars
(cid:73) Form stars by attaching each facility in to closest facility
1
F
in .
1
F
22
FF
11
FF
Li-Svensson’s rounding algorithm
(cid:73) Key property: “if some leaf is closed then its root must be
open”.
(cid:73) Pr[i is open] a for all i
1 1 1
≈ ∈ F
(cid:73) Pr[i is open] b for all i
2 2 2
≈ √∈ F
(cid:73) approximation factor = 1+ 3 1.366
2 ≈
(cid:73) #open facilities = a +b +O(1) = k +O(1).
1 2
|F | |F |
22
FF
11
FF
Distance bounds
ii
22
dd
22
dd
11
jj
dd
33
ii ii
11 33
(cid:73) There is an open facility in i ,i ,i
1 2 3
{ }
(cid:73) d d +d(i ,i ) d +2d
3 2 2 3 1 2
≤ ≤
(cid:73) The total (expected) connection cost is a (non-linear)
function of cost( ), cost( ),a,b which can be related
1 2
F F
to OPT.
Tight Instance
(cid:73) In tight case, almost all stars have exactly 1 leaf.
(cid:73) BUT still many clients near 2-leafed stars.