The Maximum Clique Problem (MCP)

Transcript

1 The Maxmum Clque Problem (MCP) You are gve: A udrected graph G = (V,E), where - V = {,.,} - E V x V ad are asked to Fd the largest complete subgraph (clque) of G The problem s kow to be NP-hard, ad so s problem of determg just the sze of the maxmum clque. Pardalos ad Xue (994) provde a revew of the MCP wth 260 refereces.

2 The Maxmum Clque Problem (MCP) Affrotado l problema MCP term d rete eurale: Trasformare MCP da problema dscreto a problema cotuo Nell esempo del TSP co l modello d Hopfeld, o è detto che c sa l percorso verso (potremmo otteere ad esempo ua matrce che o ha sgfcato); questo uovo problema MCP, la bdrezoaltà è d obblgo.

3 Some Notato Gve a arbtrary graph G = (V,E) wth odes: If C V, x c wll deote ts characterstc vector whch s defed as x c = 0, C, f otherwse C S s the stadard smplex R : S = x R : = x = ad x 0, A=(a j ) s the adjacecy matrx of G: a j =, 0, f v ~ v otherwse j

4 Ifeasble Maxma Motzk-Straus S cosder la fuzoe cotua: ( ) f x xax a x x = = = j= j j dove x ' è l vettore trasposto e A è la matrce d adaceza. Lagragao del grafo: ( ) f x = x x, j E j esempo: (,, ) f x x x = xx + xx

5 Cotuous Formulato of MAX-CLIQUE Il pote che crea Motzk-Straus è udrezoale; solo se l vettore resttuto è ella forma d vettore caratterstco allora c è bdrezoaltà. Nell esempo vsto c soo due massm global : T x =,,0 x =,0, T S dmostra che soo massm global ache tutt put del segmeto x -x T α α ovvero tutt put,, α [ 0,] ;o essedo vettor caratterstc (soluzo spure) o è possble estrarre la clque massma. La soluzoe cosste el sommare alla dagoale prcpale d A 2 A A I 2 T = + f ( x) = x Ax ( ) T f x x = A+ I x 2

6 Ifeasble Maxma Motzk-Straus Teorema Dato C V e x c vettore caratterstco allora: - C è ua clque massma d G x c è u massmo globale d f - C è ua clque massmale d G x c è u massmo locale d f S S -tutt massm local soo strett e soo vettor caratterstc

7 Evolutoary Games Developed evolutoary game theory to model the evoluto of behavor amal coflcts. Assumptos A large populato of dvduals belogg to the same speces whch compete for a partcular lmted resource Ths kd of coflct s modeled as a game, the players beg pars of radomly selected populato members Players do ot behave ratoally but act accordg to a pre-programmed behavoral patter, or pure strategy Reproducto s assumed to be asexual Utlty s measured terms of Darwa ftess, or reproductve success

8 Notatos { } J =, L, s the set of pure strateges ( ) x t t s the proporto of populato members playg strategy at tme The state of populato at a gve stat s the vector x= ( x, L, x ) Gve a populato state, the support of, deoted, s defed as the set of postve compoets of x x σ ( x) x,.e., ( ) { : 0} σ x = J x >

9 Payoffs ( ) A= a j Let be the payoff (or ftess) matrx. a j represets the payoff of a dvdual playg strategy playg strategy, j J. j ( ) agast a oppoet x If the populato s state, the expected payoff eart by a strategst s: π ( x) = a x = ( Ax) j j j= whle the mea payoff over the etre populato s: π ( ) π ( ) x = x x = xax =

10 Replcator Equatos Developed evolutoary game theory to model the evoluto of behavor amal coflcts (Hofbauer & Sgmud, 998; Webull, 995). ( ) W = w j Let be a o-egatve real-valued matrx, ad let Cotuous-tme verso: π () t w x () t = j j j= d x t x t t x j t j t dt j= () = () π () () π () Dscrete-tme verso: x ( t ) + = j = ( ) π ( ) x () t π () t x t t j j

11 Replcator Equatos & Fudametal Theorem of Selecto S s varat uder both dyamcs, ad they have the same statoary pots. W = W Theorem: If, the the fucto F( x) = xwx s strctly creasg alog ay o-costat trajectory of both cotuous-tme ad dscrete-tme replcator dyamcs

12 Mappg MCP s oto Relaxato Nets To (approxmately) solve a MCP by relaxato, smply costruct a et havg uts, ad a { 0,} -weght matrx gve by where A s the adjacecy matrx of G. W = A+ I 2 Example:

13 Mappg MCP s oto Relaxato Nets The system startg from u(0) wll maxmze the Motzk-Straus fucto ad wll coverge to a fxed pot u * whch correspods to a (local) maxmum of. f The value k = 2f u ( ) ca be regarded as a approxmato of the maxmum clque sze. Co Q measure s msura la qualtà Q = f f ave ave f α RE f dove è l terme d cofroto rspetto alla meda, è la replcator ave α equato e è l valore ottmale. Quado l rsultato è buoo. Q f RE

14 Expermetal Setup Expermets were coducted over radom graphs havg: sze: = 0, 25, 50, 75, 00 desty: = 0.0, 0.25, 0.50, 0.75, 0.90 Comparso wth Bro-Kerbosch (BK) clque-fdg algorthm (974). For each par (, ) 00 graphs geerated radomly wth sze ad desty. The case = 00 ad = 0.90 was excluded due to the hgh cost of BK algorthm. Total umber of graphs = δ δ δ δ d (54) 0.99 (36) 0.99 (53) 0.97 (59) 0.92 (82) (54) 0.99 (64) 0.99 (84).00 (98) 0.97 (2) (56) 0.99 (8) 0.99 (53) 0.96 (60) 0.90 (87) (99).00 (75).00 (268).00 (284).00 (369) (9).00 (224).00 (367) 0.99 (53) ---- Values of Q-measure for varous szes ad destes