Vertex Cover Problems

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1 Rob van Stee: Approximations- und Online-Algorithmen 1 Vertex Cover Problems Consider a graph G=(V,E) S V is a vertex cover if {u,v} E : u S v S minimum vertex cover (MIN-VCP): find a vertex cover S that minimizes S.

2 Rob van Stee: Approximations- und Online-Algorithmen 2 Motivation This problem has many applications Example: placing ATMs in a city Each additional ATM costs money Want to have an ATM in every street (block, district) Where should they be placed so that we need as little ATMs as possible?

3 Rob van Stee: Approximations- und Online-Algorithmen 3 Greedy Algorithm Function greedyvc(v,e) C:= /0 while E /0 do return C select any {u, v} E C:= C {u,v} remove all edges incident to u or v from E Exercise: explain how to implement the algorithm to run in time O( V + E ) b a b a b a b c e c d h e f g c e c d f d f d h g h g h a e f g (postprocess:)

4 Rob van Stee: Approximations- und Online-Algorithmen 4 Greedy Algorithm Function greedyvc(v,e) C:= /0 while E /0 do return C select any {u, v} E C:= C {u,v} remove all edges incident to u or v from E Exercise: explain how to implement the algorithm to run in time O( V + E ) b a b a b a b a c d h e f g c d h e f g c d h e f g c d h e f g

5 Rob van Stee: Approximations- und Online-Algorithmen 5 Greedy Algorithm Function greedyvc(v,e) C:= /0 while E /0 do return C select any {u, v} E C:= C {u,v} remove all edges incident to u or v from E Exercise: explain how to implement the algorithm to run in time O( V + E ) b a b a b a b a c d h e f g c d h e f g c d h e f g c d h e f g

6 Rob van Stee: Approximations- und Online-Algorithmen 6 Greedy Algorithm Function greedyvc(v,e) C:= /0 while E /0 do return C select any {u, v} E C:= C {u,v} remove all edges incident to u or v from E Exercise: explain how to implement the algorithm to run in time O( V + E ) b a b a b a b a c d h e f g c d h e f g c d h e f g c d h e f g

7 Rob van Stee: Approximations- und Online-Algorithmen 7 Theorem 1. Algorithm greedyvc computes a two-approximation of MIN-VCP. Proof. Correctness: trivial since only covered edges are removed. Quality: Let A denote the set of edges selected by greedyvc. We have C =2 A. A is a matching, i.e., no node covers two edges in A. Hence, any vertex cover contains at least one node from each edge in A, i.e., opt A.

8 Rob van Stee: Approximations- und Online-Algorithmen 8 Weighted Vertex Cover Consider a graph G=(V,E) S V is a vertex cover if {u,v} E : u S v S 1 1 minimum WEIGHT vertex cover (WEIGHT-VCP): find a vertex cover S that minimizes c(s) v S

9 Rob van Stee: Approximations- und Online-Algorithmen ILP Formulation Assume V ={1,...,n} Variables: x v = 1 iff v V minimize c x subject to {u,v} E : x u + x v 1 v V : x v {0,1}

10 Rob van Stee: Approximations- und Online-Algorithmen ILP Formulation Linear Relaxation Assume V ={1,...,n} Variables: x v = 1 iff v V minimize c x subject to {u,v} E : x u + x v 1 v V : x v {0,1} Assume V ={1,...,n} Variables: x v = 1 iff v V minimize c x subject to {u,v} E : x u + x v 1 v V : x v 0

11 Rob van Stee: Approximations- und Online-Algorithmen ILP Formulation Linear Relaxation Assume V ={1,...,n} Variables: x v = 1 iff v V minimize c x subject to {u,v} E : x u + x v 1 v V : x v {0,1} Assume V ={1,...,n} Variables: x v = 1 iff v V minimize c x subject to {u,v} E : x u + x v 1 v V : x v 0 LP Rounding Algorithm for WEIGHT-VCP Function lpweightedvc(v, E, c) x:= lpsolve(linearrelaxation(v, E, c)) return{v V : x v 1/2}

12 Rob van Stee: Approximations- und Online-Algorithmen 12 Theorem 2. Algorithm lpweightedvc computes a two-approximation of WEIGHT-VCP. Correctness: Consider any edge {u,v} E. We have x u + x v 1, hence, max{x u,x v } 1/2, i.e., rounding will put at least one of {u,v} into the output.

