Graph Theory and Optimization Integer Linear Programming

Transcription

1 Graph Theory ad Optimizatio Iteger Liear Programmig Nicolas Nisse Iria, Frace Uiv. Nice Sophia Atipolis, CNRS, I3S, UMR 7271, Sophia Atipolis, Frace September 2015 N. Nisse Graph Theory ad applicatios 1/1

2 Outlie N. Nisse Graph Theory ad applicatios 2/1

3 Liear Programme (remider) Liear programmes ca be writte uder the stadard form: Maximize Subject to: c j x j a ij x j b i for all 1 i m x j 0 for all 1 j. the problem is a maximizatio; all costraits are iequalities (ad ot equatios); all variables x 1,,x are o-egative. Liear Programme (Real variables) ca be solved i polyomial-time i the umber of variables ad costraits (e.g., ellipsoid method) N. Nisse Graph Theory ad applicatios 3/1

4 Liear Programme (remider) Liear programmes ca be writte uder the stadard form: Maximize Subject to: c j x j a ij x j b i for all 1 i m x j 0 for all 1 j. the problem is a maximizatio; all costraits are iequalities (ad ot equatios); all variables x 1,,x are o-egative. Liear Programme (Real variables) ca be solved i polyomial-time i the umber of variables ad costraits (e.g., ellipsoid method) N. Nisse Graph Theory ad applicatios 3/1

5 Iteger Liear programmes: Iteger Liear Programme Maximize Subject to: c j x j a ij x j b i for all 1 i m x j N for all 1 j. the problem is a maximizatio; all costraits are iequalities (ad ot equatios); all variables x 1,,x are Itegers. Iteger Liear Programme is NP-complete i geeral! N. Nisse Graph Theory ad applicatios 4/1

6 Outlie N. Nisse Graph Theory ad applicatios 5/1

7 Kapsack Problem (Weakly NP-hard) Data: a kapsack with maximum weight 15 Kg 12 objects with a weight w i a value v i Objective: which objects should be chose to maximize the value carried while ot exceedig 15 Kg? N. Nisse Graph Theory ad applicatios 6/1

8 Kapsack Problem (Weakly NP-hard) Data: a kapsack with maximum weight 15 Kg 12 objects with a weight w i a value v i Objective: which objects should be chose to maximize the value carried while ot exceedig 15 Kg? max subject to v i x i 1 i 12 w i x i 15 1 i 12 x i {0,1} N. Nisse Graph Theory ad applicatios 6/1

9 Miimum Vertex Cover (NP-hard) Let G = (V,E) be a graph Vertex Cover: set K V such that e E, e K /0 set of vertices that touch" every edge N. Nisse Graph Theory ad applicatios 7/1

10 Miimum Vertex Cover (NP-hard) Let G = (V,E) be a graph Vertex Cover: set K V such that e E, e K /0 set of vertices that touch" every edge Solutio: K V variables x v, for each v V x v = 1 if v K, x v = 0 otherwise. N. Nisse Graph Theory ad applicatios 7/1

11 Miimum Vertex Cover (NP-hard) Let G = (V,E) be a graph Vertex Cover: set K V such that e E, e K /0 set of vertices that touch" every edge Solutio: K V Objective fuctio: miimize K variables x v, for each v V x v = 1 if v K, x v = 0 otherwise. miimize x v v V N. Nisse Graph Theory ad applicatios 7/1

12 Miimum Vertex Cover (NP-hard) Let G = (V,E) be a graph Vertex Cover: set K V such that e E, e K /0 set of vertices that touch" every edge Solutio: K V variables x v, for each v V x v = 1 if v K, x v = 0 otherwise. Objective fuctio: miimize K miimize x v v V Costrait: {u,v} E, u K or v K x u + x v 1 N. Nisse Graph Theory ad applicatios 7/1

13 Miimum Vertex Cover (NP-hard) Let G = (V,E) be a graph Vertex Cover: set K V such that e E, e K /0 set of vertices that touch" every edge Solutio: K V variables x v, for each v V x v = 1 if v K, x v = 0 otherwise. Objective fuctio: miimize K miimize x v v V Costrait: {u,v} E, u K or v K x u + x v 1 Miimize x v v V Subject to: x v + x u 1 for all {u,v} E x v {0,1} for all v V N. Nisse Graph Theory ad applicatios 7/1

