Partial Vertex Cover

Transcription

1 Partial Vertex Cover Joshua Wetzel Department of Computer Science Rutgers University Camden April 2, 2009 Joshua Wetzel Partial Vertex Cover / 48

2 Vertex Cover Input: Given G = (V, E) Non-negative weights on vertices Objective: Find a least-weight collection of vertices such that each edge in G in incident on at least one vertex in the collection Joshua Wetzel Partial Vertex Cover 2 / 48

3 Vertex Cover: Example COST = 97 COST = 08 Joshua Wetzel Partial Vertex Cover / 48

4 Vertex Cover: IP Formulation x v if v is in our cover, 0 otherwise min w v x v s.t. v V x a + x b, e = (a, b) x v {0, }, v V Joshua Wetzel Partial Vertex Cover 4 / 48

5 LP Relaxation Integer programs have been shown to be NP-hard Relax the integrality constraints x v {0, }, v V min s.t. w v x v v V x a + x b, e = (a, b) x v 0, v V Joshua Wetzel Partial Vertex Cover 5 / 48

6 Constructing the Dual LP Primal LP: min s.t. w v x v v V x a + x b, e = (a, b) (y e ) x v 0, v V Joshua Wetzel Partial Vertex Cover 6 / 48

7 Primal LP and Dual LP Primal LP: Dual LP: min s.t. w v x max v v V x a + x b, e = (a, b) x v 0, v V s.t. e E y e y e w v, v V e : e hits v y e 0, e E Joshua Wetzel Partial Vertex Cover 7 / 48

8 Primal-Dual Method Dual Feasible Dual OPT = Primal OPT Primal OPT OPT IP Construct the dual LP Construct an algorithm that manually tightens dual constraints to obtain a maximal dual solution Joshua Wetzel Partial Vertex Cover 8 / 48

9 Clarkson s Algorithm Inititally all edges are uncovered. While an uncovered edge in G: raise y e for all uncovered edges simultaneouslty until a vertex, v, becomes full (i.e y e = w v ) e:e hits v C C {v} any e that touches v is covered. Return C as our vertex cover Joshua Wetzel Partial Vertex Cover 9 / 48

10 Clarkson s Algorithm: Example y e w v e:e hits v Raise each y e uniformly until a vertex is full. Joshua Wetzel Partial Vertex Cover 0 / 48

11 Clarkson s Algorithm: Example e:e hits v y e w v Joshua Wetzel Partial Vertex Cover / 48

12 Clarkson s Algorithm: Example e:e hits v y e w v Joshua Wetzel Partial Vertex Cover / 48

13 Clarkson s Algorithm: Example e:e hits v y e w v Joshua Wetzel Partial Vertex Cover / 48

14 Clarkson s Algorithm: Example y e w v e:e hits v COST = 8 Joshua Wetzel Partial Vertex Cover 4 / 48

15 Clarkson s Algorithm: Analysis Dual Obj. Fn: max e y e y e w v e:e hits v Our Cost = wt(red vertices) 2 e hits red 2 e = 2DFS y e 2OPT y e Joshua Wetzel Partial Vertex Cover / 48

16 Clarkson s Algorithm: Tight Example 6 6 COST OPT = 6 COST Clarkson = Joshua Wetzel Partial Vertex Cover 6 / 48

17 Partial Vertex Cover Input: Graph, G = (V, E) Non-negative integer weights for vertices Integer, k Objective: Find the least cost set of vertices in G that will cover at least k edges. Joshua Wetzel Partial Vertex Cover 7 / 48

18 Key Issue In full vertex cover OPT covers all edges and hence we know which edges to cover. In partial vertex cover, we do not know which k edges OPT covers. When k is part of the input, the techniques for full-coverage do not directly apply. Joshua Wetzel Partial Vertex Cover 8 / 48

19 Clarkson s Algorithm Fails Input: k = 5/6 5/6 5 5/6 5 5/6 5/6 5/6 COST OPT = COST Clarkson = 5 Joshua Wetzel Partial Vertex Cover 9 / 48

20 Related Work Bshouty & Burroughs 998: solve the LP, modify it and then round the modified solution. 2-approximation for partial vertex cover. Hochbaum 998: 2-approximation for partial vertex cover. Bar-Yehuda 999: local ratio method, 2-approximation for partial vertex cover. Mestre 2005: 2-approximation primal-dual algorithm for partial vertex cover with improved running time Joshua Wetzel Partial Vertex Cover 20 / 48

21 Vertex Cover: IP Formulation x v if v is in our cover, 0 otherwise min w v x v s.t. v V x a + x b, e = (a, b) x v {0, }, v V Joshua Wetzel Partial Vertex Cover 2 / 48

22 Partial Vertex Cover: IP formulation x v if vertex v is chosen in the cover, 0 otherwise. y e if edge e is not covered, 0 otherwise. min w v x v s.t. v V y e + x a + x b, e = (a, b) y e m k e E x v {0, }, v V y e {0, }, e E Joshua Wetzel Partial Vertex Cover 22 / 48

23 Patial Vertex Cover: LP formulation Relax the integrality constraints. x v {0, } x v 0, v V y e {0, } y e 0, e E Joshua Wetzel Partial Vertex Cover 2 / 48

24 Constructing the Dual LP min s.t. w v x v v V y e + x a + x b, e = (a, b) (u e ) y e (m k) (z) e E x v 0, v V y e 0, e E Joshua Wetzel Partial Vertex Cover 24 / 48

