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  Z n and y such that w T x  r · b T y  w T x  r · b T y  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] and 1.61-approximation algorithms both using dual fitting [JMMSV03] Motivation: combinatorial interpretations of both non standard applications

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

7 This Work Two new approximation frameworks: Combinatorial Based on weight manipulation (in the spirit of local ratio) Framework1 st  Dual Fitting ExamplesMUFL [JMMSV03] Set Cover [Chv79] * Defined in this paper 2 nd Primal Fitting* MUFL [JV01] Disk Cover [Chu]

8 An Example: Set Cover Input: C = {S 1,…,S m }, S i  U, w : C  R + Solution: C’  C s.t. Measure: Algorithm Greedy: 1. While instance is not empty do: 2. k  argmin i {w(S i ) / |S i |} 3.Add S k to the solution 4.Remove the elements in S k and discard empty sets Approximation ratio is

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 = H n

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 S k is added to the solution Elements covered by S k retract checks that were given to other sets Checks are not retracted with respect to w $

11 u4u4 u3u3 S3S3 Example S1S1 u1u1 u2u2 S2S2 w 1 =40 w 2 =104 w 3 =  =0  =2  =  =2  =6

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 |U j | ·  j (“Local Ratio” argument: one check from each element must be cached) Deletion of elements may further decrease Opt $ Also, Opt $ = 0 at termination  Assumption: w $  0 throughout execution Solution is r-approximate

14 Analysis (same as [JMMSV03]) Problem: find min{r | w $  0 at all times} z i - amount paid by u i For fixed d and w(S) S u1u1 u2u2 u3u3 udud z1z1 z2z2 z3z3 zdzd  Approx ratio H n

15 Combinatorial Interpretation of Dual Fitting Dual FittingCombinatorial Framework Increasing a dual variable Subtracting a weight function Dividing y by r (inflated dual) Defining w $ = r · w (inflated weights) wTx  bTywTx  bTyw T x  Opt $ Find r s.t. y/r is feasibleFind 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.

16 Primal Fitting Construct an infeasible primal solution x and a (feasible) dual solution y such that w T x  b T y The primal solution is non integral  r s.t. r ·x is a feasible integral solution  w T (r ·x)= r · w T x  r · b T y  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 Interpretation of Primal Fitting Primal FittingCombinatorial Framework Increasing a dual variable Subtracting a weight function Multiplying x by r (deflated primal) Defining w $ = w/r (deflated weights) wTx  bTywTx  bTy(w $ ) T (r ·x)  Opt Value of primal variableFraction of weight left

18 The End