Combinatorial Interpretations of
Dual Fitting and Primal Fitting
Ari Freund
Cesarea Rothschild Institute, University of Haifa
Dror Rawitz
Department of Computer Science, Technion
Approximation Using LP Duality


Minimization problem
LP-relaxation and dual:
(P) min wT x
s.t. Ax  b
x0
(D) max
bT y
s.t. AT x  w
y0
Find xZn and y such that wTx  r · bTy
 wTx  r · bTy  r · Opt(P)  r · Opt

Question: How do we find such solutions?
2
Primal-Dual Schema




x and y are constructed simultaneously
In each iteration: y is updated such that
relaxed dual complementary slackness
conditions are satisfied
Primal complementary slackness conditions
are obeyed
Used extensively in the last decade
(e.g., [GW95,BT98])
3
A Combinatorial Approach:
The Local Ratio Technique


Based on weight manipulation
Primal-Dual Schema  Local Ratio Technique
[BR01]



Dual update  Weight subtraction
Local Ratio Technique is more intuitive
Breakthrough results were achieved due to local
ratio (e.g., FVS [BBF99,BG96], Max [BBFNS01])
Conclusion: combinatorial approach is beneficial
4
Metric Uncapacitated Facility
Location Problem (MUFL)
Non-Standard Applications:
 3-approximation algorithm that relaxes
primal comp. slackness conditions [JV01]
 1.861 and 1.61-approximation algorithms
both using dual fitting [JMMSV03]
Motivation: combinatorial interpretations
of both non standard applications
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Dual Fitting

Construct an infeasible dual y and a
feasible primal x such that wTx  bTy

Find r s.t. y/r is feasible

wTx  bTy = r · bT(y/r)  r ·
Opt
Problem: finding the smallest
r s.t. (for all input instances)
y/r is feasible.
y
y/8
6
This Work
Two new approximation frameworks:
 Combinatorial
 Based on weight manipulation
(in the spirit of local ratio)
Framework

1st
Dual Fitting
2nd
Primal Fitting*
Examples
MUFL [JMMSV03]
MUFL [JV01]
Set Cover [Chv79]
Disk Cover [Chu]
* Defined in this paper
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An Example: Set Cover



Input: C = {S1,…,Sm}, Si U, w : C  R+
Solution: C’  C s.t.
S U
Measure:
w( S ) SC '


SC '
Algorithm Greedy:
1. While instance is not empty do:
2.
k  argmini{w(Si) / |Si|}
3.
Add Sk to the solution
4.
Remove the elements in Sk and discard empty sets
Approximation ratio is H n 
n

i 1
1
i
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Combinatorial Interpretation

Uses weight manipulation
A new weight function: w$ = r · w
Opt$ = r · Opt

w(Solution)  Opt$


 Performance ratio r

In this case r = Hn
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Combinatorial Interpretation
In each iteration:
 Uncovered elements issue checks
 Bookkeeping is performed by adjusting weights
 A weight function  is subtracted from w
(and from w$)
 A zero-weight set Sk is added to the solution
 Elements covered by Sk retract checks that were
given to other sets
$
 Checks are not retracted with respect to w
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Example
u1
=0
=2 =2
=4 =6
u2
u3
u4
S1
w1=4
S2
w2=10 4 8 4 10
S3
w3=12 6 8 0
0
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Analysis - w



Consider an element u
u is covered by S(u) in iteration j(u)
u pays
j u 

j 1
j
for S(u)
j (u )
 w(Sol) =
   U
uU j 1
j
j
j
j
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Analysis –
$
w
In the j’th iteration:

Opt$ decreases by at least |Uj| · j
(“Local Ratio” argument: one check from each
element must be cached)

Deletion of elements may further decrease Opt$
Also, Opt$ = 0 at termination

$
U


Opt
 r  Opt
 j j
j
Solution is r-approximate
Assumption: w$  0 throughout execution
13
Analysis (same as [JMMSV03])
Problem: find min{r | w$  0 at all times}


zi - amount paid by ui
For fixed d and w(S)
1 d
max
zi

w( S ) i 1
s.t. zi  zi 1
i
d  i  1zi  w( S ) i
zi  0
i
d
1
zi  d 1i 1 
zi  H d

w( S ) i 1
u
z1
1
S
u2
z2
u3
z3
.
.
.
ud
zd
 Approx ratio Hn
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Combinatorial Interpretation
of Dual Fitting
Dual Fitting
Combinatorial Framework
Increasing a dual
variable
Dividing y by r
(inflated dual)
Subtracting a weight
function
Defining w$ = r · w
(inflated weights)
wTx  bTy
wTx  Opt$
Find r s.t. y/r is feasible
Find r s.t. w$  0
Remark: problem of finding the best r can be formulated
using LP, but LP-theory is not used in its solution.
15
Primal Fitting
Construct an infeasible primal solution x and a
(feasible) dual solution y such that wTx  bTy
 The primal solution is non integral
  r s.t. r ·x is a feasible integral solution
 wT(r ·x)= r · wTx  r · bTy  r · Opt

Can be used to analyze:

3-approx algorithm for MUFL [JV01]

9-approx algorithm for a disk cover [Chu]
Both were originally designed using primal-dual
16
Combinatorial Interpretation
of Primal Fitting
Primal Fitting
Combinatorial Framework
Increasing a dual
variable
Subtracting a weight
function
Multiplying x by r
(deflated primal)
Defining w$ = w/r
(deflated weights)
wTx  bTy
(w$)T(r ·x)  Opt
Value of primal variable
Fraction of weight left
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The End
18