Hyperbolic set covering problems with competing ground-set elements

Transcription

1 g Hyperbolic set covering problems with competing ground-set elements Edoardo Amaldi, Sandro Bosio and Federico Malucelli Dipartimento di Elettronica e Informazione (DEI), Politecnico di Milano, Italy XI Workshop on Combinatorial Optimization, Aussois, 2007.

2 Outline g Problems definition The motivating application: Wireless Local Area Network design Hyperbolic integer programming formulation Complexity and Approximability results Linearizations and Lagrangean Relaxation Ongoing work and concluding remarks

3 Set Covering notation g

4 Set Covering notation g I I: a finite groundset

5 Set Covering notation g J I: a finite groundset J : a collection of subsets J = {I j I : j J}

6 Set Covering notation g i J i I: a finite groundset J : a collection of subsets J = {I j I : j J} J i J : subcollection of the subsets covering an element i I

7 Set Covering notation g S I: a finite groundset J : a collection of subsets J = {I j I : j J} J i J : cover S: subcollection of the subsets covering an element i I a subcollection indexed by S J such that j S I j = I

8 Set Covering problems g Classical Set Covering Problem (SCP): Given an instance (I, J ) and a cost c j R for each j J,

9 Set Covering problems g Classical Set Covering Problem (SCP): Given an instance (I, J ) and a cost c j R for each j J, find a cover S that minimizes the total cost c(s) = c j j S

10 Set Covering problems g Classical Set Covering Problem (SCP): Given an instance (I, J ) and a cost c j R for each j J, find a cover S that minimizes the total cost c(s) = c j j S Variants: Set Partitioning forbidden overlap Set Multicover required overlap

11 Set Covering problems g Classical Set Covering Problem (SCP): Given an instance (I, J ) and a cost c j R for each j J, find a cover S that minimizes the total cost c(s) = c j j S Variants: Set Partitioning forbidden overlap Set Multicover required overlap Also: Quadratic objective functions Maximum coverage...

12 Coverage share g Given a covering instance (I, J ), a cover S and an element i I coverage share: r(s, i) = + N i (S)

13 Coverage share g Given a covering instance (I, J ), a cover S and an element i I coverage share: r(s, i) = + N i (S) S

14 Coverage share g Given a covering instance (I, J ), a cover S and an element i I coverage share: r(s, i) = + N i (S) N i i r(s,i)= 3

15 Coverage share g Given a covering instance (I, J ), a cover S and an element i I coverage share: r(s, i) = + N i (S) i r(s,i)= 7 N i

16 Coverage share g Given a covering instance (I, J ), a cover S and an element i I coverage share: r(s, i) = + N i (S) i r(s,i)= 7 Fraction of resource received by i assuming fair allocation N i among the competing elements (neighbors of i)

17 Coverage share problems g Maximum Total Coverage Share Problem (TCSP): Given an instance (I, J ), find a cover S that maximizes f t (S) = i I + N i (S)

18 Coverage share problems g Maximum Total Coverage Share Problem (TCSP): Given an instance (I, J ), find a cover S that maximizes f t (S) = i I + N i (S) Maximum Minimum Coverage Share Problem (MCSP): Given an instance (I, J ), find a cover S that maximizes f m (S) = min i I + N i (S)

19 Coverage share problems g Maximum Total Coverage Share Problem (TCSP): Given an instance (I, J ), find a cover S that maximizes f t (S) = i I + N i (S) Maximum Minimum Coverage Share Problem (MCSP): Given an instance (I, J ), find a cover S that maximizes f m (S) = min i I + N i (S) Set covering problems with competing ground-set elements

20 Coverage share problems g

21 Coverage share problems g Instance

22 Coverage share problems g Instance SCP opt S = 2 f t (S) =.40 f m (S) =

23 Coverage share problems g Instance SCP opt TCSP opt S = 2 f t (S) =.40 f m (S) = S = 4 f t (S) = 3.64 f m (S) = 7

24 Coverage share problems g Instance SCP opt TCSP opt MCSP opt S = 2 f t (S) =.40 f m (S) = S = 4 f t (S) = 3.64 f m (S) = 7 S = 4 f t (S) = 3.42 f m (S) = 4

