Perfect Matchings via Uniform Sampling in Regular Bipartite Graphs

Transcription

1 Perfect Matchigs via Uiform Samplig i Regular Bipartite Graphs Ashish Goel Michael Kapralov Sajeev Khaa Abstract I this paper we further ivestigate the well-stuie problem of fiig a perfect matchig i a regular bipartite graph. The first o-trivial algorithm, with ruig time O(m), ates back to Köig s work i 1916 (here m = is the umber of eges i the graph, is the umber of vertices, a is the egree of each oe). The curretly most efficiet algorithm takes time O(m), a is ue to Cole, Ost, a Schirra. We improve this ruig time to O(mi{m,.5 l }); this miimum ca ever be larger tha O( 1.75 l ). We obtai this improvemet by provig a uiform samplig theorem: if we sample each ege i a -regular bipartite graph iepeetly with a probability p = O( l ) the the resultig graph has a perfect matchig with high probability. The proof ivolves a ecompositio of the graph ito pieces which are guaratee to have may perfect matchigs but o ot have ay small cuts. We the establish a correspoece betwee potetial witesses to o-existece of a matchig (after samplig) i ay piece a cuts of comparable size i that same piece. Karger s samplig theorem for preservig cuts i a graph ca ow be aapte to prove our uiform samplig theorem for preservig perfect matchigs. Usig the O(m ) algorithm (ue to Hopcroft a Karp) for fiig maximum matchigs i bipartite graphs o the sample graph the yiels the state ruig time. We also provie a ifiite family of istaces to show that our uiform samplig result is tight up to poly-logarithmic factors (i fact, up to l ). 1 Itrouctio A bipartite graph G =(U, V, E) with vertex set U V a ege set E U V is sai to be regular if every vertex has the same egree. We use m = to eote the umber of eges i G a to represet the umber of vertices i U (as a cosequece of regularity, U a V have the same size). Regular bipartite graphs have bee the subject of much stuy. Raom regular bipartite graphs represet some of the simplest examples of expaer graphs [1]. These graphs are also use to Departmets of Maagemet Sciece a Egieerig a (by courtesy) Computer Sciece, Stafor Uiversity. ashishg@stafor.eu. Research supporte by NSF ITR grat , NSF CAREER awar 03396, a a Alfre P. Sloa fellowship. Istitute for Computatioal a Mathematical Egieerig, Stafor Uiversity. kapralov@stafor.eu. Departmet of Computer a Iformatio Sciece, Uiversity of Pesylvaia, Philaelphia PA. sajeev@cis.upe.eu. Supporte i part by a Guggeheim Fellowship, a IBM Faculty Awar, a by NSF Awar CCF moel scheulig, routig i switch fabrics, a taskassigmet problems (sometimes via ege colorig, as escribe below) [1, 6]. A regular bipartite graph of egree ca be ecompose ito exactly perfect matchigs, a fact that is a easy cosequece of Hall s theorem [4]. Fiig a matchig i a regular bipartite graph is a well-stuie problem, startig with the algorithm of Köig i 1916, which is ow kow to ru i time O(m) [11]. The well-kow bipartite matchig algorithm of Hopcroft a Karp [8] ca be use to obtai a ruig time of O(m ). I graphs where is a power of, the followig simple iea, ue to Gabow a Kariv [7], leas to a algorithm with O(m) ruig time. First, compute a Euler tour of the graph (i time O(m)) a the follow this tour i a arbitrary irectio. Exactly half the eges will go from left to right; these form a regular bipartite graph of egree /. The total ruig time T (m) thus follows the recurrece T (m) = O(m) + T (m/) which yiels T (m) = O(m). Exteig this iea to the geeral case prove quite har, a after a series of improvemets (eg. by Cole a Hopcroft [5], a the by Schrijver [13] to O(m)), Cole, Ost, a Schirra [6] gave a O(m) algorithm for the case of geeral. The mai iterest of Cole, Ost, a Schirra was i ege colorig of geeral bipartite graphs of maximum egree, where fiig perfect matchigs i regular bipartite graphs is a importat subroutie. Fiig perfect matchigs i regular bipartite graphs is also closely relate to the problem of fiig a Birkhoff vo Neuma ecompositio of a oubly stochastic matrix [3, 16]. I this paper we preset a algorithm for fiig a perfect matchig i a regular bipartite graph that rus i time O(mi{m,.5 l }). It is easy to see that this miimum ca ever be larger tha O( 1.75 l ). This is a sigificat improvemet over the ruig time of Cole, Ost, a Schirra whe the bipartite graph is relatively ese. We first prove (Theorem.1 i sectio ) that if we sample the eges of a regular bipartite graph iepeetly a uiformly at rate p = O( l ), the the resultig graph has a perfect matchig with high probability. The resultig graph has O(mp) eges i expectatio, a ruig the bipartite

