Combinatorial Interpretations of Dual Fitting and Primal Fitting

1. Combinatorial Interpretations of Dual Fitting and Primal Fitting Ari Freund Cesarea Rothschild Institute, University of Haifa Dror Rawitz Department of Computer Science, Technion

2. Approximation Using LP Duality • Minimization problem • LP-relaxation and dual: • 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?

3. 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])

4. 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

5. 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

6. Find r s.t. y/r is feasible • wTxbTy = r· bT(y/r) r · Opt Dual Fitting • Construct an infeasible dual y and a feasible primal x such that wTxbTy Problem: finding the smallest r s.t. (for all input instances) y/r is feasible. y y/8

7. This Work Two new approximation frameworks: • Combinatorial • Based on weight manipulation (in the spirit of local ratio) * Defined in this paper

8. Approximation ratio is An Example: Set Cover • Input: C = {S1,…,Sm}, SiU, w : C  R+ • Solution: C’  C s.t. • Measure: 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

9. 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

10. 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$) • A zero-weight set Sk is added to the solution • Elements covered by Skretract checks that were given to other sets • Checks are not retracted with respect to w$

11. Example u1 S1 w1=4 0 =0 u2 S2 w2=10 4 8 4 10 =2 =2 u3 S3 =4 =6 w3=12 6 8 0 u4

12. Analysis - w • Consider an element u • u is covered by S(u) in iteration j(u) • u pays for S(u) w(Sol) =

13. 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  Solution is r-approximate Assumption:w$ 0 throughout execution

14. u1 z1 • zi - amount paid by ui • For fixed d and w(S) u2 z2 S u3 z3 . . . ud zd Analysis (same as [JMMSV03]) Problem: find min{r | w$ 0 at all times}  Approx ratio Hn

15. Combinatorial Interpretation of Dual Fitting Remark: problem of finding the best r can be formulated using LP, but LP-theory is not used in its solution.

16. 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

17. Combinatorial Interpretationof Primal Fitting

18. The End