1. www.cs.technion.ac.il/~reuven ISMP2006 1 LP Rounding using Fractional Local Ratio Reuven Bar-Yehuda www.cs.technion.ac.il/~reuven

2. www.cs.technion.ac.il/~reuven ISMP2006 2 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* ] An algorithm is an r-approximation if for any w, F it returns an r-approximation

3. www.cs.technion.ac.il/~reuven ISMP2006 3 15 Min 5x Bisli +8x Tea +12x Water +10x Bamba +20x Shampoo +15x Popcorn +6x Chocolate +$4x WaterShampoo + s.t. x Shampoo + x Water + x WaterShampoo  1 5 8 12 20 6 10 $4 $1 $3 $1 $2 $1

4. www.cs.technion.ac.il/~reuven ISMP2006 4 The generalized vertex cover problem Minimize w·x Subject to:x u + x v + x e  1  e={u,v}  E x  {0,1} |V|+|E|

5. www.cs.technion.ac.il/~reuven ISMP2006 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} 1 w 1 (x) = 0 else Notice:w 1 x  2 w 1 x for Good(x) VC(G, w-w 1 ) REC= GVC(G, w 2 = w-w 1 ) Induction hyp is: w 2 REC  2 w 2 x so if Good(REC): w 1 REC  2 w 1 x we are done If REC-e is a cover thenREC=REC-e If REC-e is a cover thenREC=REC-e Return REC   

6. www.cs.technion.ac.il/~reuven ISMP2006 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:x u + x v + x e  1  e=(u,v)  E x  [0,1] |V|+|E|

7. www.cs.technion.ac.il/~reuven ISMP2006 7 “d integral for the price of fractional”: 2-2/(Δ+1)-Approx GVC(G,w) “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.t x v is minimum and Let  =min(w(i) : i  N[v]}  if i  N[v] 1 w 1 (i) = 0 else Claim:w 1 x  r Δ w 1 x for Good(x) VC(G, w-w 1 ) REC= GVC(G, w 2 = w-w 1 ) Induction hyp is: w 2 REC  r Δ w 2 x so if Good(REC): w 1 REC  r Δ w 1 x we are done If REC is not a minimal cover then make REC minimal If REC is not a minimal cover then make REC minimal Return REC           Min x v

8. www.cs.technion.ac.il/~reuven ISMP2006 8 “d integral for the price of fractional”: “d integral for the price of ½(d+1) fractional”: Claim: w 1 x  r Δ w 1 x for Good(x)           Min x v If Min x v ≥ ½ Then x (N[v]) ≥ ½(d+1) Else x (N[v]) ≥ ½(d+1) Thus w 1 x ≥ ½(d+1)  But w 1 x  d  Hence : w 1 x/ w 1 x  2-2/(d+1) Δ  2-2/( Δ +1) = r Δ

9. www.cs.technion.ac.il/~reuven ISMP2006 9 A Generalized Local-Ratio Schema for M inimization [ M aximization] problems: Let x 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  Set w(v)  0 then return r-ApproxMax(Set-{v},w ) ;] Define “good” w 1 ; i.e.  Good(x): w 1 x  [  ] r w 1 x REC = r-ApproxMin [Max](Set, w 2 ) ; Induction hyp is: w 2 REC  [  ] r w 2 x so if Good(REC): w 1 REC  [  ] r w 1 x we are done, otherwise “fix it”; return REC’;

10. www.cs.technion.ac.il/~reuven ISMP2006 10 The maximum independent set problem Maximize w·x Subject to:x u + x v ≤ 1  e=(u,v)  E x  {0,1} |V|

11. www.cs.technion.ac.il/~reuven ISMP2006 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:x u + x v ≤ 1  e=(u,v)  E x  [0,1] |V|

12. www.cs.technion.ac.il/~reuven ISMP2006 12 Gain 1 integral, lose fractional 2/(Δ+1)-Approx IS(G,w) Gain 1 integral, lose ½(d+1) fractional 2/(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.t x v is maximum and Let  = w(v)  if i  N[v] 1 w 1 (i) = 0 else Claim:w 1 x ≥r Δ w 1 x for Good(x) (G, w-w 1 ) REC= IS(G, w 2 = w-w 1 ) Induction hyp is: w 2 REC ≥ r Δ w 2 x so if Good(REC): w 1 REC ≥ r Δ w 1 x we are done If REC+v is an independent set then REC=REC+v If REC+v is an independent set then REC=REC+v Return REC       Max x v