25 Coverage share problems g Instance SCP opt TCSP opt MCSP opt S = 2 f t (S) =.40 f m (S) = S = 4 f t (S) = 3.64 f m (S) = 7 S = 4 f t (S) = 3.42 f m (S) = 4 Privilege covers whose subsets have small cardinality and limited overlaps.

26 Wireless Local Area Network g IEEE 802. WLAN: a set of Access Points each able of serving a set of users

27 Wireless Local Area Network g IEEE 802. WLAN: a set of Access Points each able of serving a set of users WLANs are becoming pervasive in airports, trains and train stations, private companies, universities, hotels,...

28 Wireless Local Area Network g IEEE 802. WLAN: a set of Access Points each able of serving a set of users Medium Access Control (MAC) Protocol: A user can access the network if and only if no other user is interfering directly or indirectly

29 Wireless Local Area Network g IEEE 802. WLAN: a set of Access Points each able of serving a set of users Medium Access Control (MAC) Protocol: A user can access the network if and only if no other user is interfering directly or indirectly Assuming uniform peak traffic and fair access after collision, coverage share of element i fraction of time used by user i

30 Wireless Local Area Network g IEEE 802. WLAN: a set of Access Points each able of serving a set of users Due to protocol issues, increasing sizes of deployed WLANs and limited resources, and optimization models and methods can be very useful to support the planning decisions.

31 Previous and related work g WLAN design: Large-scale WLAN design (Hills 0,...) Max average signal quality in test points (Rodrigues, Mateus and Loureiro 00/0) Max coverage level (Kamenetsky and Unbehaun 02) Max capacity based on constraint satisfaction (Prommak et al. 02)... First hyperbolic model and heuristics (Amaldi, Capone, Cesana and Malucelli 04)

32 Integer programming formulations g max i I + N i (S) ( TCSP ) s.t. I j = I complete coverage j S S J select subcollection

33 Integer programming formulations g max i I + N i (S) ( TCSP ) s.t. I j = I complete coverage j S S J select subcollection Variables: x j = if subset I j is selected (0 otherwise) y ih = if elements i and h are neighbors (0 otherwise)

34 Integer programming formulations g max ( TCSP ) s.t. i I + j J i x j y ih x j y ih x j {0, } h N i y ih j J i J h x j y ih {0, } i I i I, h N i, j J i J h i I, h N i j J i I, h N i

35 Integer programming formulations g max ( TCSP ) s.t. i I + j J i x j h N i y ih 0- hyperbolic sum problem i I y ih x j y ih j J i J h x j x j {0, } y ih {0, } i I, h N i, j J i J h i I, h N i j J i I, h N i

36 Integer programming formulations g (MCSP) max s.t. min i I + j J i x j h N i y ih i I y ih x j y ih j J i J h x j i I, h N i, j J i J h i I, h N i x j {0, } j J y ih {0, } i I, h N i

37 Integer programming formulations g (MCSP) max s.t. min i I + j J i x j h N i y ih = i I + min max i I h N i y ih y ih x j y ih j J i J h x j x j {0, } y ih {0, } i I, h N i, j J i J h i I, h N i j J i I, h N i

38 Connection with Quadratic SCP g Quadratic Set Covering Problem (QSCP): Given (I, J ), Q = {q jl R : j, l J} (wlog symmetric with zero diagonal) and c = {c j R : j J}, find a cover S J that maximizes q(s) = 2 j S l S q jl + j S c j