2 matchig algorithm of Hopcroft a Karp gives a expecte ruig time of O(.5 l ). Sice we kow this ruig time i avace, we ca choose the better of m a.5 l i avace. It is worth otig that uiform samplig ca easily be implemete i O(1) time per sample ege assumig that the ata is give i ajacecy list format, with each list store i a array, a assumig that log bit raom umbers ca be geerate i oe time step 1. We believe that our samplig result is also iepeetly iterestig as a combiatorial fact. The proof of our samplig theorem relies o a sequetial ecompositio proceure that creates a vertex-isjoit collectio of subgraphs, each subgraph cotaiig may perfect matchigs o its uerlyig vertex set. We the show that if we uiformly sample eges i each ecompose subgraph at a suitably chose rate, with high probability at least oe perfect matchig survives i each ecompose subgraph. This is establishe by usig Karger s samplig theorem [9, 10] i each subgraph. A effective use of Karger s samplig theorem requires the mi-cuts to be large, a property that is ot ecessarily true i the origial graph. For istace, G coul be a uio of two isjoit -regular bipartite graphs, i which case the mi-cut is 0; o-pathological examples are also easy to obtai. However, our serial ecompositio proceure esures that the mi-cuts are large i each ecompose subgraph. We the establish a 1-1 correspoece betwee possible Hall s theorem couter-examples i each subgraph a cuts of comparable size i that subgraph. Sice Karger s samplig theorem is base o coutig cuts of a certai size, this couplig allows us to claim (with high probability) that o possible couterexample to Hall s theorem exists i the sample graph. O a relate ote, Beczur [] presete aother samplig algorithm which geerates O( l ) eges that approximate all cuts; however this samplig algorithm, as well as recet improvemets [15, 14] take Ω(m) time to geerate the sample graph. Hece these approaches o ot irectly help i improvig upo the alreay kow O(m) ruig time for fiig perfect matchigs i - regular bipartite graphs. The samplig rate we provie may seem couterituitive; a superficial aalogy with Karger s samplig theorem or Beczur s work might suggest that samplig a total of O( l ) eges shoul suffice. We show (Theorem 4.1, sectio 4) that this is ot the case. I particular, we preset a family of graphs where uiform samplig at rate o( ) results i a vaishigly low prob- l 1 Eve if we assume that oly oe raom bit ca be geerate i oe time step, the ruig time of our algorithm remais ualtere sice the Hopcroft-Karp algorithm icurs a overhea of per sample ege ayway. ability that the sample subgraph has a perfect matchig. Thus, our samplig rate is tight up to factors of O(l ). This lower bou suggests two promisig irectios for further research: esigig a efficietly implemetable o-uiform samplig scheme, a esigig a algorithm that rus faster tha Hopcroft-Karp s algorithm for ear-regular bipartite graphs (sice the egree of each vertex i the sample subgraph will be cocetrate arou the expectatio). Uiform Samplig for Perfect Matchigs: A Upper Bou I this sectio, we will establish our mai samplig theorem state below. We will the show i Sectio 3 that this theorem immeiately yiels a O( 1.75 l ) time algorithm for fiig a perfect matchig i regular bipartite graphs. Theorem.1. There exists a costat c such that give a -regular bipartite graph G(U, V, E), a subgraph G of G geerate by samplig the eges i G uiformly at raom with probability p = matchig with high probability. c l cotais a perfect Our proof is base o a ecompositio proceure that partitios the give graph ito a vertex-isjoit collectio of subgraphs such that (i) the miimum cut i each subgraph is large, a (ii) each subgraph cotais Ω() perfect matchigs o its vertices. We the show that for a suitable choice of samplig rate, w.h.p. at least oe perfect matchig survives i each subgraph. The uio of these perfect matchigs the gives us a perfect matchig i the origial graph. We emphasize here that the ecompositio proceure is merely a artifact for our proof techique. Note that the theorem is trivially true whe is O( l ). follows, we assume that is Ω( l ). So i what.1 Hall s Theorem Witess Sets Let G(U, V, E) be a bipartite graph. We shall use the followig otatio. For a graph G a a set of vertices W we eote the umber of eges crossig the bouary of W i G by δ G (W ). Also, we eote the vertex set of G by V (G). A pair (A, B) with A U a B V is sai to be a left relevat pair to Hall s theorem if A > B. Similarly, a pair (A, B) with A U a B V is sai to be a right relevat pair to Hall s theorem if A < B. Give a left relevat pair (A, B), we eote by E(A, B) the set of eges i E (A (V \B)). Similarly, give a right relevat pair (A, B), we eote by E(A, B) the set of eges i E ((U \A) B). We refer to the set E(A, B) as a witess ege set if (A, B) is a left or right relevat pair. By Hall s theorem (see, for istace, [4]),

