CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i, e j M = e i, e j re vertex isjoint. Definition A mximl mthing is mthing tht nnot e improve. By vertex isjoint, we men tht no two eges in the set my e inient with the sme vertex. Figure 1 shows two possile mthings in simple grph. A mthing is improve when one or more eges is e to it. Definition 3 A mximum mthing is mthing of mximum rinlity (over ll possile mthings in grph G); ν(g) enotes the rinlity of mximum mthing in G. Figure : mximl ut not mximum mthing; mximum mthing. 1
A mximl mthing is not neessrily mximum mthing. Figure shows mximl mthing (tht is not mximum mthing) n mximum mthing. All mximum mthings re mximl mthings, ut not ll mximl mthings re mximum mthings. In generl, it is hrer to fin mximum mthing thn to fin mximl mthing. Definition 4 Given mthing M, n M-expose vertex is vertex not inient with ny ege in M; n M-overe vertex is vertex inient with some ege in M. Given mximum mthing M the numer of M-overe verties = ν(g); the numer of M-expose verties = V ν(g). The numer of M-expose verties is lso lle the efiieny, or ef(g). In other wors, ef(g) is the minimum numer of noes tht will e expose in ny mthing. Definition 5 Given grph G = V, E, perfet mthing is mthing with efiieny ef(g) = V ν(g) = 0. Interesting prolems onerning mthings inlue: oes G hve lrge mthing? oes G hve perfet mthing? wht is mximum mthing of G? et. Up to now we hve onsiere only unweighte grphs. If we onsier weighte grphs n sum the weights of eges inlue in mthings, we might lso try to fin the minimum-weight perfet mthing of prtiulr grph (if one exists). Tehniques for Fining Mximum Mthing Definition 6 Given mthing M in grph G, pth P ompose of eges tht lterntely elong to n o not elong to M is lle n M-lternting pth. Definition 7 An M-lternting pth P is n M-ugmenting pth if the first n lst verties re M- expose. Figure 3 shows simple grph with mthing M = {e }. The pth from to to to is n M- lternting pth, euse the eges in the pth lterntely elong to the mthing (e / M, e M, e / M). The sme pth is lso n M-ugmenting pth euse its enpoints, n re M-expose; tht is, they re not inient with ny ege in M. Figure 3: mthing M = {e }. Theorem 1 A mthing M in grph G = V, E is mximum if n only if there is no M-ugmenting pth. In the textook, Theorem 1 is lle the Augmenting Pth Theorem of Mthings [CO98]. For the proof of Theorem 1, we nee the set opertion symmetri ifferene. We efine the symmetri ifferene etween two sets S n T s S T = { elements S or T ut not oth } (this is like the exlusive or opertion ).
even yle M-ugmenting pth M N M N Figure 4: mthings M, N, n their symmetri ifferene M N, whih inlues n even yle n n M-ugmenting pth. To prove Theorem 1, we first show tht if there is n M-ugmenting pth then M is not mximum mthing. Let P e n M-ugmenting pth. Consier the set S = M E(P ), whih is the set of ll elements from either the mthing M or the pth P ut not oth. Sine the set of S-overe verties inlues ll M-overe verties, while lso inluing the two Mexpose verties, S improves on M; therefore M is not mximum mthing. We now show tht if M is not mximum mthing then there must e n M-ugmenting pth. Sine M is not mximum, let N e nother mthing suh tht N > M ; tht is, N improves on M. Consier S = M N. Beuse M n N re mthings, verties in the grph G my e inient to t most one ege from M n t most one ege from N. Therefore every vertex in the grph G is inient to t most two eges in S. From this we n infer struturl hrteristis of S; speifilly, S ontins only even yles n M- lternting pths. Eges in the yles lternte etween eges from N n eges from M. Sine the yles re even, they ontin the sme numer of eges from N s from M. But N > M, so we know there must e t lest one M-lternting pth with more eges from N thn from M. This pth woul egin n en with eges from N n woul hve M-expose enpoints; therefore it woul e n M-ugmenting pth. Figure 4 shows two mthings, M n N, for grph. The symmetri ifferene M N shown inlues n even yle with lternting eges (the ol eges ome from N) n n M-ugmenting pth. Theorem 1 gives us generl ie for esigning n lgorithm to fin mximum mthing: s long s there is n M-ugmenting pth, we know tht M is not mximum mthing. 3 O yles n the Tutte-Berge Formul Definition 8 Given grph G = V, E, over is set of verties A V suh tht for every ege e = vw E, either v A or w A. Suppose A is over of some grph for whih we hve mthing M. We know tht ll eges in M must hve t lest one enpoint in A, euse ll eges in the grph hve t lest one enpoint in A (A is over). Therefore the mximum size of the mthing nnot exee A ( M A ). In iprtite grph mximum mthing n minimum over will e equl in size (there is priml-ul reltionship etween the two prolems when formulte s liner progrms). Grphs tht re not iprtite my hve o yles. Figure 5 shows grph of n o yle. Clerly, the size of mximum mthing will e stritly less thn the size of minimum over for o yles. Also, while 3