14 Vertex Colorig (NP-hard) Let G = (V,E) be a graph k-proper colorig: c : V {1,,k} s.t. c(u) c(v) for all {u,v} E. color the vertices s ( k colors) s.t. adjacet vertices receive colors N. Nisse Graph Theory ad applicatios 8/1

15 Vertex Colorig (NP-hard) Let G = (V,E) be a graph k-proper colorig: c : V {1,,k} s.t. c(u) c(v) for all {u,v} E. color the vertices s ( k colors) s.t. adjacet vertices receive colors Solutio: c : V {1,,} variables y j, is color j {1,,} used? variable c j v for color j ad vertex v: c j v = 1 if v colored j, c j v = 0 otherwise N. Nisse Graph Theory ad applicatios 8/1

16 Vertex Colorig (NP-hard) Let G = (V,E) be a graph k-proper colorig: c : V {1,,k} s.t. c(u) c(v) for all {u,v} E. color the vertices s ( k colors) s.t. adjacet vertices receive colors Solutio: c : V {1,,} variables y j, is color j {1,,} used? variable cv j for color j ad vertex v: cv j = 1 if v colored j, cv j = 0 otherwise Objective fuctio: miimize # of used colors miimize y j 1 j N. Nisse Graph Theory ad applicatios 8/1

17 Vertex Colorig (NP-hard) Let G = (V,E) be a graph k-proper colorig: c : V {1,,k} s.t. c(u) c(v) for all {u,v} E. color the vertices s ( k colors) s.t. adjacet vertices receive colors Solutio: c : V {1,,} variables y j, is color j {1,,} used? variable cv j for color j ad vertex v: cv j = 1 if v colored j, cv j = 0 otherwise Objective fuctio: miimize # of used colors miimize y j 1 j Costraits: each vertex v has 1 color 1 j cv j = 1 eds of each edge {u,v} E have colors: cv j + cu j 1 for all j {1,,} color j used if 1 vertex colored with j cv j y j for all v V N. Nisse Graph Theory ad applicatios 8/1

18 Vertex Colorig (NP-hard) Let G = (V,E) be a graph k-proper colorig: c : V {1,,k} s.t. c(u) c(v) for all {u,v} E. color the vertices s ( k colors) s.t. adjacet vertices receive colors Miimize Subject to: y j 1 j 1 j c j v = 1 for all v V c j v + c j u 1 for all j {1,,},{u,v} E cv j y j for all j {1,,},v V y j {0,1} for all j {1,,} cv j {0,1} for all j {1,,},v V N. Nisse Graph Theory ad applicatios 8/1

19 Outlie N. Nisse Graph Theory ad applicatios 9/1

20 Iteger Programme vs. Liear Programme Iteger Liear programme (ILP): Max. s.t.: c j x j a ij x j b i 1 i m x j N 1 j. NP-hard i geeral N. Nisse Graph Theory ad applicatios 10/1

21 Iteger Programme vs. Liear Programme Iteger Liear programme (ILP): Fractioal Relaxatio: Liear Programme Max. s.t.: c j x j a ij x j b i 1 i m x j N 1 j. Max. s.t.: c j x j a ij x j b i 1 i m x j 0 1 j. NP-hard i geeral Polyomial-time solvable N. Nisse Graph Theory ad applicatios 10/1

22 Iteger Programme vs. Liear Programme Iteger Liear programme (ILP): Fractioal Relaxatio: Liear Programme Max. s.t.: c j x j a ij x j b i 1 i m x j N 1 j. Max. s.t.: c j x j a ij x j b i 1 i m x j 0 1 j. NP-hard i geeral Polyomial-time solvable What is the differece betwee Optimal solutios of LP ad of ILP? N. Nisse Graph Theory ad applicatios 10/1