13. www.cs.technion.ac.il/~reuven ISMP2006 13 Gain 1 integral, lose fractional Gain 1 integral, lose ½(d+1) fractional Claim: w 1 x ≥ r Δ w 1 x for Good(x) Max x v If Max x v ≤ ½ Then x (N[v]) ≤ ½(d+1) Else x (N[v]) ≤ ½(d+1) Thus w 1 x ≤ ½(d+1)  But w 1 x ≥  Hens : w 1 x/ w 1 x ≥ 2/(d+1) Δ ≥ 2/( Δ +1) = r Δ      

14. www.cs.technion.ac.il/~reuven ISMP2006 14 Single Machine Scheduling : 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: 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

15. www.cs.technion.ac.il/~reuven ISMP2006 15 Î, and the weight decomposition: Let Î be the interval which ends first.  I in conflict with Î, Define w 1 (I) = w 2 = w-w 1 0 otherwise, w 1 =  w 1 = 0 time Activity9 Activity8 Activity7 Activity6 Activity5 Activity4 Activity3 Activity2 Activity1

16. www.cs.technion.ac.il/~reuven ISMP2006 16 4-approximation for 2 Dimentional Interval graphs

17. www.cs.technion.ac.il/~reuven ISMP2006 17 4-approximation for 2 Dimentional Interval graphs

18. www.cs.technion.ac.il/~reuven ISMP2006 18 4-approximation for 2 Dimentional Interval graphs

19. www.cs.technion.ac.il/~reuven ISMP2006 19 4-approximation for 2 Dimentional Interval graphs

20. www.cs.technion.ac.il/~reuven ISMP2006 20 4-approximation for 2 Dimentional Interval graphs

21. www.cs.technion.ac.il/~reuven ISMP2006 21 2t-approximation for t- Dimentional Interval graphs

22. www.cs.technion.ac.il/~reuven ISMP2006 22 2t-approximation for t-Interval Graphs Maximize w·x Subject to:  v  C x v ≤ 1  C Clique x  {0,1} |V|

23. www.cs.technion.ac.il/~reuven ISMP2006 23 2t-approximation for t- Interval Graphs finding x Maximize w·x Subject to:  v  C x v ≤ 1  C Interval Clique x  [0,1] |V| e.g. x 1 +x 4 +x 5 ≤ 1

24. www.cs.technion.ac.il/~reuven ISMP2006 24 2t-approximation for t- Interval Graphs finding more relaxed x Maximize w·x Subject to:  v  C x v ≤ t  C t-Interval Clique x  [0,1] |V| e.g. x 1 +x 3 +x 4 +x 5 ≤ 3

25. www.cs.technion.ac.il/~reuven ISMP2006 25 Gain 1 integral, lose fractional 1/(2t)-Approx IS(G,w) Gain 1 integral, lose 2 t fractional 1/(2t)-Approx IS(G,w) If  v  V w(v)  0 return IS(G-v, w) If E=  return V Let v  V s.t x (N[v]) is minimum and Let  = w(v)  if i  N[v] 1 w 1 (i) = 0 else Claim:w 1 x ≥ r t w 1 x for Good(x) (G, w-w 1 ) REC= IS(G, w 2 = w-w 1 ) Induction hyp is: w 2 REC ≥ r t w 2 x so if Good(REC): w 1 REC ≥ r t w 1 x we are done If REC+v is an independent set then REC=REC+v If REC+v is an independent set then REC=REC+v Return REC       Min x (N[v])  2t

26. www.cs.technion.ac.il/~reuven ISMP2006 26 Gain 1 integral, lose fractional Gain 1 integral, lose 2t fractional Claim: w 1 x ≥ r t w 1 x for Good(x) Min x (N[v]) We need to show that (next slide) x (N[v]) ≤ 2t Thus w 1 x ≤ 2t  But w 1 x ≥ 1  Hence : w 1 x/ w 1 x ≥  /(2t  ) = r t      