e Figure 5: n o yle; rkene eges mthing M n gry verties over A; M (stritly) < A. even yles mit only two possile mximum mthings, o yles of size > 3 will hve more (mny more, s the size of the yle inreses). So it is o yles tht use prolem we nee to fin tehnique of hnling o yles. H_1 H_k A Figure 6: H 1... H k re k o omponents extrte from grph G; A is the prt of the grph left fter the H s hve een extrte. We ivie grph G into setions s shown in Figure 6 (G\A is ivie into o omponents H i... H k ). Suppose M is mthing from G. For eh H i, if there re no M-expose verties H i, there must e n ege of M with one en in H i n the other en in A. So the totl numer of M-expose verties in the grph will e t lest k A. Let the numer of o omponents (k until now) tken from G\A = o(g\a). So the numer of M-expose verties is o(g \ A) A. From this we onlue tht, for ny A V : ν(g) 1 ( V o(g \ A) + A ) Tht is, the size of mximum mthing is t most hlf of the totl numer of verties minus the numer of M-expose verties. If the set A hosen is over of G, there will e V A o omponents in G \ A (eh one will e single noe). In tht se the right sie of the inequlity will reue to A, whih gives us the sme oun stte ove, tht the size of the mximum mthing is t most the size of the minimum over. This mens our new oun is t lest s goo s the over oun ove. For the o-yle grph shown in Figure 5, the size of mximum mthing =. We hoose the smllest possile A, A =, so tht V = 5, A = 0, G \ A = G, n (therefore) o(g \ A) = 1; for this se we hve: 4
ν(g) 1 (5 1 + 0) In this se the inequlity is stritly equl. In ft, the inequlity will e stritly equl every time mximum mthing n minimum right sie re use this is the Tutte-Berge Formul, stte more extly elow in Theorem. Theorem (Tutte-Berge Formul) given grph G = V, E, mx{ M : M is mthing } = min{ 1 ( V o(g \ A) + A ) : A V }. Theorem oul e use to fin the size of mximum mthing n therey stopping onition for some mximum mthing lgorithm. We oul simply try every possile A n tke the minimum right sie. But there will e n A s, so this woul e very ineffiient. Tutte s originl proof of Theorem ws extremely omplex. Here we will present some of the kgroun for Berge s simpler proof, pulishe yers fter Tutte s. Theorem 3 follows from the Tutte-Berge Formul ove (Theorem ). Theorem 3 A grph G = V, E hs perfet mthing if n only if A V, o(g \ A) A The min ie of Berge s proof involves shrinking opertion one on o yles, in whih the o yle eomes single vertex. Formlly, we let C e n o yle in grph G. We efine G = G C s the grph forme from G fter shrinking the o yle C. The set of verties in G = (V \ V (C)) {C}, where V (C) is the set of verties in the yle C, n the new vertex rete y shrinking the yle C is lle C lso. The set of eges in G = E \ γ(v (C)), where γ(v (C)) is the set of eges in the o yle C. For eh ege e G, with vertex v one of e s ens, if v V (C) then vertex C reples v s the en of e in G. All other eges remin the sme in G s they were in G. This proess n e thought of s series of noe ientifitions, s esrie on pge 73 of the text [CO98] in referene to the Minimum Cut Prolem for unirete grphs. An illustrtion of the shrinking opertion ppers on pge 131 of the text [CO98]. Theorem 4 Let C e n o iruit of grph G, let G = G C, n let M e mthing of G. There is mthing M of G suh tht M M E(C), n the numer of M-expose verties of G is the sme s the numer of M -expose verties of G. Theorem 4 sys tht ny mthing on the reue grph G C n e extene into mthing in the originl grph G. For the proof of Theorem 4, we let M = some mthing in the grph G = G C. Either C is M -overe or M -expose in the grph G. We go kwrs from G to moify the originl grph G in the following wy: if C is M -overe in G, we remove from G the vertex through whih C ws M -overe; if C is M -expose in G, we remove ny (ritrry) vertex from the o yle in G. So wht we hve one, in either se, is to remove two eges from n o yle. If two eges re remove from n o yle, the result is o-length pth, whih must hve perfet mthing (hoose the o eges of the pth: 1, 3, 5...); let M e this perfet mthing. The union of the mthing M from G n this M gives us mthing mthing M in the originl grph G (M M = M), stisfying the properties of the theorem. Sine M is perfet mthing of n o-length pth, it ontriutes no M -expose noes. Therefore the numer of M -expose noes must e equl to the numer of M-expose noes. The size of mximum mthing from grph G is t lest the sum of the size of the mximum mthing in G C (fter shrinking n o yle C) plus the numer of eges in mximum mthing of the o yle C (whih will lwys e 1 ( V (C) 1)), or: ν(g) ν(g C) + 1 ( V (C) 1) 5