39 Connection with Quadratic SCP g Quadratic Set Covering Problem (QSCP): Given (I, J ), Q = {q jl R : j, l J} (wlog symmetric with zero diagonal) and c = {c j R : j J}, find a cover S J that maximizes q(s) = 2 j S l S q jl + j S c j Choice: c j = i I j I j q jl = i I j I l ( I j I l I j I l ) (for j l)

40 Connection with Quadratic SCP g Quadratic Set Covering Problem (QSCP): Given (I, J ), Q = {q jl R : j, l J} (wlog symmetric with zero diagonal) and c = {c j R : j J}, find a cover S J that maximizes q(s) = 2 j S l S q jl + j S c j Choice: c j = i I j I j q jl = i I j I l ( I j I l I j I l ) (for j l) Then we can verify that: f t (S) = q(s) if at most two subsets overlap f t (S) q(s) otherwise (overhestimated penalty)

41 Previous and related work g Unconstrained 0- Hyperbolic Programming Single-ratio: NP-hard, poly with positive denominator (Hammer and Rudeanu 68) Multiple-ratio: NP-hard; tackled by SA, Tabu, and decomposing into independent polynomial single-ratio problems (Hansen, Poggi de Aragao and Ribeiro 90/9) Constrained 0- Hyperbolic Programming Single-ratio (Stancu-Minasian 97) Multiple-ratio: MILP convex reformulations (Tawarmalani, Ahmed and Sahinidis 02) Quadratic Set Covering Problem Various application oriented works (Bazaara 75, Boros, Hammer et al. 00,...) Generic: not 2 p( I ) -approximable for any polynomial p() (Escoffier and Convex: approximable within O(ln 2 ( I )) but not within ρ ln 2 ( I ) Hammer 05)

42 Complexity and Approximability (TCSP) g

43 Complexity and Approximability (TCSP) g Generic instances

44 Complexity and Approximability (TCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Not approximable within ρ ( I ) ε or ρ ( J ) ε for a given ρ > 0 and any ε > 0, unless NP = ZPP Reduction from Max Independent Set

45 Complexity and Approximability (TCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Not approximable within ρ ( I ) ε or ρ ( J ) ε for a given ρ > 0 and any ε > 0, unless NP = ZPP Reduction from Max Independent Set Euclidean 2D Instances

46 Complexity and Approximability (TCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Not approximable within ρ ( I ) ε or ρ ( J ) ε for a given ρ > 0 and any ε > 0, unless NP = ZPP Reduction from Max Independent Set Euclidean 2D Instances Strongly NP-hard (does not admit a FPTAS unless P = NP) Adapting and extending a reduction for Disc-Cover (Fowler et al. 8) Under a reasonable restriction, admits a PTAS Using the shifting lemma

47 Complexity and Approximability (TCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Not approximable within ρ ( I ) ε or ρ ( J ) ε for a given ρ > 0 and any ε > 0, unless NP = ZPP Reduction from Max Independent Set Euclidean 2D Instances Strongly NP-hard (does not admit a FPTAS unless P = NP) Adapting and extending a reduction for Disc-Cover (Fowler et al. 8) Under a reasonable restriction, admits a PTAS Using the shifting lemma Euclidean D Instances (or instances with CC covering matrix)

48 Complexity and Approximability (TCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Not approximable within ρ ( I ) ε or ρ ( J ) ε for a given ρ > 0 and any ε > 0, unless NP = ZPP Reduction from Max Independent Set Euclidean 2D Instances Strongly NP-hard (does not admit a FPTAS unless P = NP) Adapting and extending a reduction for Disc-Cover (Fowler et al. 8) Under a reasonable restriction, admits a PTAS Using the shifting lemma Euclidean D Instances (or instances with CC covering matrix) Polynomial-time solvable Longest path on an appropriate directed acyclic digraph

49 Complexity and Approximability (MCSP) g

50 Complexity and Approximability (MCSP) g Generic instances

51 Complexity and Approximability (MCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Polynomial-time solvable if I j = 2 Reduction to perfect b-matching

