Formulations and Algorithms for Minimum Connected Dominating Set Problems

Transcription

1 Formulations and Algorithms for Minimum Connected Dominating Set Problems Abilio Lucena 1 Alexandre Salles da Cunha 2 Luidi G. Simonetti 3 1 Universidade Federal do Rio de Janeiro 2 Universidade Federal de Minas Gerais 3 Universidade Federal Fluminense Aussois, January 2012

2 Graph G = (V, E)

3 Dominating Set W V such that every vertex of V is in W or has a neighbor in W.

4 Connected Dominating Set

5 Connected Dominating Set W such that G W = (W, E(W )) is connected

6 Minimum Connected Dominating Set Problem Find connected dominating set W with as few vertices as possible

7 Closely Related Problem The Maximum Leaf Spanning Tree Problem

8 Closely Related Problem The Maximum Leaf Spanning Tree Problem

9 Closely Related Problem The Maximum Leaf Spanning Tree Problem

10 Spanning Tree Connected Dominating Set

11 Spanning Tree Connected Dominating Set Drop the leaves from the tree!

12 Spanning Tree Connected Dominating Set

13 Spanning Tree Connected Dominating Set Connected Dominating Set W

14 Connected Dominating Set Spanning Tree

15 Connected Dominating Set Spanning Tree Find Spanning Tree for G W = (W, E(W ))

16 Connected Dominating Set Spanning Tree Now, connect every vertex j V \ W to a neighbor vertex in W

17 Connected Dominating Set Spanning Tree Spanning tree of G

18 k-connected Dominating Set G W = (W, E(W )) is k-edge-connected (resp. k-vertex-connected) if it remains connected when fewer than k edges (resp. vertices) are removed from it. Minimum k-connected Dominating Set Problem: Find k-connected G W = (W, E(W )) with W as small as possible.

19 k-connected Dominating Set G W = (W, E(W )) is k-edge-connected (resp. k-vertex-connected) if it remains connected when fewer than k edges (resp. vertices) are removed from it. Minimum k-connected Dominating Set Problem: Find k-connected G W = (W, E(W )) with W as small as possible.

20 k-connected Dominating Set G W = (W, E(W )) is k-edge-connected (resp. k-vertex-connected) if it remains connected when fewer than k edges (resp. vertices) are removed from it. Minimum k-connected Dominating Set Problem: Find k-connected G W = (W, E(W )) with W as small as possible.

21 Minimum Dominating Tree Problem

22 Minimum Dominating Tree Problem Edge weights {c e R : e E} are associated with G = (V, E): find least weight tree T = (W, E W ), W V, E W E(W ), where W is dominating for G.

23 Minimum Dominating Circuit Problem

24 Minimum Dominating Circuit Problem Edge weights {c e R : e E} are associated with G = (V, E): find least weight circuit C = (W, E W ), W V, E W E(W ), where W that is dominating for G.

25 The Cycle Problem

26 The Cycle Problem Edge weights {c e R : e E} are associated with G = (V, E): find a circuit of G with the least weight.

27 Why Investigate These Problems? Intrinsically interesting and quite challenging NP-hard problems {1,2}-CDSPs: model to some telecommunication network design and circuit layout design applications Cycle Problem: generalizes TSP and is the basic underlying structure in problems such as Prize Collecting TSP Dominating Circuit Problem: no solution algorithms available and is the basic underlying structure in problems such as Ring-star Problem

28 Why Investigate These Problems? Intrinsically interesting and quite challenging NP-hard problems {1,2}-CDSPs: model to some telecommunication network design and circuit layout design applications Cycle Problem: generalizes TSP and is the basic underlying structure in problems such as Prize Collecting TSP Dominating Circuit Problem: no solution algorithms available and is the basic underlying structure in problems such as Ring-star Problem

29 Why Investigate These Problems? Intrinsically interesting and quite challenging NP-hard problems {1,2}-CDSPs: model to some telecommunication network design and circuit layout design applications Cycle Problem: generalizes TSP and is the basic underlying structure in problems such as Prize Collecting TSP Dominating Circuit Problem: no solution algorithms available and is the basic underlying structure in problems such as Ring-star Problem

30 Why Investigate These Problems? Intrinsically interesting and quite challenging NP-hard problems {1,2}-CDSPs: model to some telecommunication network design and circuit layout design applications Cycle Problem: generalizes TSP and is the basic underlying structure in problems such as Prize Collecting TSP Dominating Circuit Problem: no solution algorithms available and is the basic underlying structure in problems such as Ring-star Problem

31 Why Investigate These Problems? Intrinsically interesting and quite challenging NP-hard problems {1,2}-CDSPs: model to some telecommunication network design and circuit layout design applications Cycle Problem: generalizes TSP and is the basic underlying structure in problems such as Prize Collecting TSP Dominating Circuit Problem: no solution algorithms available and is the basic underlying structure in problems such as Ring-star Problem

