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LP Rounding using Fractional Local Ratio

LP Rounding using Fractional Local Ratio. Reuven Bar-Yehuda www.cs.technion.ac.il/~reuven. General framework:. Given a weight vector w. Minimize [Maximize] w·x Subject to: feasibility constraints F(x) x is an r -approximation if F(x) and w·x  r  w·x* [ w·x  r  w·x* ]

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LP Rounding using Fractional Local Ratio

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  1. LP RoundingusingFractional Local Ratio Reuven Bar-Yehuda www.cs.technion.ac.il/~reuven

  2. General framework: Given a weight vector w. Minimize [Maximize]w·x Subject to: feasibility constraints F(x) x is anr-approximationif F(x) and w·x  rw·x* [w·x  rw·x* ] An algorithm is anr-approximation if for any w, F it returns an r-approximation

  3. 8 12 5 20 10 6 $1 $1 $4 $1 $1 $3 15 $2 $1 Min 5xBisli+8xTea+12xWater+10xBamba+20xShampoo+15xPopcorn+6xChocolate +$4xWaterShampoo+ • • • s.t. xShampoo + xWater + xWaterShampoo  1

  4. The generalized vertex cover problem Minimize w·x Subject to: xu + xv + xe 1  e={u,v} E x{0,1}|V|+|E|

  5. 2-Approx GVC(G,w) If E= return  If  e E w(e)=0 return {e}+GVC(G-e, w) If  v V w(v)=0 return {v}+GVC(G-E(v), w) Let e={u,v} E s.t = min {w(u), w(v), w(e)}>0.  if x{u,v,e} w1(x)= 0 else Notice:w1x  2w1x for Good(x) REC= GVC(G, w2= w-w1) Induction hyp is: w2REC 2w2x so if Good(REC):w1REC 2w1x we are done If REC-e is a cover thenREC=REC-e Return REC   

  6. “2 integral for the price of 1 fractional”: The local ratio technique for rounding Let xbe the the fractional solution Minimize w·x Subject to: xu + xv + xe 1  e=(u,v) E x[0,1]|V|+|E|

  7. “d integral for the price of ½(d+1) fractional”:2-2/(Δ+1)-Approx GVC(G,w)  If E= return  If  e E w(e)=0 return {e}+GVC(G-e, w) If  v V w(v)=0 return {v}+GVC(G-E(v)-v, w) Let v V s.txv is minimum and Let =min(w(i) : i  N[v]}  if i  N[v] w1(i)= 0 else Claim:w1x rΔw1x for Good(x) REC= GVC(G, w2= w-w1) Induction hyp is: w2RECrΔw2x so if Good(REC):w1RECrΔw1x we are done If REC is not a minimal cover then make REC minimal Return REC     Min xv      

  8. “d integral for the price of ½(d+1) fractional”:Claim: w1x rΔw1x for Good(x) IfMin xv ≥ ½ Thenx(N[v]) ≥ ½(d+1) Elsex(N[v]) ≥ ½(d+1) Thus w1x ≥ ½(d+1)  But w1x d Hence: w1x/ w1x  2-2/(d+1)  2-2/(Δ +1) = rΔ      Min xv      

  9. A Generalized Local-Ratio Schema for Minimization [Maximization] problems: Letxbe any “fisible?” vector (e.g. an optimal solution) Algorithm r-ApproxMin [Max](Set, w) If Set =then return  ; If v  Set w(v) = 0 then return {v} r-ApproxMin(Set-{v},w ) ; [If  v  Setw(v) 0 then returnr-ApproxMax(Set-{v},w ) ;] Define “good” w1 ; i.e. Good(x):w1x  []rw1x REC = r-ApproxMin [Max](Set, w2) ; Induction hyp is: w2REC []rw2x so if Good(REC):w1REC []rw1x we are done, otherwise “fix it”; return REC’;

  10. The maximum independent set problem Maximize w·x Subject to: xu + xv≤ 1  e=(u,v) E x{0,1}|V|

  11. The maximum independent set problem “1 integral for the gain of r fractional”: Let xbe the the fractional solution Maximize w·x Subject to: xu + xv≤ 1  e=(u,v) E x[0,1]|V|

  12. Gain 1 integral, lose ½(d+1) fractional2/(Δ+1)-Approx IS(G,w)   If  v V w(v)  0 return IS(G-v, w) If E= return V Let v V s.txv is maximum and Let = w(v)  if i  N[v] w1(i)= 0 else Claim:w1x ≥rΔw1x for Good(x) REC= IS(G, w2= w-w1) Induction hyp is: w2REC≥rΔw2x so if Good(REC):w1REC≥rΔw1x we are done If REC+v is an independent set then REC=REC+v Return REC  Max xv   

  13. Gain 1 integral, lose ½(d+1) fractionalClaim: w1x ≥rΔw1x for Good(x) IfMax xv≤ ½ Then x(N[v]) ≤ ½(d+1) Elsex (N[v]) ≤ ½(d+1) Thus w1x ≤½(d+1) But w1x ≥ Hens: w1x/ w1x ≥2/(d+1) ≥2/(Δ +1) = rΔ    Max xv   

  14. Single Machine Scheduling : Bar-Noy, Guha, Naor and Schieber STOC 99: 1/2 LP Berman, DasGupta, STOC 00: 1/2 Bar-Noy at al, STOC 00  1/2 Activity9 Activity8 Activity7 Activity6 Activity5 Activity4 Activity3 Activity2 Activity1 ????????????? time Maximize s.t.For each instance I: For each time t: For each activity A:

