LR at Haifa 1

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• Reuven Bar-Yehuda
• CS Technion IIT
• Slides and papers at
• http//www.cs.technion.ac.il/reuven

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Example VC

• Given a graph G(V,E) penalty pv? Z for each v ?
  V
• Min ? pvxv
• S.t. xv? 0,1
• xv xu ? 1 ?v,u? E

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Linear Programming (LP) Integer
Programming (IP)

• Given a profit penalty vector p.
• MaximizeMinimize px
• Subject to Linear Constraints F(x)
• IP where x is an integer vector is a constraint

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Example VC

• Given a graph G(V,E) and penalty vector p? Zn
• Minimize px
• Subject to x? 0,1n
• xi xj ? 1 ?i,j? E

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Example SC

• Given a Collection S1, S2,,Sn of all subsets of
  1,2,3,,m and penalty vector p? Zn
• Minimize px
• Subject to x? 0,1n
• ?xi ? 1 ?j1..m
• j? Si

S1
1
2
S2
3
S3
m
Sn
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Example Min Cut

• Given Network N(V,E) s,t ? V and capacity vector
  p? ZE
• Minimize px
• Subject to x? 0,1E
• ?xe ? 1 ?s?t path P
• e? P

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Example Shortest Path

• Given digraph G(V,E) s,t ? V and length vector p?
  ZE
• Minimize px
• Subject to x? 0,1E
• ?xe ? 1 ?s?t cut P
• e? P

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Example MST (Minimum Spanning Tree)

• Given graph G(V,E) s,t ? V and length vector p?
  ZE
• Minimize px
• Subject to x? 0,1E
• ?xe ? 1 ?cut P
• e? P

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Example Minimum Steiner Tree

• Given graph G(V,E) T?V and length vector p? ZE
• Minimize px
• Subject to x? 0,1E
• ?xe ? 1 ?Ts cut P
• e? P

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Example Generalized Steiner Forest

• Given graph G(V,E) T1T1Tk ? V
• and length vector p? ZE
• Min px
• S.t. x? 0,1E
• ?xe ? 1 ?i ?Tis cut P
• e? P

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Example IS (Maximum Independent Set)

• Given a graph G(V,E) and profit vector p? Zn
• Maximaize px
• Subject to x? 0,1n
• xi xj ? 1 ?i,j? E

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Maximum Independent Set in Interval Graphs

• Activity9
• Activity8
• Activity7
• Activity6
• Activity5
• Activity4
• Activity3
• Activity2
• Activity1
• 
  
  
  time
• Maximize
  s.t. For each instance I
• For each time t

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The Local-Ratio Technique Basic
definitions

• Given a profit penalty vector p.
• Minimize Maximize px
• Subject to feasibility constraints F(x)
• x is r-approximation if F(x) and px ? r px
• An algorithm is r-approximation if for any p, F
• it returns an r-approximation

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The Local-Ratio Theorem

• x is an r-approximation with respect to p1
• x is an r-approximation with respect to p- p1
• ?
• x is an r-approximation with respect to p
• Proof (For minimization)
• p1 x ? r p1
• p2 x ? r p2
• ?
• p x ? r ( p1
  p2)
• ? r ( p1 p2 )

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Special case Optimization is
1-approximation

• x is an optimum with respect to p1
• x is an optimum with respect to p- p1
• x is an optimum with respect to p

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A Local-Ratio Schema for
MinimizationMaximization problems

• Algorithm r-ApproxMinMax( Set, p )
• If Set F then return F
• If ?I ? Set p(I)0 then return I ?
  r-ApproxMin( Set-I, p )
• If ? I ? Set p(I) ? 0 then return
  r-ApproxMax( Set-I, p )
• Define good p1
• REC r-ApproxMaxMin( Set, p- p1 )
• If REC is not an r-approximation w.r.t. p1 then
  fix it
• return REC

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The Local-Ratio Theorem Applications

• Applications to some optimization algorithms (r
  1)
• ( MST) Minimum Spanning Tree (Kruskal)
• ( SHORTEST-PATH) s-t Shortest Path (Dijkstra)
• (LONGEST-PATH) s-t DAG Longest Path (Can be
  done with dynamic programming)
• (INTERVAL-IS) Independents-Set in Interval
  Graphs Usually done with dynamic programming)
• (LONG-SEQ) Longest (weighted) monotone
  subsequence (Can be done with dynamic
  programming)
• ( MIN_CUT) Minimum Capacity s,t Cut (e.g.
  Ford, Dinitz)
• Applications to some 2-Approximation algorithms
  (r 2)
• ( VC) Minimum Vertex Cover (Bar-Yehuda and
  Even)
• ( FVS) Vertex Feedback Set (Becker and Geiger)
  