32 Why Investigate These Problems? Intrinsically interesting and quite challenging NP-hard problems {1,2}-CDSPs: model to some telecommunication network design and circuit layout design applications Cycle Problem: generalizes TSP and is the basic underlying structure in problems such as Prize Collecting TSP Dominating Circuit Problem: no solution algorithms available and is the basic underlying structure in problems such as Ring-star Problem

33 Why Investigate These Problems? Intrinsically interesting and quite challenging NP-hard problems {1,2}-CDSPs: model to some telecommunication network design and circuit layout design applications Cycle Problem: generalizes TSP and is the basic underlying structure in problems such as Prize Collecting TSP Dominating Circuit Problem: no solution algorithms available and is the basic underlying structure in problems such as Ring-star Problem

34 Outline of the Presentation (1) Formulations, valid inequalities, algorithms and preliminary computational results for Minimum Connected Dominating Set Problem Cycle Problem Minimum Dominating Circuit Problem (2) Future Work

35 Minimum Connected Dominating Set Problem

36 Minimum Connected Dominating Set Problem Previous Work Approximation Algorithms: Solis-Oba [1998]: factor of 2. Exact Solution Algorithm: V 150 Lucena, Maculan and Simonetti [2010]: Maximum Leaf STP. Chen, Ljubić and Raghavan [2010]: Regenerator Location Problem (Minimum Connected DSP). The lower the graph density the harder the instance is!

37 A Formulation for the Minimum Connected DSP Basic idea: simultaneously exhibit W and a spanning tree for G W = (W, E(W )) {y i {0, 1} : i V }: for W vertices {x e R + : e E}: for spanning tree of G W = ( W, E(W ) ) edges y: also impose the dominance condition

38 A Formulation for the Minimum Connected DSP Additional notation: (1) Γ i V : set of neighbor vertices to i V (2) E(S) E: edges e = (i, j) E such that i, j S, S V

39 A Formulation for the Minimum Connected DSP Part One: formulation for a tree of G. x e = y i 1 e E i V x e y i, S V, j S e E(S) i S\{j} x e 0, e E y i {0, 1}, i V

40 A Formulation for the Minimum Connected DSP Part One: formulation for a tree of G. x e = y i 1 e E i V x e y i, S V, j S e E(S) i S\{j} x e 0, e E y i {0, 1}, i V Part Two: vertex i or a neighbor vertex to i is in W. y i + j Γ i y j 1, i V

41 A Formulation for the Minimum Connected DSP Polyhedral region R 0 : x e = y i 1 e E i V x e y i, S V, j S e E(S) i S\{j} y i + j Γ i y j 1, i V x e 0, e E 0 y i 1, i V Formulation: { min } i V y i : (x, y) R 0 (R E +, Z V )

42 Strengthening the Formulation

43 Strengthening the Formulation If S W : e E(S) x e i S\{j} y i, S V, j S, is dominated by e E(S) x e i S y i 1, S V

44 Particular Case

45 Particular Case For S = {i} Γ i, condition S W applies: y i + j Γ i y j 1, i V could be lifted into the strengthened GSEC y i + j Γ i y j e E(Γ i {i}) x e 1, i V i

46 Reinforced Linear Programming Relaxation Bounds V Density (%) Optimal R 0 R

47 Further Strengthening the Formulation

48 Further Strengthening the Formulation For S V such that ( ) ( ) S j S Γ j V and (V \ S) j V \S Γ j V, cutset inequality e E(S,V \S) x e 1 is valid

49 Further Strengthening the Formulation For S V such that ( ) ( ) S j S Γ j V and (V \ S) j V \S Γ j V, cutset inequality e E(S,V \S) x e 1 is valid

50 Further Strengthening the Formulation For S V such that ( ) ( ) S j S Γ j V and (V \ S) j V \S Γ j V, cutset inequality e E(S,V \S) x e 1 is valid

51 Further Strengthening the Formulation For S V such that ( ) ( ) S j S Γ j V and (V \ S) j V \S Γ j V, cutset inequality e E(S,V \S) x e 1 is valid

52 Further Strengthening the Formulation For S V such that ( ) ( ) S j S Γ j V and (V \ S) j V \S Γ j V, cutset inequality e E(S,V \S) x e 1 is valid

53 Further Strengthening the Formulation For S V such that ( ) ( ) S j S Γ j V and (V \ S) j V \S Γ j V, cutset inequality e E(S,V \S) x e 1 is valid (since S W 1 and (V \ S) W 1)

54 Particular Case

55 Particular Case The inequality applies to non intersecting vertex sets S i = {i} Γ i and S j = {j} Γ j and therefore e E(S,V \S) x e 1, for S i S and S j V \ S is valid.