23 Iteger Programme vs. Liear Programme Iteger Liear programme (ILP): Fractioal Relaxatio: Liear Programme Max. s.t.: c j x j a ij x j b i 1 i m x j N 1 j. Max. s.t.: c j x j a ij x j b i 1 i m x j 0 1 j. NP-hard i geeral Polyomial-time solvable What is the differece betwee Optimal solutios of LP ad of ILP? OPT(LP) OPT (ILP) (for a maximizatio problem) OPT (LP) OPT (ILP) (for a miimizatio problem) If OPT (LP) is closed" to OPT(ILP), the solvig the Fractioal Relaxatio (i polyomial-time) gives a good boud for the ILP N. Nisse Graph Theory ad applicatios 10/1

24 Fractioal Relaxatio of Vertex Colorig Iteger Liear programme (ILP): Miimize Subject to: y j 1 j cv j = 1 1 j cv j + cu j 1 cv j y j y j {0,1} cv j {0,1} Fractioal Relaxatio (LP): Miimize y j 1 j Subject to: cv j = 1 1 j cv j + cu j 1 cv j y j y j 0 c j v 0 y red = y blue = y gree = 1 ca red = cb blue = cc gree = 1 OPT(ILP) = c y c = 3 y red = y blue = 1/2, y gree = 0 ca red = cb red = cc red = 1/2 ca blue = cb blue = cc blue = 1/2 OPT(LP) = c y c = 1 N. Nisse Graph Theory ad applicatios 11/1

25 Fractioal Relaxatio of Vertex Colorig Iteger Liear programme (ILP): Miimize Subject to: y j 1 j cv j = 1 1 j cv j + cu j 1 cv j y j y j {0,1} cv j {0,1} Fractioal Relaxatio (LP): Miimize y j 1 j Subject to: cv j = 1 1 j cv j + cu j 1 cv j y j y j 0 c j v 0 b b a a c c y red = y blue = y gree = 1 ca red = cb blue = cc gree = 1 OPT(ILP) = c y c = 3 y red = y blue = 1/2, y gree = 0 ca red = cb red = cc red = 1/2 ca blue = cb blue = cc blue = 1/2 OPT(LP) = c y c = 1 N. Nisse Graph Theory ad applicatios 11/1

26 Fractioal Relaxatio of Kapsac Iteger Liear programme (ILP): max v i x i 1 i subject to w i x i W 1 i x i {0,1} Fractioal Relaxatio (LP): max v i x i 1 i subject to w i x i W 1 i x i 0 Example: Sac: W = Objects: oe object (O 1 ) of weight + 0,1 ad value 1 objects (O 2,,O ) of weight 1 ad value 1/ x 1 = 0,x 2 = = x = 1 OPT(ILP) = c v i x i = ( 1)/ x 1 = +0,1,x 2 = = x = 0 OPT(LP) = c v i x i = 2 +0,1 the ratio betwee the LP optimal solutio ad the Itegral opt. solutio may be arbitrary large N. Nisse Graph Theory ad applicatios 12/1

27 Fractioal Relaxatio of Kapsac Iteger Liear programme (ILP): max v i x i 1 i subject to w i x i W 1 i x i {0,1} Fractioal Relaxatio (LP): max v i x i 1 i subject to w i x i W 1 i x i 0 Example: Sac: W = Objects: oe object (O 1 ) of weight + 0,1 ad value 1 objects (O 2,,O ) of weight 1 ad value 1/ x 1 = 0,x 2 = = x = 1 OPT(ILP) = c v i x i = ( 1)/ x 1 = +0,1,x 2 = = x = 0 OPT(LP) = c v i x i = 2 +0,1 the ratio betwee the LP optimal solutio ad the Itegral opt. solutio may be arbitrary large N. Nisse Graph Theory ad applicatios 12/1

28 Fractioal Relaxatio of Kapsac Iteger Liear programme (ILP): max v i x i 1 i subject to w i x i W 1 i x i {0,1} Fractioal Relaxatio (LP): max v i x i 1 i subject to w i x i W 1 i x i 0 Example: Sac: W = Objects: oe object (O 1 ) of weight + 0,1 ad value 1 objects (O 2,,O ) of weight 1 ad value 1/ x 1 = 0,x 2 = = x = 1 OPT(ILP) = c v i x i = ( 1)/ x 1 = +0,1,x 2 = = x = 0 OPT(LP) = c v i x i = 2 +0,1 the ratio betwee the LP optimal solutio ad the Itegral opt. solutio may be arbitrary large N. Nisse Graph Theory ad applicatios 12/1