52 Complexity and Approximability (MCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Polynomial-time solvable if I j = 2 Reduction to perfect b-matching Euclidean 2D Instances

53 Complexity and Approximability (MCSP) g Generic instances Strongly NP-hard (Amaldi et al. 04) Polynomial-time solvable if I j = 2 Reduction to perfect b-matching Euclidean 2D Instances Strongly NP-hard (does not admit a FPTAS unless P = NP) Adapting the reduction for TCSP Not approximable within 3/2 unless P = NP Consequence of the above reduction Under a reasonable restriction, approximable within a factor 3 Tiling with hexagons

54 Linearization g For each ratio + is introduced a variable r i 0 and the quadratic constraint y ih h i r i = + r i + y ih h i h i r i y ih =

55 Linearization g For each ratio + is introduced a variable r i 0 and the quadratic constraint y ih h i r i = + r i + y ih h i h i r i y ih = r i y ih is standardly linearized with a variable z ih 0 and the constraints z ih u i y ih z ih l i y ih z ih r i + u i (y ih ) z ih r i + l i (y ih )

56 Linearization g For each ratio + is introduced a variable r i 0 and the quadratic constraint y ih h i r i = + r i + y ih h i h i r i y ih = r i y ih is standardly linearized with a variable z ih 0 and the constraints z ih u i y ih z ih l i y ih z ih r i + u i (y ih ) z ih r i + l i (y ih ) NB: r i is continuous and bounded, and y is binary

57 Tightening linearization of bilinear terms g z y Z = {(r, y, z) : z = r y, r [l, u], y {0, }} { } z ly, z r + u(y ), conv(z) = z uy, z r + l(y ) 0 l u r

58 Tightening linearization of bilinear terms g z y Z = {(r, y, z) : z = r y, r [l, u], y {0, }} { } z ly, z r + u(y ), conv(z) = z uy, z r + l(y ) 0 l u r Z = {(r, y, z) : z = 0, r [l, u ], y = 0} Z = {(r, y, z) : z = r, r [l, u ], y = }

59 Tightening linearization of bilinear terms g z y Z = {(r, y, z) : z = r y, r [l, u], y {0, }} { } z ly, z r + u(y ), conv(z) = z uy, z r + l(y ) 0 l u r Z = {(r, y, z) : z = 0, r [l 0, u 0 ], y = 0} Z = {(r, y, z) : z = r, r [l, u ], y = }

60 Tightening linearization of bilinear terms g z y Z = {(r, y, z) : z = r y, r [l, u], y {0, }} { } z ly, z r + u(y ), conv(z) = z uy, z r + l(y ) 0 l Z = {(r, y, z) : z = 0, r [l 0, u 0 ], y = 0} Z = {(r, y, z) : z = r, r [l, u ], y = } { z l y, z r + u 0 (y ), } 0 z u r y conv(z ) = z u y, z r + l 0 (y ) l 0 l u 0 u r

61 Lagrangean relaxation g By applying Lagragean relaxation to an appropriate reformulation the problem is decomposed into smaller and easier subproblems.

62 Lagrangean relaxation g By applying Lagragean relaxation to an appropriate reformulation the problem is decomposed into smaller and easier subproblems. Expanded formulation obtained by adding for each i I a vector χ i = {χ ij : j J i } of binary variables, one for each covering subset. Incidence vector of a local covering solution for i.

63 Lagrangean relaxation g By applying Lagragean relaxation to an appropriate reformulation the problem is decomposed into smaller and easier subproblems. Expanded formulation obtained by adding for each i I a vector χ i = {χ ij : j J i } of binary variables, one for each covering subset. Incidence vector of a local covering solution for i. J i i

64 Lagrangean relaxation g By applying Lagragean relaxation to an appropriate reformulation the problem is decomposed into smaller and easier subproblems. Expanded formulation obtained by adding for each i I a vector χ i = {χ ij : j J i } of binary variables, one for each covering subset. Incidence vector of a local covering solution for i. i