25 Primal LP and Dual LP Primal LP: Dual LP: min s.t. w v x v v V y e + x a + x b, e y e m k e E x v 0, v V y e 0, e E max s.t. u e z(m k) e E u e w v, v V e : e hits v u e z, e E u e 0, e E z 0 Joshua Wetzel Partial Vertex Cover 25 / 48

26 Intuition for Primal-Dual Input: k = 5/6 5/6 5 5/6 5 5/6 5/6 5/6 COST OPT = COST Clarkson = 5 In last iteration, Clarkson s Alg. may choose more edges than we need at a very high cost We must somehow bound the cost of last vertex chosen Joshua Wetzel Partial Vertex Cover 26 / 48

27 Primal-Dual Algorithm For each vertex, v, in G Guess v to be the heaviest vertex in OPT, called v h C v {v h } Raise weight of all heavier vertices in G to Remove all edges touching v h from G k k deg(v h ) Run Clarkson on this instance until k edges are covered. C choices C choices C V Return the lowest cost cover, C Joshua Wetzel Partial Vertex Cover 27 / 48

28 Primal-Dual Algorithm: Example Input: G = (V, E), k = 8 Joshua Wetzel Partial Vertex Cover 28 / 48

29 Primal-Dual Algorithm: Example u e w v e:e hits v V h k = 6 Joshua Wetzel Partial Vertex Cover 29 / 48

30 Primal-Dual Algorithm: Example e:e hits v u e w v V h Joshua Wetzel Partial Vertex Cover 0 / 48

31 Primal-Dual Algorithm: Example e:e hits v u e w v V h Joshua Wetzel Partial Vertex Cover / 48

32 Primal-Dual Algorithm: Example e:e hits v u e w v 6 0 V h Joshua Wetzel Partial Vertex Cover 2 / 48

33 Primal-Dual Algorithm: Example u e w v e:e hits v 6 0 V h COST = Joshua Wetzel Partial Vertex Cover / 48

34 Primal-Dual Algorithm: Example u e w v e:e hits v 0 0 V h COST = Joshua Wetzel Partial Vertex Cover 4 / 48

35 Primal-Dual Algorithm: Example u e w v e:e hits v V h 4 COST = 69 Joshua Wetzel Partial Vertex Cover 5 / 48

36 Primal-Dual Algorithm: Example u e w v e:e hits v 6 0 V h COST = Joshua Wetzel Partial Vertex Cover 6 / 48

37 Primal-Dual Algorithm: Example e:e hits v u e w v 0 V h COST = Joshua Wetzel Partial Vertex Cover 7 / 48

38 Primal-Dual Algorithm: Example u e w v e:e hits v 8 7 V h COST = 60 Joshua Wetzel Partial Vertex Cover 8 / 48

39 Primal-Dual Algorithm: Example u e w v e:e hits v 24 V h COST = 76 Joshua Wetzel Partial Vertex Cover 9 / 48

40 Primal-Dual Algorithm: Example u e w v e:e hits v V h COST = 80 Joshua Wetzel Partial Vertex Cover 40 / 48

41 Primal-Dual Algorithm: Example Heaviest Vertex Cost Return cover with cost of 60 Joshua Wetzel Partial Vertex Cover 4 / 48

42 Analysis: Cost of OPT I h : inst. in which we correctly guess heaviest vertex in OPT V h 7 OPT I h OPT = OPT(I h ) + w(v h ) Joshua Wetzel Partial Vertex Cover 42 / 48

43 Analysis: Our Cost OPT = OPT(I h ) + w(v h ) V h 7 V h OPT(I h ) + w(v h ) = 45 COST(I h ) + w(v h ) = 60 Joshua Wetzel Partial Vertex Cover 4 / 48

44 Analysis: What is z? V l 8 V h 7 0 Z Z Z V l 8 V h 7 0 I h u e z, e z = Num of edges for which u e = z is at least m k Joshua Wetzel Partial Vertex Cover 44 / 48

45 Analysis: Our Cost OPT = OPT(I h ) + w(v h ) Z Z Z V l 8 V h 7 0 Our Cost Cost(I h ) + w(v h ) = w(redvert) + w(v l ) + w(v h ) w(redvert) + 2w(v h ) 2 u e + 2w(v h ) e hits red 2[ u e z(m k )] + 2w(v h ) e I h 2DFS(I h ) + 2w(v h ) 2OPT(I h ) + 2w(v h ) 2OPT Joshua Wetzel Partial Vertex Cover 45 / 48

46 Primal LP and Dual LP Primal LP: Dual LP: min s.t. w v x v v V y e + x a + x b, e y e m k e E x v 0, v V y e 0, e E max s.t. u e z(m k) e E u e w v, v V e : e hits v u e z, e E u e 0, e E z 0 Joshua Wetzel Partial Vertex Cover 46 / 48

47 Reference K. L. Clarkson. A modification of the greedy algorithm for the vertex cover. Information Processing Letters 6:2-25, 98. R. Gandhi, S. Khuller, and A. Srinivasan. Approximation Algorithms for Partial Covering Problems. Journal of Algorithms, 5():55-84, October Joshua Wetzel Partial Vertex Cover 47 / 48

48 Thank You. Joshua Wetzel Partial Vertex Cover 48 / 48