• ( GSF) Generalized Steiner Forest (Williamson,
  Goemans, Mihail, and Vazirani)
• ( Min 2SAT) Minimum Two-Satisfibility
  (Gusfield and Pitt)
• ( 2VIP) Two Variable Integer Programming
  (Bar-Yehuda and Rawitz)
• ( PVC) Partial Vertex Cover (Bar-Yehuda)
• ( GVC) Generalized Vertex Cover (Bar-Yehuda
  and Rawitz)
• Applications to some other Approximations
• ( SC) Minimum Set Cover (Bar-Yehuda and Even)
• ( PSC) Partial Set Cover (Bar-Yehuda)
• ( MSP) Maximum Set Packing (Arkin and Hasin)

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The creative partfind ?-Effective weights

• p1 is ?-Effective if every feisible solution is
  ?-approx w.r.t. p1
• i.e. p1 x ? ? p1
• VC (vertex cover)
• Edge
• Matching
• Greedy
• Homogeneous

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VC Recursive implementation (edge by edge)

• VC (V, E, p)
• If E? return ?
• If ? p(v)0 return vVC(V-v, E-E(v), p)
• Let (x,y)?E
• Let ? minp(x), p(y)
• Define p1 (v) ? if vx or vy and 0 otherwise
• Return VC(V, E, p- p1 )

0
0
0
?
?
0
0
0
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VC Iterative implementation (edge by edge)
? ?

• VC (V, E, p)
• for each e ? E
• let ? minp(v) v ? e
• for each v ? e
• p(v) p(v) - ?
• return v p(v)0

0
0
0
?
?
0
0
0
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Min 5xBisli8xTea12xWater10xBamba20xShampoo15x
Popcorn6xChocolate s.t. xShampoo xWater ? 1

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Movie1 4 the price of 2
?
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VC Iterative implementation (edge by edge)
? ?

• VC (V, E, p)
• for each e ? E
• let ? minp(v) v ? e
• for each v ? e
• p(v) p(v) - ?
• return v p(v)0

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15
90
10
50
100
80
2
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VC Greedy ( O(H(?)) - approximation)H(?)1/21/3
1/? O(ln ?)

• Greedy_VC (V, E, p)
• C ?
• while E??
• let varc min p(v)/d(v)
• C C v
• V V v
• return C

n/? n/4 n/3 n/2 n
?
?
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VC LR-Greedy (star by star)

• LR_Greedy_VC (V, E, p)
• C ?
• while E??
• let varc min p(v)/d(v)
• let ? p(v)/d(v)
• C C v
• V V v
• for each u ? N(v)
• p(v) p(v) - ?
• return C

?
?
4?
?
?
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VC LR-Greedy by reducing 2-effective
homogeniousHomogenious all vertices have the
same greedy value

• LR_Greedy_VC (V, E, p)
• C ?
• Repeat
• Let ? Min p(v)/d(v)
• For each v ? V
• p(v) p(v) ? d(v)
• Move from V to C all zero weight vertices
• Remove from V all zero degree vertices
• Until E?
• Return C

3?
4?
6?
4?
5?
3?
3?
2?
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Example MST (Minimum Spanning Tree)

• Given graph G(V,E) s,t ? V and length vector p?
  ZE
• Minimize px
• Subject to x? 0,1E
• ?xe ? 1 ?cut P
• e? P

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MST Recursive implementation (Homogenious)

• MST (V, E, p)
• If V? return ?
• If ? self-loop e return MST(V, E-e, p)
• If ? p(e)0 return eMST(Vshrink(e),
  Eshrink(e), p)
• Let ? minp(e) e?E
• Define p1 (e) ? for all e?E
• Return MST(V, E, p- p1 )

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MST Iterative implementation (Homogenious)

• MST (V, E, p)
• Kruskal

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Some effective weights
VC IS MST S. Path Steiner FVS Min Cut
Edge 2 ? ? ? ? ? ?
Matching 2 ? ? ? ? ? ?
Odd Cycle 2-1/k ? ? ? ? ? ?
Clique (k-1)/k ? ? ? ? ? ?
Star 2 (k1)/2 ? 1 ? ? ?
Homogenious 2 k 1 ? 2 2 ?
Special trik 1