Bipartite Matching. Matching. Bipartite Matching. Maxflow Formulation

Transcription

1 Mching Inpu: undireced grph G = (V, E). Biprie Mching Inpu: undireced, biprie grph G = (, E).. Mching Ern Myr, Hrld äcke Biprie Mching Inpu: undireced, biprie grph G = (, E). Mflow Formulion Inpu: undireced, biprie grph G = ( {, }, E ). Direc ll edge from o. Add ource nd connec i o ll node on he lef. Add nd connec ll node on he righ o. All edge hve uni cpciy.. Mching Ern Myr, Hrld äcke. Mching Ern Myr, Hrld äcke 6

2 Proof M mching in G vlue of mflow in G Given mimum mching M of k. Conider flow f h end one uni long ech of k ph. f i flow nd h k. Proof M mching in G vlue of mflow in G e f be mflow in G of vlue k Inegrliy heorem k inegrl; we cn ume f i 0/. Conider M= e of edge from o wih f (e) =. Ech node in nd pricipe in mo one M = k, he flow mu ue le k middle edge. G G G G. Mching Ern Myr, Hrld äcke 7. Mching Ern Myr, Hrld äcke 8. Mching Bebll Eliminion Which flow lgorihm o ue? Generic ugmening ph: O(m vl(f )) = O(mn). Cpciy cling: O(m log C) = O(m ). Shore ugmening ph: O(mn ). em win loe remining gme i w i l i Al Phi NY Mon Aln Phildelphi New York Monrel For uni cpciy imple grph hore ugmening ph cn be implemened in ime O(m n). A grph i uni cpciy imple grph if every edge h cpciy node h eiher mo one leving edge or mo one enering edge Which em cn end he eon wih mo win? Monrel i elimined, ince even fer winning ll remining gme here re only 80 win. Bu lo Phildelphi i elimined. Why?. Mching Ern Myr, Hrld äcke 9. Bebll Eliminion Ern Myr, Hrld äcke 0

3 Bebll Eliminion Bebll Eliminion Flow nework for =. M i number of win Tem cn ill obin. Forml definiion of he problem: Given e S of em, nd one pecific em S. Tem h lredy won w gme. Tem ill h o ply em y, r y ime. Doe em ill hve chnce o finih wih he mo number of win. r r r r r r M w M w M w M w - Ide. Diribue he reul of remining gme in uch wy h no em ge oo mny win.. Bebll Eliminion Ern Myr, Hrld äcke. Bebll Eliminion Ern Myr, Hrld äcke Cerifice of Eliminion e T S be ube of em. Define w(t ) := w i, r (T ) := win of em in T i T remining gme mong em in T i,j T,i<j w(t )+r (T ) If T > M hen one of he em in T will hve more hn M win in he end. A em h cn win mo M gme i herefore elimined. r ij Theorem A em i elimined if nd only if he flow nework for doe no llow flow of vlue ij S\{},i<j r ij. Proof ( ) Conider he mincu A in he flow nework. e T be he e of em-node in A. If for node -y no boh em-node nd y re in T, hen -y A ow. he cu would cu n infinie cpciy edge. We don find flow h ure ll ource edge: r (S \ {}) > cp(a, V \ A) r ij + (M w i ) i<j: i T j T i T r (S \ {}) r (T ) + T M w(t ). Bebll Eliminion Ern Myr, Hrld äcke Thi give M < (w(t ) + r (T ))/ T, i.e., i elimined.

4 Bebll Eliminion Projec Selecion Proof ( ) Suppoe we hve flow h ure ll ource edge. We cn ume h hi flow i inegrl. For every piring -y i define how mny gme em nd em y hould win. The flow leving he em-node cn be inerpreed he ddiionl number of win h em will obin. Thi i le hn M w becue of cpciy conrin. Hence, we found e of reul for he remining gme, uch h no em obin more hn M win in ol. Hence, em i no elimined. Projec elecion problem: Se P of poible projec. Projec v h n ocied profi (cn be poiive or negive). Some projec hve requiremen (king coure EA require coure EA). Dependencie re modelled in grph. Edge (u, v) men cn do projec u wihou lo doing projec v. A ube A of projec i feible if he prerequiie of every projec in A lo belong o A. Gol: Find feible e of projec h mimie he profi.. Bebll Eliminion Ern Myr, Hrld äcke. Projec Selecion Ern Myr, Hrld äcke 6 Projec Selecion The prerequiie grph: {,, } i feible ube. {, } i infeible. Projec Selecion Mincu formulion: Edge in he prerequiie grph ge infinie cpciy. Add edge (, v) wih cpciy for node v wih poiive profi. Cree edge (v, ) wih cpciy for node v wih negive profi. prerequiie grph u p u p v w p w p p. Projec Selecion Ern Myr, Hrld äcke 7. Projec Selecion Ern Myr, Hrld äcke 8

5 Theorem A i mincu if A \ {} i he opiml e of projec. Proof. A i feible becue of cpciy infiniy edge. cp(a, V \ A) = + ( ) v Ā: >0 = v: >0 v A v A: Ā: <0 v >0 For he formul we define p := 0. prerequiie grph u The ep follow by dding v A: >0 v A: >0 = 0. p u v w p p w Noe h minimiing he cpciy of he cu (A, V \ A) correpond o mimiing profi of projec in A. p p