56 Generalization

57 Generalization Let (V 1,..., V k ), for k 2, be a partition of V such that V l W for l {1,..., k}. The previous inequalities could be generalized into x(δ(v 1,..., V k ) k 1, where δ(v 1,..., V k ) is the set of edges with endpoints in different partition sets. (similar to the Steiner Partition Facets of Chopra and Rao [1994])

58 Preliminary Computational Experiments

59 Preliminary Computational Experiments Test Instances: same as in Lucena, Maculan and Simonetti [2010] New formulation: R 1 : LP relaxation bounds Stronger dominance condition inequalities used. (none of the additional valid inequalities were used) Reformulations for Maximum Leaf Spanning Tree Problem (Lucena, Maculan and Simonetti [2010]): DGR: directed graph reformulation (weaker but computationally cheap) STN: Steiner Problem in Graphs reformulation (stronger but computationally expensive) Processor: Intel Quadcore 2 GHz, 8 Gb RAM memory.

60 Linear Programming Relaxation Lower Bounds Lower Bounds OPT n density (%) R 1 DGR STN ? ? ? ?? ?? ?? ?

61 Branch and Cut Algorithm: Optimal Solutions Branch and Cut n den (%) t(s) BLB BUB gap (%) t DGR (s) OPT > >? 10 > >? 20 > >? 30 > >?

62 Summary of the Results LP relaxation bounds: Stronger than Directed Graph Reformulation (higher CPU times) Weaker than Steiner Problem in Graphs Reformulation (much lower CPU times) Branch and Cut algorithm compared to Directed Graph Reformulation s: Faster for higher density instances Slower for lower density instances

63 Bounds for a Directed Graph 1-MCDSP Reformulation Lower Bounds OPT n density (%) R 1d R 1 DGR STN ? ? ? ?? ?? ?? ?

64 Cycle Problem

65 Cycle Problem Given G = (V, E) and edge weights {c e : e E}: find elementary cycle of G with the smallest total weight

66 Literature for Cycle Problem

67 Literature for Cycle Problem (1) Circuit Polytope (Bauer [1994]). (2) Cycle Polytope for undirected graphs (Salazar González [1994] and Fischetti, Salazar González, Toth [2002]) and directed graphs (Balas and Oosten [2000] and Balas and Stephan [2009] ). (3) Computational results?

68 Cycle Problem (Undirected Graphs): Modeling Difficulties How to impose solution connectivity? (1) Generalized subtour elimination constraints? e E(S) x e i S\{j} y i, S V, j S (2) Cut-set inequalities: x(δ(s)) 2(y i + y j 1), S V, i S, j V \ S (Fischetti, Salazar González, Toth [2002])

69 Extended Formulation for the Cycle Problem How to use { e E(S) x e i S\{j} y i, S V, j S} to prevent illegal cycles?

70 Extended Formulation for the Cycle Problem How to use { e E(S) x e i S\{j} y i, S V, j S} to prevent illegal cycles? Cycle: path (restricted tree) + additional edge (almost like Held & Karp)

71 Extended Formulation for the Cycle Problem How to use { e E(S) x e i S\{j} y i, S V, j S} to prevent illegal cycles? {y i {0, 1} : i V }: for cycle vertices {x e {0, 1} : e E}: for tree edges (1 cycle edge is out) {z e R + : e E}: for the missing edge

72 Extended Formulation for the Cycle Problem Polyhedral region R 2 : (x e + z e) = 2y i, i V e δ(i) x e e E(S) i S\{j} y i 3 i V y i, j S, S V, S 3 z e = 1 e E x e + z e y k, e = (i, j) E, k {i, j} x e, z e 0, e E 0 y i 1, i V Formulation: min { e E c e(x e + z e ) : (x, y, z) R 2 (B E, B V, R E )}

73 Cycle Problem: Highlights So Far (1) Algorithmically easy to use formulation. (2) Inequalities derived from facet defining TSP inequalities (Salazar González [1994]). Very preliminary computational results: (a) Only generalized subtour elimination constraints and generalized 2-matching inequalities used. (b) No primal heuristics used. (c) TSPLIB instances with V 100: edge weights positive or negative with probabilities 0.95 e 0.05 All instances solved to optimality in fractions of a second.

74 Minimum Dominating Circuit Problem

75 Minimum Dominating Circuit Problem Hamiltonian cycle for G W = (W, E(W )) + dominance conditions Inequalities {y i + j Γ i y j e E(Γ i ) x e 1, i V } are not valid anymore. { y i + j Γ i y j 1, i V } dominates {y i + j Γ i y j 1, i V }.

76 Minimum Dominating Circuit Problem: Highlights So Far (1) Algorithmically easy to use formulation. (2) Valid Inequalities for Cycle Problem are valid. (3) Valid Inequalities for Min. 1-Connect. DSP are valid. Very preliminary computational results: (a) Only generalized subtour elimination constraints and generalized 2-matching inequalities used. (b) No primal heuristics used. (c) Same instances as Min. 1-Connect. DSP with V 100. All instances solved to optimality in 120 CPU secs of an Intel Quadcore 2 GHz, com 8 Gb RAM memory.

77 Future Work (1) Investigate the theoretical strength of proposed valid inequalities. (2) Design separation algorithms for valid inequalities. (3) Design and implement primal heuristics for the Cycle Problem and the Minimum Dominating Circuit Problem. (4) Investigate Prize-Collecting TSP, Ring-Star Problem,...