65 Lagrangean relaxation g Expanded formulation: max s.t. X i I X + P h N i y ih j J i x j i I ( y ih x j i I, h N i, j J i J h (2 X χ ij i I (3 j J i y ih χ ij i I, h N i, j J i J h (4 X y ih χ ij i I, h N i (5 j J i J h x j = χ ij i I, j J i (6 y ih = y hi i I, h N i : h > i (7 x j, χ ij, y ih {0,} Several possibilities, depending on which constraints are deleted/dualized

66 Lagrangean relaxation g Expanded formulation: max s.t. X i I X + P h N i y ih j J i x j i I ( y ih x j i I, h N i, j J i J h (2 X χ ij i I (3 j J i y ih χ ij i I, h N i, j J i J h (4 X y ih χ ij i I, h N i (5 j J i J h x j = χ ij i I, j J i (6 y ih = y hi i I, h N i : h > i (7 x j, χ ij, y ih {0,} Several possibilities, depending on which constraints are deleted/dualized

67 Lagrangean relaxation g Expanded formulation: max s.t. X i I X + P h N i y ih j J i x j i I ( y ih x j i I, h N i, j J i J h (2 X χ ij i I (3 j J i y ih χ ij i I, h N i, j J i J h (4 X y ih χ ij i I, h N i (5 j J i J h x j = χ ij i I, j J i (6 y ih = y hi i I, h N i : h > i (7 x j, χ ij, y ih {0,} Without (2), (6) and (7): one SCP and I independent hyperbolic subproblems

68 Lagrangean relaxation g Expanded formulation: max s.t. X i I X + P h N i y ih j J i x j i I ( y ih x j i I, h N i, j J i J h (2 X χ ij i I (3 j J i y ih χ ij i I, h N i, j J i J h (4 X y ih χ ij i I, h N i (5 j J i J h x j = χ ij i I, j J i (6 y ih = y hi i I, h N i : h > i (7 x j, χ ij, y ih {0,} LAG a : remove (2) and dualize (6) and (7) NP-hard hyperbolic subproblems

69 Lagrangean relaxation g Expanded formulation: max s.t. X i I X + P h N i y ih j J i x j i I ( y ih x j i I, h N i, j J i J h (2 X χ ij i I (3 j J i y ih χ ij i I, h N i, j J i J h (4 X y ih χ ij i I, h N i (5 j J i J h x j = χ ij i I, j J i (6 y ih = y hi i I, h N i : h > i (7 x j, χ ij, y ih {0,} LAG b : remove (5), (6) and dualize (2), (7) polynomial hyperbolic subproblems

70 Lagrangean subproblem for LAG b g Problem for a given element i: max s.t. + P + y h h N i X χ j j J i X h N i c h y h y h χ j χ j {0,} y h {0,} h N i, j J i J h j J i h N i

71 Lagrangean subproblem for LAG b g Problem for a given element i: max s.t. + P + y h h N i X χ j j J i X h N i c h y h y h χ j χ j {0,} y h {0,} h N i, j J i J h j J i h N i

72 Lagrangean subproblem for LAG b g Fix one variable χ l to (try all). This covers all h I l. max s.t. + P + y h h N i X χ j j J i X h N i c h y h y h χ j χ j {0,} y h {0,} h N i, j J i J h j J i h N i

73 Lagrangean subproblem for LAG b g Fix one variable χ l to (try all). This covers all h I l. max + P h N i y h + s.t. χ l = X h N i c h y h y h χ j χ j {0,} y h {0,} h N i, j J i J h j J i h N i

74 Lagrangean subproblem for LAG b g Fix one variable χ l to (try all). This covers all h I l. max + P h N i y h + s.t. χ l = y h = X h N i c h y h h I l y h χ j χ j {0,} y h {0,} h N i \ I l, j J i J h j J i h N i