  15. w1=  w1= 0 Activity9 Activity8 Activity7 Activity6 Activity5 Activity4 Activity3 Activity2 Activity1 w1=  w1= 0 w1= 0 w1= 0 w1= 0 w1= 0 w1= 0 w1= 0 w1=  w1 = 0 w1=  w1= 0 w1= 0 w1=  w1=  w1=  w1=  w1=  time Î, and the weight decomposition: • Let Î be the interval which ends first. I in conflict with Î, • Define w1(I) = w2= w-w1 0 otherwise,

  16. 4-approximation for2 Dimentional Interval graphs

  17. 4-approximation for2 Dimentional Interval graphs

  18. 4-approximation for2 Dimentional Interval graphs

  19. 4-approximation for2 Dimentional Interval graphs

  20. 4-approximation for2 Dimentional Interval graphs

  21. 2t-approximation fort- Dimentional Interval graphs

  22. 2t-approximation for t-Interval Graphs Maximize w·x Subject to:  vCxv≤ 1  C Clique x{0,1}|V|

  23. 2t-approximation for t- Interval Graphsfinding x Maximize w·x Subject to:  vCxv≤ 1  CInterval Clique x[0,1]|V| e.g. x1+x4+x5≤ 1

  24. 2t-approximation for t- Interval Graphsfinding more relaxedx Maximize w·x Subject to:  vCxv≤ t  Ct-Interval Clique x[0,1]|V| e.g. x1+x3+x4+x5≤3

  25. Gain 1 integral, lose 2t fractional1/(2t)-Approx IS(G,w)   If  v V w(v)  0 return IS(G-v, w) If E= return V Let v V s.tx(N[v])is minimum and Let = w(v)  if i  N[v] w1(i)= 0 else Claim:w1x ≥ rtw1x for Good(x) REC= IS(G, w2= w-w1) Induction hyp is: w2REC≥rtw2x so if Good(REC):w1REC≥rtw1x we are done If REC+v is an independent set then REC=REC+v Return REC  Min x(N[v])2t   

  26. Gain 1 integral, lose 2t fractionalClaim: w1x ≥rtw1x for Good(x) We need to show that (next slide) x(N[v]) ≤ 2t Thus w1x ≤2t But w1x ≥1 Hence: w1x/ w1x ≥ /(2t) = rt    Min x(N[v])   

  27. Claim: v u N[v] xu≤ 2t Define a directed graph G(V,E) V = Set of t-splits E = {ij : A right endpoind of i “hits” interval j} Define xij = xi xj yi+ =  ij xij and yi- =  ji xji Thus yi+ txi i yi =i yi+ + i yi-  2tixi Thus i yi 2txi and therefore i  i-j xj 2t

  28. 6t-apx for t-Interval Graphs with demandsfinding x Maximize w·x Subject to:  vCdvxv≤ 1  CInterval Clique x[0,1]|V| e.g. d1x1+d4x4+d5x5≤ 1

  29. t-Interval Graphs with demands 6t = (fat)2t+(thin)4t (Assign zi=dixi ) R.Bar-Yehuda and D. Rawitz.ESA2005andDiscrete Optimization 2006.

  30. 2- Dimentional Interval graphs rectangles packing

  31. MIS on axix-parallel rectangles: • NP-Hard even on unit squares [Asano91] • Divide and conquare O(logn)-apx [AKS98] • PTAS where all heights are the same [AKS98] • log(n)/ apx for any constant  [BDMR01] • 4c-apx where c=max #rects covering a point [LNO04] • 12c-apx with demands [Rawitz06]

  32. 4c-apx Liane Lewin-Eytan, Joseph (Seffi) Naor, and Ariel Orda1 Admission Control in Networks with Advance Reservations Algorithmica (2004) 40: 293–304

  33. 4c-apx for rectangle packing Types of intersections: Stabbing: Crossing:

  34. 4c-apx for rectangle packing Algorithm: Partition the input into c crossing free sets Apply 4-apx for each and pick the maximum. . Result: 4c-apx

  35. 4-approximation for MIS on axix-parallel rectanglesfinding x Maximize w·x Subject to:  vCxv≤ 1  C right upper corner Clique x[0,1]|V| 2 3 1 5 4 e.g. x1+x3+x4≤ 1

  36. 4-approximation for MIS on axix-parallel rectanglesfinding more relaxedx Maximize w·x Subject to:  vCxv≤ 2  C right segment Cliques x[0,1]|V| 2 3 1 5 4 e.g. x1+x3+x4+x5≤2

  37. Gain 1 integral, lose 4 fractional4-apx for crossing free recangles  If  v V w(v)  0 return IS(G-v, w) If E= return V Let v V s.tx(N[v])is minimum and Let = w(v)  if i  N[v] w1(i)= 0 else Claim:w1x ≥ ¼ w1x for Good(x) REC= IS(G, w2= w-w1) Induction hyp is: w2REC≥¼ w2x so if Good(REC):w1REC≥¼ w1x we are done If REC+v is an independent set then REC=REC+v Return REC 0 0 Min x(N[v])4  0   0 0

  38. Claim: v u N[v] xu≤4 Define a directed graph G(V,E) V = Set of rectangles E = {ij : Rectangle i “right-stubs” rectangle j} Define xij = xi xj yi+ =  ij xij and yi- =  ji xji Thus yi+ 2*xi i yi =i yi+ + i yi- 2*2ixi Thus i yi 4 xi and therefore i  i-j xj 4

  39. Max IS RECT with demand • Admission Control with Advance Reservation in Simple Networks Dror Rawitz 2006 Thin: Color with C colors Each factor 8 12c= fat4c + thin8c

  40. Thankyou!

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