13 Rob van Stee: Approximations- und Online-Algorithmen 13 Theorem 2. Algorithm lpweightedvc computes a two-approximation of WEIGHT-VCP. Quality: Let x := the solution computed by lpweightedvc x := the optimal solution, and x := the optimal solution of the linear relaxation c x= c i 2 x i c i 2 x i 1/2 x i 1/2 n i=1 x i c i = 2c x 2c x

14 Rob van Stee: Approximations- und Online-Algorithmen 14 Theorem 2. Algorithm lpweightedvc computes a two-approximation of WEIGHT-VCP. Quality: Let x := the solution computed by lpweightedvc x := the optimal solution, and x := the optimal solution of the linear relaxation c x= c i 2 x i c i 2 x i 1/2 x i 1/2 n i=1 x i c i = 2c x 2c x

15 Rob van Stee: Approximations- und Online-Algorithmen 15 Theorem 2. Algorithm lpweightedvc computes a two-approximation of WEIGHT-VCP. Quality: Let x := the solution computed by lpweightedvc x := the optimal solution, and x := the optimal solution of the linear relaxation c x= c i 2 x i c i 2 x i 1/2 x i 1/2 n i=1 x i c i = 2c x 2c x

16 Rob van Stee: Approximations- und Online-Algorithmen 16 Theorem 2. Algorithm lpweightedvc computes a two-approximation of WEIGHT-VCP. Quality: Let x := the solution computed by lpweightedvc x := the optimal solution, and x := the optimal solution of the linear relaxation c x= c i 2 x i c i 2 x i 1/2 x i 1/2 n i=1 x i c i = 2c x 2c x

17 Rob van Stee: Approximations- und Online-Algorithmen 17 Theorem 2. Algorithm lpweightedvc computes a two-approximation of WEIGHT-VCP. Quality: Let x := the solution computed by lpweightedvc x := the optimal solution, and x := the optimal solution of the linear relaxation c x= c i 2 x i c i 2 x i 1/2 x i 1/2 n i=1 x i c i = 2c x 2c x

18 Rob van Stee: Approximations- und Online-Algorithmen 18 Iterated Rounding [Vazirani Section 23.2] Function iteratedlpweightedvc(v, E, c) M:= /0 while E >0 do x:= lpsolve(linearrelaxation(v, E, c)) let v denote the node which maximizes x v M:= M {v} V:= V \{v} return M E:= E\{{u,v} E}

19 Rob van Stee: Approximations- und Online-Algorithmen 19 Iterated Rounding: Discussion Might give better solutions for many inputs No better approximation guarantees for VC Larger (still polynomial) execution time But: Resolving an LP is often quite fast Important technique for other problems

20 Rob van Stee: Approximations- und Online-Algorithmen 20 A Randomized Algorithm [Ausiello et al. Section 5.1] Function randweightedvc(v, E, c) C:= /0 while E /0 do b a b a c e c e d f d f h g h g select any {v,t} E b c d h return C flip a coin with sides {v,t} and P[v]= c t c v + c t x:= upper side of coin C:= C {x} remove all edges incident to x from E a b a b a e c e c e f d f d f g h g h g

21 Rob van Stee: Approximations- und Online-Algorithmen 21 Theorem 3. Algorithm randweightedvc computes a vertex cover x with E[c x] 2c x. Correctness: as for greedyvc.

22 Rob van Stee: Approximations- und Online-Algorithmen 22 Theorem: Algorithm randweightedvc computes a vertex cover x with E[c x] 2c x. Quality: Define the random variables c v if v x X v := 0 otherwise c v if {v,t} is selected and v x X {v,t},v := 0 otherwise (1) (2) Note that X v = X {v,t},v {t:{v,t} E}

23 Rob van Stee: Approximations- und Online-Algorithmen 23 Lemma 4. E[X {v,t},v ]=E[X {v,t},t ] Proof. E[X {v,t},v ]=c v P[{v,t} is selected ]P[v x] E[X {v,t},v ]=c v P[{v,t} is selected ] c v + c t c v = c t P[{v,t} is selected ] c v + c t =E[X {v,t},t ] c t