75 Lagrangean subproblem for LAG b g Fix all other χ j to 0 (no o.f. contribution). max + P h N i y h + s.t. χ l = y h = X h N i c h y h h I l y h χ j χ j {0,} y h {0,} h N i \ I l, j J i J h j J i h N i

76 Lagrangean subproblem for LAG b g Fix all other χ j to 0 (no o.f. contribution). max + P h N i y h + s.t. y h = y h {0,} X h N i c h y h h I l h N i

77 Lagrangean subproblem for LAG b g Remains an unconstrained problem with hyperbolic+linear o.f.. max + P h N i y h + s.t. y h = y h {0,} X h N i c h y h h I l h N i

78 Lagrangean subproblem for LAG b g Remains an unconstrained problem with hyperbolic+linear o.f.. max + P h N i y h + s.t. y h = y h {0,} X h N i c h y h h I l h N i Since hyperbolic depends only on how many and not on which:

79 Lagrangean subproblem for LAG b g Remains an unconstrained problem with hyperbolic+linear o.f.. max + P h N i y h + s.t. y h = y h {0,} X h N i c h y h h I l h N i Since hyperbolic depends only on how many and not on which: ) sort c h coefficients in nonincreasing order. f

80 Lagrangean subproblem for LAG b g Remains an unconstrained problem with hyperbolic+linear o.f.. max + P h N i y h + s.t. y h = y h {0,} X h N i c h y h h I l h N i Since hyperbolic depends only on how many and not on which: ) sort c h coefficients in nonincreasing order. f 2) fix the first k variables to, the remaining to 0 (try all k).

81 Comparison - our department g Our Department

82 Comparison - our department g Our Department TCSP best solution Standard Linearization Improved Linearization LAG b J I den gap time gap time gap time (%) (sec) (%) (sec) (%) (sec) : time limit exceeded

83 Comparison - our department g Our Department TCSP best solution SCP optimal solution

84 Comparison - our department g Our Department TCSP best solution SCP optimal solution Tests with a WLAN simulator (ns-2): 2.58 Mb/s for SCP solution, 5.8 Mb/s for TCSP solution

85 Comparison - synthetic instances g Standard Linearization Improved Linearization LAG b J I den gap stdev time stdev gap stdev time stdev gap stdev time stdev GEOMETRIC INSTANCES (LOW DENSITY) GEOMETRIC INSTANCES (HIGH DENSITY) STANDARD SCP INSTANCES (CLASS SCP4*) STANDARD SCP INSTANCES (CLASS SCPE*) : the primal-dual gap is zero (proven optimality) : time limit exceeded for all instances of the class

86 Concluding remarks g This presentation:

87 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements

88 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions

89 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions Improved linearization and efficient Lagrangean relaxation

90 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions Improved linearization and efficient Lagrangean relaxation Ongoing work:

91 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions Improved linearization and efficient Lagrangean relaxation Ongoing work: Linearization by Dantzig-Wolfe decomposition

92 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions Improved linearization and efficient Lagrangean relaxation Ongoing work: Linearization by Dantzig-Wolfe decomposition Refined hyperbolic models, accounting for relevant features of WLANs

93 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions Improved linearization and efficient Lagrangean relaxation Ongoing work: Linearization by Dantzig-Wolfe decomposition Refined hyperbolic models, accounting for relevant features of WLANs Direct interference and node association

94 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions Improved linearization and efficient Lagrangean relaxation Ongoing work: Linearization by Dantzig-Wolfe decomposition Refined hyperbolic models, accounting for relevant features of WLANs Direct interference and node association Multiple frequencies and adaptive rate

95 Concluding remarks g This presentation: New interesting class: set covering problems with competing ground-set elements Complexity and approximability for generic and geometric versions Improved linearization and efficient Lagrangean relaxation Ongoing work: Linearization by Dantzig-Wolfe decomposition Refined hyperbolic models, accounting for relevant features of WLANs Direct interference and node association Multiple frequencies and adaptive rate