Approximation Algorithms for LP/IP Optimization Problems by Reuven Bar-Yehuda

Seminar 236813: Approximationalgorithms for LP/IP optimization problems • Reuven Bar-Yehuda • Technion IIT • Slides and papers at: • http://www.cs.technion.ac.il/~cs236813

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

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 Zn • Minimizep·x • Subject to: x {0,1}n • xi + xj  1 {i,j} E

Example SC • Given a Collection S1, S2,…,Sn of all subsetsof {1,2,3,…,m} and penalty vector p Zn • Minimizep·x • Subject to: x {0,1}n • xi  1 j=1..m • j Si

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

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

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

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

Example Generalized Steiner Forest • Given graph G(V,E) T1T1…Tk  V • and length vector p Z|E| • Minp·x • S.t.: x {0,1}|E| • xe  1 i Ti’s cut P • e P

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

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 • Given a penalty [profit] vector p. • Minimize[Maximize]p·x • Subject to: feasibility constraints F(x) • x isr-approximationif F(x) and p·x r·p·x* • An algorithm is r-approximationif for any p, F • it returns an r-approximation

The Local-Ratio Theorem: • xis an r-approximation with respect to p1 • xis an r-approximation with respect to p- p1 •  • xis 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 )*

The Local-Ratio Theorem: (Proof2) • xis an r-approximation with respect to p1 • xis an r-approximation with respect to p- p1 •  • xis an r-approximation with respect to p • Proof2: (For minimization) • Let x*, x1*,x2*be optimal solutions for p, p1, p2 respectively • p1 · x  r ×p1 x1* • p2 · x  r ×p2x2* •  • p · x  r ×( p1 x1*+ p2x2*) •  r ×(p1x*, + p2 x*) = px*

Special case: Optimization is 1-approximation • xis an optimum with respect to p1 • xis an optimum with respect to p- p1 • xis an optimum with respect to p

A Local-Ratio Schema for Minimization[Maximization] problems: • Algorithm r-ApproxMin[Max]( Set, p ) • If Set = Φ then returnΦ ; • If I  Setp(I)=0 then return {I} r-ApproxMin( Set-{I}, p ) ; • [If  I  Setp(I)  0 then returnr-ApproxMax( Set-{I}, p ) ;] • Define “good” p1 ; • REC = r-ApproxMax[Min]( Set, p- p1) ; • If REC is not an r-approximation w.r.t. p1 then “fix it”; • return REC;

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) Applications Resource Allocation and Scheduling: ….

The creative part…find -Effective weights • p1 is -Effective if every feasible solution is -approx w.r.t. p1 • i.e. p1 ·x  p1* • VC (vertex cover) • Edge • Matching • Greedy • Homogenious

VC: Recursive implementation (edge by edge) • 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 p1(v) =  if v=x or v=y and 0 otherwise; • Return VC(V, E, p- p1) 0 0 0   0 0 0

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}; 0 0 0   0 0 0

8 12 5 20 10 6 Min 5xBisli+8xTea+12xWater+10xBamba+20xShampoo+15xPopcorn+6xChocolate s.t. xShampoo + xWater  1 15

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}; 30 15 90 10 50 100 80 2

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  

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 3 5 3 2

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

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 {e}+MST(Vshrink(e), Eshrink(e), p); • Let  = min{p(e) : eE}; • Define p1(e) =  for all eE; • Return MST(V, E, p- p1)

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

Some effective weights

Approximation Algorithms for LP/IP Optimization Problems by Reuven Bar-Yehuda