24 Rob van Stee: Approximations- und Online-Algorithmen 24 Lemma 4. E[X {v,t},v ]=E[X {v,t},t ] Proof. E[X {v,t},v ]=c v P[{v,t} is selected ]P[v x] E[X {v,t},v ]=c v P[{v,t} is selected ] c v + c t c v = c t P[{v,t} is selected ] c v + c t =E[X {v,t},t ] c t

25 Rob van Stee: Approximations- und Online-Algorithmen 25 Lemma 4. E[X {v,t},v ]=E[X {v,t},t ] Proof. E[X {v,t},v ]=c v P[{v,t} is selected ]P[v x] E[X {v,t},v ]=c v P[{v,t} is selected ] c v + c t c v = c t P[{v,t} is selected ] c v + c t =E[X {v,t},t ] c t

26 Rob van Stee: Approximations- und Online-Algorithmen 26 Lemma 5. v x E[X v ] t x E[X t ] Proof. E[X v ]= E v x v x But also t x E[X t ]= t x Every term in (*) shows up in (**). [ ] X {v,t},v {t:{v,t} E} (X v = X {v,t},v ) {t:{v,t} E} = E[X {v,t},v ] Linearity ofe[ ] v x {t:{v,t} E} = E[X {v,t},t ](*). Lemma 4 v x {t:{v,t} E} E[X {v,t},t ](**). {v:{v,t} E}

27 Rob van Stee: Approximations- und Online-Algorithmen 27 Lemma 5. v x E[X v ] t x E[X t ] Proof. E[X v ]= E v x v x But also t x E[X t ]= t x Every term in (*) shows up in (**). [ ] X {v,t},v {t:{v,t} E} (X v = X {v,t},v ) {t:{v,t} E} = E[X {v,t},v ] Linearity ofe[ ] v x {t:{v,t} E} = E[X {v,t},t ](*). Lemma 4 v x {t:{v,t} E} E[X {v,t},t ](**). {v:{v,t} E}

28 Rob van Stee: Approximations- und Online-Algorithmen 28 Lemma 5. v x E[X v ] t x E[X t ] Proof. E[X v ]= E v x v x But also t x E[X t ]= t x Every term in (*) shows up in (**). [ ] X {v,t},v {t:{v,t} E} (X v = X {v,t},v ) {t:{v,t} E} = E[X {v,t},v ] Linearity ofe[ ] v x {t:{v,t} E} = E[X {v,t},t ](*). Lemma 4 v x {t:{v,t} E} E[X {v,t},t ](**). {v:{v,t} E}

29 Rob van Stee: Approximations- und Online-Algorithmen 29 Lemma 5. v x E[X v ] t x E[X t ] Proof. E[X v ]= E v x v x [ ] X {v,t},v {t:{v,t} E} (X v = X {v,t},v ) {t:{v,t} E} = E[X {v,t},v ] Linearity ofe[ ] v x {t:{v,t} E} But also t x E[X t ]= t x Every term in (*) shows up in (**). = E[X {v,t},t ](*). Lemma 4 v x {t:{v,t} E} E[X {v,t},t ](**). {v:{v,t} E} t... t v t

30 Rob van Stee: Approximations- und Online-Algorithmen 30 Theorem: Algorithm randweightedvc computes a vertex cover x with E[c x] 2c x. Quality: (Finishing Up) v V E[X v ] = v x E[X v ]+ t x E[X t ] Lemma 5 2 t x c t X t = 0 or X t = c t

31 Rob van Stee: Approximations- und Online-Algorithmen 31 Theorem: Algorithm randweightedvc computes a vertex cover x with E[c x] 2c x. Quality: (Finishing Up) v V E[X v ] = v x E[X v ]+ t x E[X t ] Lemma 5 2 t x E[X t ] 2 t x c t X t = 0 or X t = c t

32 Rob van Stee: Approximations- und Online-Algorithmen 32 Theorem: Algorithm randweightedvc computes a vertex cover x with E[c x] 2c x. Quality: (Finishing Up) v V E[X v ] = v x E[X v ]+ t x E[X t ] Lemma 5 2 t x E[X t ] 2 t x c t X t = 0 or X t = c t