3 to prove Theorem.1 it suffices to show that w.h.p. i the sample graph G, at least oe ege is chose from each witess set. We will focus o a sub-class of relevat pairs, referre to as miimal relevat pairs. A left relevat pair (A, B) is miimal if there oes ot exist aother left relevat pair (A,B ) with A A a E(A,B ) E(A, B). Miimal right relevat pairs are similarly efie. A witess ege set correspoig to a miimal left relevat pair or a miimal right relevat pair is calle a miimal left witess set or a miimal right witess set, respectively. If a graph G has a perfect matchig, every miimal witess set must be o-empty. It also follows that ay subgraph of G that iclues at least oe ege from every miimal witess set must have a perfect matchig. A key iea uerlyig our proof is a mappig from miimal witess sets i G to istict cuts i G. I particular, we will map each miimal left witess set E(A, B) to the cut δ G (A B). The theorem below shows that this is a oe-to-oe mappig. The aalogous theorem hols for miimal right witess sets. Theorem.. Let G(U, V, E) be a bipartite graph that has at least oe perfect matchig. If (A, B) a (A,B ) are miimal left relevat pairs i G with E(A, B) E(A,B ), the δ G (A B) δ G (A B ). Proof. Assume by way of cotraictio that there exist miimal left relevat pairs (A, B) a (A,B ) i G with E(A, B) E(A,B ) but δ G (A B) =δ G (A B ). The the followig coitios must be satisfie for ay ege (u, v) E : A1. If u (A\A ) (A \A) the v (B \B ) (B \B). To see this, assume w.l.o.g. that u A \ A, a the ote that if v B B, the (u, v) δ G (A B ) but (u, v) δ G (A B). A cotraictio. Similarly, if v V \(B B ), the (u, v) δ G (A B) but (u, v) δ G (A B ). A cotraictio. A. If u (A A ) the v (B \ B ) (B \ B). To see this, cosier w.l.o.g. that v (B \ B ). The (u, v) δ G (A B ) but (u, v) δ G (A B). A cotraictio. I what follows, we slightly abuse the otatio a give ay (ot ecessarily relevat) pair (C, D) with C U a D V, we eote by E(C, D) the set of eges i E (C (V \D)). As a immeiate corollary of the properties A1 a A, we ow obtai the followig cotaimet results: B1. E(A \ A,B\ B ) E(A, B). This follows irectly from property A1 above. B. E(A A,B B ) E(A, B). This follows irectly from property A above. We ow cosier three possible cases base o the relatioship betwee A a A, a establish a cotraictio for each case. Case 1: A A =. By property A1, if u A A the v B B. I other wors, there are o eges from A A to vertices outsie B B. Sice A A = A + A > B + B, this cotraicts our assumptio that G has at least oe perfect matchig. Case : A = A. For ay ege (u, v) with u A, property A shows that v (B \ B ) (B \ B). The E(A, B) =E(A,B ). A cotraictio. Case 3: A A a A A. Assume w.l.o.g. that A \ A. Sice A > B, it must be that either A \ A > B \ B or A A > B B. If A \ A > B \ B, the (A \ A,B\ B ) is a left relevat pair, a by B1, it cotraicts the fact that (A, B) is a miimal left relevat pair. If A A > B B, the (A A,B B ) is a left relevat pair set, a by B, it cotraicts the fact that (A, B) is a miimal left relevat pair.. A Decompositio Proceure Give a -regular bipartite graph o vertices, we will first show that it ca be partitioe ito k = O(/) vertex isjoit graphs G 1 (U 1,V 1,E 1 ),G (U,V,E ),..., G k (U k,v k,e k ) such that each graph G i satisfies the followig properties: P1. the size of a miimum cut i G i (U i,v i,e i ) is strictly greater tha α = 4. P. δ G (U i V i ) / (hece G i cotais at least / ege-isjoit perfect matchigs). The ecompositio proceure is as follows. Iitialize H 1 = G, a set i = Fi a smallest subset X i V (H i ) such that δ Hi (X i ) α. If o such set X i exists, the the ecompositio proceure termiates.. Defie G i to be the subgraph of H i iuce by the vertices i X i. Also, let M i eote the umber of eges i the cut δ Hi (X i ). 3. Defie H i+1 as H i with vertices from X i remove. 4. Icremet i, a go to step (1). We ow prove the followig properties of the ecompositio proceure.