29 Outlie N. Nisse Graph Theory ad applicatios 13/1

30 No itegrality gap Iteger Liear programme (ILP): Fractioal Relaxatio: Liear Programme Max. s.t.: c j x j a ij x j b i 1 i m x j N 1 j. Max. s.t.: c j x j a ij x j b i 1 i m x j 0 1 j. NP-hard i geeral Polyomial-time solvable I some cases: OPT (ILP) = OPT (LP). there exists a itegral solutio with value OPT (LP). I such case, Polyomial-time solvable: solve the Fractioal Relaxatio N. Nisse Graph Theory ad applicatios 14/1

31 Iteger Programme Example: Shortest path D = (V,A) be a digraph with legth l : A R +, ad s,t V. Problem: Compute a shortest directed path from s to t. N. Nisse Graph Theory ad applicatios 15/1

32 Iteger Programme Example: Shortest path D = (V,A) be a digraph with legth l : A R +, ad s,t V. Problem: Compute a shortest directed path from s to t. Solutio: A path P from s to t variables x a for each a A x a = 1 if a A(P), x a = 0 otherwise. N. Nisse Graph Theory ad applicatios 15/1

33 Iteger Programme Example: Shortest path D = (V,A) be a digraph with legth l : A R +, ad s,t V. Problem: Compute a shortest directed path from s to t. Solutio: A path P from s to t variables x a for each a A x a = 1 if a A(P), x a = 0 otherwise. Miimize Subject to: l(a)x a a A u N + (s) u N (t) u N + (v) x(su) = 1 x(tu) = 1 x(uv) = x(vu) for all v V \ {s,t} u N (v) x(a) {0,1} for all a A N. Nisse Graph Theory ad applicatios 15/1

34 Iteger Programme Example: Shortest path Miimize Subject to: l(a)x a a A u N + (s) u N (t) u N + (v) x(su) = 1 x(tu) = 1 x(uv) = x(vu) for all v V \ {s,t} u N (v) x(a) 0 for all a A 0,25 0,25 0,25 0,25 Example: all edges have legth 1 0,5 0,5 s 0,5 0,5 0,5 0,5 t s t a fractioal solutio OPT(LP)=4 a itegral solutio OPT(ILP)=4 Exercise: Prove that this LP always admits a itegral optimal solutio N. Nisse Graph Theory ad applicatios 15/1

35 Iteger Programme Example: Maximum Matchig G = (V,E) be a graph Problem: Compute a maximum matchig Solutio: a set M E of pairwise disjoit edges variables x e for each e E x e = 1 if e M, x e = 0 otherwise. Maximize Subject to: x e e E e E,v e x e 1 for all v V x e {0,1} for all e E Exercise: Prove that the fractioal relaxatio of this ILP always admits a itegral optimal solutio N. Nisse Graph Theory ad applicatios 16/1

36 Totally uimodular matrices uimodular matrix: square matrix with determiat +1 or 1 totally uimodular matrix: every square o-sigular submatrix is uimodular Iteger Liear programme (ILP): Fractioal Relaxatio: Liear Programme Max. s.t.: c j x j a ij x j b i 1 i m x j N 1 j. Max. s.t.: c j x j a ij x j b i 1 i m x j 0 1 j. NP-hard i geeral Polyomial-time solvable Theorem [Hoffma,Kruskal, 1956] If the matrix A = [a ij ] is totally uimodular the every basic feasible solutio (the corer" of the polytope) is itegral exist itegral optimal solutio of the LP OPT (ILP) ca be computed by solvig the Fractioal relaxatio N. Nisse Graph Theory ad applicatios 17/1

37 Outlie N. Nisse Graph Theory ad applicatios 18/1

38 Iteger Programme Example: Miimum Spaig Tree G = (V,E) be a graph with weight w : E R +, ad s,t V. Problem: Compute a miimum spaig tree Solutio: A spaig tree T variables x e for each e E x E = 1 if e E(T ), x e = 0 otherwise. Miimize Subject to: w(e)x e e E x e 1 for all S V e={u,v} E,u S,v / S x e {0,1} for all e E Remark: The umber of costraits is expoetial N. Nisse Graph Theory ad applicatios 19/1