33 Rob van Stee: Approximations- und Online-Algorithmen 33 Theorem: Algorithm randweightedvc computes a vertex cover x with E[c x] 2c x. Quality: (Finishing Up) v V E[X v ] = v x E[X v ]+ t x E[X t ] Lemma 5 2 t x E[X t ] 2 c t t x = 2c x X t = 0 or X t = c t

34 Rob van Stee: Approximations- und Online-Algorithmen 34 More on Vertex Cover There are simple deterministic linear time 2-approximations. (Special case of set covering) Best known algorithm: ratio 2 Θ(1/ logn) fixed parameter algorithms: [Niedermeyer Rossmanith] find optimal solution in time O ( kn+k k) if x k. Key idea: (clever) exhaustive search+problem reductions. Example: include nodes of degree k. include neighbors of degree 1 nodes

35 Rob van Stee: Approximations- und Online-Algorithmen 35 Scheduling on Unrelated Parallel Machines [Vazirani Chapter 17] J: set of n jobs M: set of m machines p i j : processing time of job j on machine i x( j): Machine where job j is executed L i : { j:x( j)=i} p i j, load of machine i Objective: Minimize Makespan L max = max i L i

36 Rob van Stee: Approximations- und Online-Algorithmen 36 A Misguided ILP model minimize t subject to j J : x i j = 1 i M i M : x i j p i j t j J i M, j J : x i j {0,1}

37 Rob van Stee: Approximations- und Online-Algorithmen 37 The problem with this formulation minimize t subject to j J : x i j = 1 i M i M : x i j p i j t j J i M, j J : x i j {0,1} One Job, size m everywhere. Linear relaxation: makespan 1 Optimal solution: makespan m m The linear relaxation is far away from the optimal solution and hence yields little useful information LP-speak: integrality gap m

38 Rob van Stee: Approximations- und Online-Algorithmen 38 The problem with this formulation minimize t subject to j J : x i j = 1 i M i M : x i j p i j t j J i M, j J : x i j {0,1} In ILP, we always have x i j = 0 if p i j > t This is lost in the linear relaxation: some x i j may get small values We cannot add this constraint since it is not a linear constraint

39 Rob van Stee: Approximations- und Online-Algorithmen 39 A Refined LP Relaxation (Parametric Pruning) guess makespan T feasible assignments: S T := { (i, j) : p i j T } no objective function (feasibility only) LP(T): j J : i M : x i j = 1 {i:(i, j) S T } { j:(i, j) S T } (i, j) S T : x i j 0 x i j p i j T e.g., binary search

40 Rob van Stee: Approximations- und Online-Algorithmen 40 A Refined LP Relaxation (Parametric Pruning) guess makespan T feasible assignments: S T := { (i, j) : p i j T } e.g., binary search LP(T): j J : i M : x i j = 1 {i:(i, j) S T } { j:(i, j) S T } (i, j) S T : x i j 0 x i j p i j T No objective function! We only look for a feasible solution

41 Rob van Stee: Approximations- und Online-Algorithmen 41 More LP-speak Consider a solution x of a given LP. x is an extreme point solution if it cannot be expressed as a convex combination αx +(1 α)x with α (0,1) of two other feasible solutions x and x. convex combination x x Theorem 6. x R r is an extreme point solution iff it corresponds to setting r linearly independent constraints to equality. Proof. not here.

42 Rob van Stee: Approximations- und Online-Algorithmen 42 S T := { (i, j) : p i j T } LP(T): j J : i M : x i j = 1 {i:(i, j) S T } { j:(i, j) S T } (i, j) S T : x i j 0 x i j p i j T Lemma 7. An extreme point solution of LP(T) has at most n+m nonzero variables. Proof. r= S T variables n+m constraints (except 0) Thm6 r (n+m) of the 0 constraints are tight.

43 Rob van Stee: Approximations- und Online-Algorithmen 43 Lemma 7. An extreme point solution of LP(T) has at most n+m nonzero variables. Corollary 8. An extreme point solution of LP(T) sets n m jobs integrally. Proof. a integrally set jobs a nonzero entries in x n a fractionally set jobs 2(n a) nonzero entries in x Lemma 7 2(n a)+a n+m a n m

44 Rob van Stee: Approximations- und Online-Algorithmen 44 One Reason why LP Relaxation is Useful Theorem 6. x R r is an extreme point solution iff it corresponds to setting r linearly independent constraints to equality. Theorem 6 often implies that only few variables need to be rounded to obtain an solution of the ILP.... this does not mean rounding the remaining ones is easy.

