Seminar : Approximation algorithms for LP/IP optimization problems Reuven Bar-Yehuda Technion IIT Slides and papers at:

Example VC Given a graph G=(V,E) penalty p v  Z for each v  V Min  p v ·x v S.t.: x v  {0,1} x v + x u  1  {v,u}  E

Linear Programming (LP) Integer Programming (IP) Linear Programming (LP) Integer Programming (IP) Given a profit [penalty] vector p. Maximize[Minimize] p·x Subject to:Linear Constraints F(x) IP: where “x is an integer vector” is a constraints

Example VC Given a graph G=(V,E) and penalty vector p  Z n Minimize p·x Subject to: x  {0,1} n x i + x j  1  {i,j}  E

Example SC Given a Collection S 1, S 2,…,S n of all subsetsof {1,2,3,…,m} and penalty vector p  Z n Minimize p·x Subject to: x  {0,1} n  x i  1  j=1..m j  Si

Example Min Cut Given Network N(V,E) s,t  V and capasity vector p  Z |E| Minimize p·x Subject to: x  {0,1} |E|  x e  1  s  t path P e  P

Example Min Path Given digraph G(V,E) s,t  V and length vector p  Z |E| Minimize p·x Subject to: x  {0,1} |E|  x e  1  s  t cut P e  P

Example MST (Minimum Spanning Tree) Given graph G(V,E) and length vector p  Z |E| Minimize p·x Subject to: x  {0,1} |E|  x e  1  cut P e  P

Example Minimum Steiner Tree Given graph G(V,E) T  V and length vector p  Z |E| Minimize p·x Subject to: x  {0,1} |E|  x e  1  T’s cut P e  P

Example Generalized Steiner Forest Given graph G(V,E) T 1 T 1 …T k  V and length vector p  Z |E| Min p·x S.t.: x  {0,1} |E|  x e  1  i  T i ’s cut P e  P

Example IS (Maximum Independent Set) Given a graph G=(V,E) and profit vector p  Z n Maximaize p·x Subject to: x  {0,1} n x i + x j  1  {i,j}  E

Maximum Independent Set in Interval Graphs 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:

The Local-Ratio Technique: Basic definitions The Local-Ratio Technique: Basic definitions Given a penalty [profit] vector p. Minimize [Maximize] p·x Subject to:feasibility constraints F(x) x is r-approximation if F(x) and p·x  r · p·x* An algorithm is r-approximation if for any p, F it returns an r-approximation

The Local-Ratio Theorem: The Local-Ratio Theorem: x is an r-approximation with respect to p 1 x is an r-approximation with respect to p- p 1  x is an r-approximation with respect to p Proof: ( For minimization) p 1 · x  r × p 1 * p 2 · x  r × p 2 *  p · x  r × ( p 1 *+ p 2 *)  r × ( p 1 + p 2 )*

The Local-Ratio Theorem: (Proof2) The Local-Ratio Theorem: (Proof2) x is an r-approximation with respect to p 1 x is an r-approximation with respect to p- p 1  x is an r-approximation with respect to p Proof2: ( For minimization) Let x*, x 1 *, x 2 * be optimal solutions for p, p 1, p 2 respectively p 1 · x  r × p 1 x 1 * p 2 · x  r × p 2 x 2 *  p · x  r × ( p 1 x 1 *+ p 2 x 2 * )  r × ( p 1 x*, + p 2 x*) = px*

Special case: Optimization is 1-approximation Special case: Optimization is 1-approximation x is an optimum with respect to p 1 x is an optimum with respect to p- p 1 x is an optimum with respect to p

A Local-Ratio Schema for Minimization[Maximization] problems: A Local-Ratio Schema for Minimization[Maximization] problems: Algorithm r-ApproxMin[Max]( Set, p ) If Set = Φ then return Φ ; 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” p 1 ; REC = r-ApproxMax[Min]( Set, p- p 1 ) ; If REC is not an r-approximation w.r.t. p 1 then “fix it”; return REC;

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

The creative part… find  -Effective weights p 1 is  -Effective if every feasible solution is  -approx w.r.t. p 1 i.e. p 1 ·x   p 1 * VC (vertex cover) Edge Matching Greedy Homogenious

VC (V, E, p) If E=  return  ; If  p(v)=0 return {v}+VC(V-{v}, E-E(v), p); Let (x,y)  E; Let  = min{p(x), p(y)}; Define p 1 (v) =  if v=x or v=y and 0 otherwise; Return VC(V, E, p- p 1 ) VC: Recursive implementation (edge by edge)  

VC: Iterative implementation (edge by edge)   VC (V, E, p) for each e  E; let  = min{p(v)| v  e}; for each v  e p(v) = p(v) -  ; return {v| p(v)=0};  

Min 5x Bisli +8x Tea +12x Water +10x Bamba +20x Shampoo +15x Popcorn +6x Chocolate s.t. x Shampoo + x Water  1

Movie: 1 4 the price of 2

VC: Iterative implementation (edge by edge)   VC (V, E, p) for each e  E; let  = min{p(v)| v  e}; for each v  e p(v) = p(v) -  ; return {v| p(v)=0};

VC: Greedy ( O(H(  )) - approximation) H(  )=1/2+1/3+…+1/  = O(ln  ) Greedy_VC (V, E, p) C =  ; while E  let v=arc min p(v)/d(v) C = C + {v}; V = V – {v}; return C;  n/  n/4 n/3 n/2 n … …  … … 

VC: LR-Greedy (star by star) LR_Greedy_VC (V, E, p) C =  ; while E  let v=arc 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4    

VC: LR-Greedy by reducing 2-effective homogenious Homogenious = 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; 44 66 44 55 33 33 33 22

Example MST (Minimum Spanning Tree) Given graph G(V,E) and length vector p  Z |E| Minimize p·x Subject to: x  {0,1} |E|  x e  1  cut P e  P

MST (V, E, p) If V=  return  ; If  self-loop e return MST(V, E-{e}, p); If  p(e)=0 return {e}+MST(V shrink(e), E shrink(e), p); Let  = min{p(e) : e  E}; Define p 1 (e) =  for all e  E; Return MST(V, E, p- p 1 ) MST: Recursive implementation (Homogenious)

MST (V, E, p) Kruskal MST: Iterative implementation (Homogenious)

Some effective weights VCISMSTS. PathSteinerFVSMin Cut Edge 2  Matching 2  Cycle 2-1/k  Clique (k-1)/k  Star 2 (k+1)/2  1  Homogenious 2k 1  22  Special trik1