39 Optical Networks (WDM) Optical etwork: optical fiber coectig e.g. routers Wavelegth-divisio Multiplexig (WDM): techology which multiplexes a umber of optical carrier sigals oto a sigle optical fiber by usig differet wavelegths (i.e., colors) of laser light [Wikipedia] : differet sigals o the same lik must have differet wavelegths (colors) N. Nisse Graph Theory ad applicatios 20/1

40 Optical Networks (WDM) Optical etwork: optical fiber coectig e.g. routers Wavelegth-divisio Multiplexig (WDM): techology which multiplexes a umber of optical carrier sigals oto a sigle optical fiber by usig differet wavelegths (i.e., colors) of laser light [Wikipedia] : differet sigals o the same lik must have differet wavelegths (colors) N. Nisse Graph Theory ad applicatios 20/1

41 Optical Networks (WDM) RWA problem RWA: Routig ad Waveletgh Assigmet Problem Give a graph G = (V,E) with capacity o liks, ad a traffic-demad matrix T, where T [u,v] is the amout of traffic that must trasit from u to v, for ay u,v V. Fid a set of paths ad oe wavelegth assigmet for each path such that: all demads are routed capacity of each lik caot be exceeded total umber of wavelegth is as small as possible demad matrix: a b c d e f g a b c d e f g a 1 f 2 3 b 2 g e c 2 d N. Nisse Graph Theory ad applicatios 21/1

42 Optical Networks (WDM) RWA problem RWA: Routig ad Waveletgh Assigmet Problem Give a graph G = (V,E) with capacity o liks, ad a traffic-demad matrix T, where T [u,v] is the amout of traffic that must trasit from u to v, for ay u,v V. Fid a set of paths ad oe wavelegth assigmet for each path such that: all demads are routed capacity of each lik caot be exceeded total umber of wavelegth is as small as possible demad matrix: a b c d e f g a b c d e f g a 1 2 f 1 3 b g e c d N. Nisse Graph Theory ad applicatios 21/1

43 Optical Networks (WDM) RWA problem Let us simplify the problem cosider oly Wavelegth assigmet WA: Waveletgh Assigmet Problem Give a graph G = (V,E) with capacity o liks, ad a set of paths Give Oe color to each path s.t. o two paths with the same color cross a same lik Miimize the umber of colors a b c g f d e N. Nisse Graph Theory ad applicatios 22/1

44 Optical Networks (WDM) Let us simplify the problem WA: Waveletgh Assigmet Problem RWA problem cosider oly Wavelegth assigmet Give a graph G = (V,E) with capacity o liks, ad a set of paths Give Oe color to each path s.t. o two paths with the same color cross a same lik Miimize the umber of colors a b c ac af bf g ad f d dg e gb It is the problem of PROPER COLORING i graphs!! the simplified" problem is already NP-complete :( N. Nisse Graph Theory ad applicatios 22/1

45 Optical Networks (WDM) Let us simplify the problem WA: Waveletgh Assigmet Problem RWA problem cosider oly Wavelegth assigmet Give a graph G = (V,E) with capacity o liks, ad a set of paths Give Oe color to each path s.t. o two paths with the same color cross a same lik Miimize the umber of colors a b c ac af bf g ad f d dg e gb It is the problem of PROPER COLORING i graphs!! the simplified" problem is already NP-complete :( N. Nisse Graph Theory ad applicatios 22/1

46 Optical Networks (WDM) Let us simplify the problem WA: Waveletgh Assigmet Problem RWA problem cosider oly Wavelegth assigmet Give a graph G = (V,E) with capacity o liks, ad a set of paths Give Oe color to each path s.t. o two paths with the same color cross a same lik Miimize the umber of colors a b c ac af bf g ad f d dg e gb Exercise: Write a (Iteger) Liear Programme that solves the RWA problem N. Nisse Graph Theory ad applicatios 22/1

47 Summary: To be remembered ILP allow to model may problems there may be a huge itegrality gap (betwee OPT (LP) ad OPT (ILP)). if o itegrality gap (e.g., totally uimodular matrices) Fractioal Relaxatio gives Optimal Itegral Solutio N. Nisse Graph Theory ad applicatios 23/1