45 Rob van Stee: Approximations- und Online-Algorithmen 45 The Algorithm: Top Level α:= makespan one gets by assigning each job to the fastest machine for it α is an upper bound for the optimal makespan

46 Rob van Stee: Approximations- und Online-Algorithmen 46 The Algorithm: Top Level α:= makespan one gets by assigning each job to the fastest machine for it α is an upper bound for the optimal makespan Use binary search in the range α/m,α to find the smallest T such that LP(T) has a feasible solution x

47 Rob van Stee: Approximations- und Online-Algorithmen 47 The Algorithm: Top Level α:= makespan one gets by assigning each job to the fastest machine for it α is an upper bound for the optimal makespan Use binary search in the range α/m,α to find the smallest T such that LP(T) has a feasible solution For this T, find an extremal point solution x

48 Rob van Stee: Approximations- und Online-Algorithmen 48 The Algorithm: Top Level α:= makespan one gets by assigning each job to the fastest machine for it α is an upper bound for the optimal makespan Use binary search in the range α/m,α to find the smallest T such that LP(T) has a feasible solution For this T, find an extremal point solution x assign integrally set jobs in x

49 Rob van Stee: Approximations- und Online-Algorithmen 49 The Algorithm: Top Level α:= makespan one gets by assigning each job to the fastest machine for it α is an upper bound for the optimal makespan Use binary search in the range α/m,α to find the smallest T such that LP(T) has a feasible solution For this T, find an extremal point solution x assign integrally set jobs in x deal with the fractionally set jobs // Rounding

50 Rob van Stee: Approximations- und Online-Algorithmen 50 Example p i j Four machines, five jobs For each job, the best machine for it is marked in blue.

51 Rob van Stee: Approximations- und Online-Algorithmen 51 Example p i j Each job on fastest machine: =: α Initial guess for the makespan is 6 Using binary search, we find smallest makespan in the range [6/4,6] that can be achieved using a fractional assignment

52 Rob van Stee: Approximations- und Online-Algorithmen 52 Example p i j Each job on fastest machine: =: α x i j Solution of LP(3):

53 Rob van Stee: Approximations- und Online-Algorithmen 53 Dealing with Fractionally Set Jobs Consider the bipartite graph H:=(J M,E ) where J := { j J : i : 0<x i j < 1 } M := { i M : j : 0<x i j < 1 } E := { {i, j} : x i j 0,i M, j J } Idea: Find a perfect matching in H assign jobs according to that matching matching

54 Rob van Stee: Approximations- und Online-Algorithmen 54 Matching A set of edges M that do not have any nodes in common, i.e., (V,M) has maximum degree one. Perfect Matching A matching of size V /2, i.e., all nodes are matched

55 Rob van Stee: Approximations- und Online-Algorithmen 55 Lemma 8. H is a pseudo forest, i.e., each connected component H C =(V C,E C ) has E C V C (a tree plus, possibly, one edge) Proof. It suffices to show this for the larger graph G:=(J M,E) where E:= { {i, j} : x i j 0,i M, j J } Consider a connected component H C of G. restrict x and LP(T) to H C : x C, LP C (T) x C is extreme point solution of LP C (T) (Otherwise, x itself could not be extreme point solution) Lemma 7 LP C (T) has V C nonzero vars., i.e., H C has V C edges.

56 Rob van Stee: Approximations- und Online-Algorithmen 56 Example p i j x i j T =

57 Rob van Stee: Approximations- und Online-Algorithmen 57 Lemma 9. H has a perfect matching Proof. We give an algorithm: M := /0 invariant H is a bipartite pseudo forest invariant all degree one nodes are machines while i M with degree one do e={i, j}:= the sole edge incident to i M := M {e} remove i, j and incident edges assert H is a collection of disjoint even cycles foreach cycle C H do match alternating edges in C

58 Rob van Stee: Approximations- und Online-Algorithmen 58 Theorem 10. The algorithm achieves an approximation guarantee of factor 2 for scheduling unrelated parallel machines. Proof. Consider solution x of LPT(T ) makespan due to jobs set integrally in x is T opt. In addition, each machine i receives 1 job j from the matching M H. p i, j T opt since otherwise {i, j} H

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