27. www.cs.technion.ac.il/~reuven ISMP2006 27 Claim:  v  u  N[v] x u ≤ 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 x ij = x i x j y i + =  i  j x ij and y i - =  j  i x ji Thus y i +  t x i  i y i =  i y i + +  i y i -  2t  i x i Thus  i y i  2t x i and therefore  i  i-j x j  2t

28. www.cs.technion.ac.il/~reuven ISMP2006 28 6t-apx for t-Interval Graphs with demands finding x Maximize w·x Subject to:  v  C d v x v ≤ 1  C Interval Clique x  [0,1] |V| e.g. d 1 x 1 +d 4 x 4 +d 5 x 5 ≤ 1

29. www.cs.technion.ac.il/~reuven ISMP2006 29 6t-apx for t-Interval Graphs with demands t-split Fat=2t+ thin4t Assign z i =d i x i for thin Reuven Bar-Yehuda and Dror Rawitz. Using fractional primal-dual to schedule split intervals with demands. To appear in Discrete Optimization. appeared in 13th ESA, LNCS 3669:714-725, 2005. rawitz.psUsing fractional primal-dual to schedule split intervals with demandsLNCS 3669rawitz.ps

30. www.cs.technion.ac.il/~reuven ISMP2006 30 2- Dimentional Interval graphs  rectangles packing

31. www.cs.technion.ac.il/~reuven ISMP2006 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. www.cs.technion.ac.il/~reuven ISMP2006 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. www.cs.technion.ac.il/~reuven ISMP2006 33 4c-apx for rectangle packing Types of intersections: Stabbing: Crossing:

34. www.cs.technion.ac.il/~reuven ISMP2006 34 4c-apx for rectangle packing. Result: 4c -apx Algorithm: Partition the input into c crossing free sets Apply 4 -apx for each and pick the maximum.

35. www.cs.technion.ac.il/~reuven ISMP2006 35 MIS on axix-parallel rectangles 4-approximation for MIS on axix-parallel rectangles finding x Maximize w·x Subject to:  v  C x v ≤ 1  C right upper corner Clique x  [0,1] |V| e.g. x 1 +x 3 +x 4 ≤ 1 2 1 3 5 4

36. www.cs.technion.ac.il/~reuven ISMP2006 36 MIS on axix-parallel rectangles 4-approximation for MIS on axix-parallel rectangles finding more relaxed x Maximize w·x Subject to:  v  C x v ≤ 2  C right segment Cliques x  [0,1] |V| e.g. x 1 +x 3 +x 4 +x 5 ≤ 2 2 1 3 5 4

37. www.cs.technion.ac.il/~reuven ISMP2006 37 Gain 1 integral, lose fractional Gain 1 integral, lose 4 fractional 4-apx for crossing free recangles If  v  V w(v)  0 return IS(G-v, w) If E=  return V Let v  V s.t x (N[v]) is minimum and Let  = w(v)  if i  N[v] 1 w 1 (i) = 0 else Claim:w 1 x ≥ ¼ w 1 x for Good(x) (G, w-w 1 ) REC= IS(G, w 2 = w-w 1 ) Induction hyp is: w 2 REC ≥ ¼ w 2 x so if Good(REC): w 1 REC ≥ ¼ w 1 x we are done If REC+v is an independent set then REC=REC+v If REC+v is an independent set then REC=REC+v Return REC Min x (N[v])  4   0 0   0 0 0

38. www.cs.technion.ac.il/~reuven ISMP2006 38 Claim:  v  u  N[v] x u ≤ 4 Define a directed graph G(V,E) V = Set of rectangles E = {i  j : Rectangle i “right-stubs” rectangle j} Define x ij = x i x j y i + =  i  j x ij and y i - =  j  i x ji Thus y i +  2*x i  i y i =  i y i + +  i y i -  2*2  i x i Thus  i y i  4 x i and therefore  i  i-j x j  4

39. www.cs.technion.ac.il/~reuven ISMP2006 39 Thank you !

40. www.cs.technion.ac.il/~reuven ISMP2006 40 12c= fat 4c thin 8c Admission Control with Advance Reservation in Simple Networks Dror Rawitz 2006 Max IS RECT with demand Color C Each factor 8

41. www.cs.technion.ac.il/~reuven ISMP2006 41 MIS on Admission Control in Networks with Advance Reservations Liane Lewin-Eytan, Joseph (Seffi) Naor, and Ariel Orda1 Algorithmica (2004) 40: 293–304