When this reltionship hols with equlity, it mens there is mximum mthing in G tht uses ll 1 ( V (C) 1) eges (the eges in mximum mthing of the o yle C). In tht se we sy the o yle C is tight, n the prolem of fining mximum mthing in G reues to the prolem of fining mximum mthing in G C. Figure 7 elow shows grph with n o yle tht is not tight. The mximum mthing size ν(g) for the grph in Figure 7 is 3 (the mximum mthing is the three eges tht re not ol). But if we shrink the o yle, we hve grph in whih the mximum mthing size ν(g C) is only 1. Sine the size of the o yle V (C) = 3 (if we sustitute into the eqution ove), we hve: 3 1 + 1 (3 1) 3 Sine the reltionship oesn t hol with equlity, the o yle is not tight. It turns out tht heking for tightness of o yles is very iffiult prolem, ut the onept of tightness is importnt for the proof of the Tutte-Berge Formul (Theorem ). Figure 7: the o yle C in this grph (shown in ol) is not tight. Definition 9 Given grph G = V, E, vertex v V is essentil if every mximum mthing of G must inlue v; v is inessentil if v is not essentil. We return to the inequlity shown elow: ν(g) 1 ( V o(g \ A) + A ) (1) Suppose we hoose set A for whih the reltionship ove hols with equlity. If we kik out vertex v A from G to get some moifie grph G, wht is the numer of o omponents o(g \ A \ {v})? It is the sme numer of o omponents s there re in G \ A, or o(g \ A). Therefore ν(g ) < ν(g), so v is essentil s re ll the verties v from the set A (A hosen so tht the reltionship (1) hols with equlity). Lemm 1 Given grph G = V, E, let ege vw E. If oth v n w re inessentil, then there exists tight o yle C tht inlues vw, n C is n inessentil vertex of G C. 6
Proof: Sine v n w re oth inessentil, there exist mximum mthings M 1 n M uner whih v n w re expose respetively. Note tht M 1 must over w n M must over v; otherwise the ege vw n e e to M 1 (or M ) to inrese the size of tht mthing. Consier the sugrph H =< V, M 1 M >. The omponent ontining v onsists of pth P originting from v; sine v is M 1 -expose. If P ens t nother M 1 -expose noe, then, we hve n M 1 -ugmenting pth, ontriting the mximum rinlity of M 1. Likewise, if it ens t n M -expose noe other thn w, then ing the ege vw to this pth, we get n M -expose noe. Thus, P must en t w; it thus forms n iruit C with ege vw. Note tht P strts with n ege vx M n ens with n ege yw M 1, for some verties x, y V. Thus the iruit forme with ege vw is neessrily o. Further, M 1 is mximum mthing of G, tht ontins 1 ( V (C) 1) eges from E(C) (elete vertex v from C!) n thus C is tight. Finlly, M 1 \ E(C) is mximum mthing of G C not overing C (hek?!), proving tht C is n inessentil noe of G C. Proof: (of Tutte-Berge Formul): Let us restte the formul: mx{ M : M is mthing } = min{ 1 ( V o(g \ A) + A ) : A V } () We hve lrey shown tht the LHS of () is less thn or equl to the RHS for ll A V. Hene, ll tht is require is to proue mthing M n set A, suh tht the numer of M expose noes is extly o(g \ A) A. We provie n inutive proof. The result is ertinly true, if G hs no eges. Simply hoose A = φ; the mximum mthing hs rinlity zero n so is V o(g \ A) + A, proving tht the LHS n RHS of () re equl. Now onsier grph hving t lest one ege, sy vw. We nee to onsier two ses: 1. Let v e n essentil noe, i.e. ν(g \ v) = ν(g) 1. Set G = G \ v. Using the inutive hypothesis on G, we note tht there exists set A n mthing M of G suh tht there re extly l 1 = o(g \ A ) A M expose noes. Set A = A {v}. From the previous sttement n the ft tht v is n essentil noe, we know tht there must exist mthing M of G, suh tht the numer of M expose noes is l 1 + 1 = o(g \ A {v}) A {v}. Setting A = A {v}, we re one. The proof for the se in whih w is n essentil noe is ientil to the ove proof.. We now onsier the se in whih oth v n w re oth inessentil. From Lemm (1), we know tht G hs tight iruit C. We pply inution to G = G C to get mthing M n set A of noes, suh tht the numer of M expose noes is extly o(g \ A ) A. The noe C of G nnot e in A, sine it is n inessentil noe n if A is suset for whih () is met with equlity, then ll noes re essentil. We lso know from Theorem (4), tht M n e extene into mthing M of G, hving the sme numer of expose noes. Oserve tht eleting A from G results in the sme numer of o omponents s eleting A from G. If C elongs to n o omponent of G \ A, then it elongs to the sme omponent in G; however it is reple y V (C) noes. Sine V (C) is o, the omponent in G is lso o. The sme rgument works when C oes not elong to n o omponent of G \ A. Thus, we get mthing M in G n set A, suh tht the numer of M expose noes is extly o(g \ A ) A s require. Note tht in this se, the set A of G C is use s the set A of G. Referenes [CO98] Willim Cook, Willim H. Cunninghm, Willim Pulleylnk, n Alexner Shrijver. Comintoril Optimiztion. John Wiley & Sons, 1998. 7