4 Propositio.1. The ecompositio proceure outlie above satisfies properties P1 a P. Proof. We start by provig that property P1 is satisfie. Suppose that there exists a cut (C, D) i G i of value less tha α, i.e. C D = X i a δ Gi (C) =δ Gi (D) α. We have δ Hi (C) \ δ Gi (C) + δ Hi (D) \ δ Gi (D) α by the choice of X i i (1). Suppose without loss of geerality that δ Hi (C) \ δ Gi (C) α. The δ Hi (C) α a C X i, which cotraicts the choice of X i as the smallest cut of value at most α i step (1) of the proceure. It remais to show that δ G (U i V i ) / for all i. I orer to establish this property, it suffices to show that k i=1 M i / (recall that M i = δ Hi (X i ) ). We prove the followig statemets by iuctio o k, the umber of ecompositio steps: 1. U k V k ;. k i=1 M i /; 3. k /. Base: k = 1 Sice α = /, we have M 1 /. It remais to show that G 1 (U 1,V 1,E 1 ) has at least vertices. Cosier ay vertex u U 1. Let j / be the umber of eges i δ H1 (U 1 V 1 ) that are iciet o vertex u. The u must have exactly ( j) eighbors i V 1. Sice δ H1 (U 1 V 1 ) /, at least oe vertex amog the eighbors of u i V 1 must have all its eighbors isie U 1. Thus U 1. Similarly, we ca show that V 1. Iuctive step: k k + 1 Suppose that the k-th step has bee execute a the algorithm has ot termiate yet. Sice k / by the iuctive hypothesis, we have k ( ) i=1 M i (/) (α) = (/) /. Cosier the cut (X k,h k \ X k ) of H k. It follows from the previous estimate that δ Hk (X k ) δ G (X k ) /. Hece, we coclue as i the base case that X k a H k \X k. Sice at every ecompositio step j k at least vertices were remove from the graph, we have k +1 /..3 Proof of Theorem.1 We ow argue that if the graph G is obtaie by uiformly samplig the eges of G with probability p = Θ ( ) l α, the w.h.p. G cotais a perfect matchig. It suffices to show that i each graph G i obtaie i the ecompositio proceure, every miimal witess set is hit w.h.p. i the sample graph (that is, at least oe ege i each miimal witess set is chose i the sample graph). This esures that at least oe perfect matchig survives isie each G i. A uio of these perfect matchigs the gives us a perfect matchig of G i the sample graph G. Fix a graph G i (U i,v i,e i ). Let (A, B) be a left or a right relevat pair i G i. Usig the fact that our startig graph G is -regular, we get δ G (A B) E(A, B). Let m A,m B eote the umber of eges i G that coect oes i A, B respectively to oes outsie G i. The δ Gi (A B) E(A, B) m A m B. By property P, sice δ G (U i V i ) /, it follows that E(A, B) E i E(A, B) /. Also, by efiitio, E(A, B) E i E(A, B) m A m B. Combiig, we obtai: δ Gi (A B) E(A, B) E i /. Thus the set E(A, B) E i cotais at least half as may eges as the the cut δ Gi (A B). We will ow utilize the followig samplig result ue to Karger [10]: Theorem.3. [10] Let G i be a uirecte graph o at most vertices, a let κ be the size of a miimum cut i G i. There exists a positive costat c such that for ay ɛ (0, 1), if we sample the eges i G i uiformly with probability at least p = c ( ) l κɛ, the every cut i G i is preserve to withi (1 ± ɛ) of its expecte value with probability at least 1 1/ Ω(1). Thus the samplig probability eee to esure that all cuts are preserve close to their expecte value, is iversely relate to the size of a miimum cut i the graph. We ow show use the theorem above to prove that at least oe perfect matchig survives i each graph G i whe eges are sample with probability specifie i Theorem.1. By Property P1, we kow that the size of a miimum cut i G i is at least α = /4. Fix a ɛ (0, 1). The theorem above implies that if we sample eges i G i with probability p = Θ ( ) l αɛ, the for every relevat pair (A, B), w.h.p. the sample graph cotais (1 ± ɛ)p δ Gi (A B) = Ω(l ) eges from the set δ Gi (A B). Note that the set δ Gi (A B) is ot a Hall s theorem witess ege set. However, by Theorem.1, we kow that for every left (right) miimal witess ege set

5 E(A, B) E i, we ca associate a istict cut, amely δ Gi (A B), of size at most twice E(A, B) E i. We ow show that this correspoece ca be use to irectly aapt Karger s proof of Theorem.3 to claim that every witess ege set i G i is preserve to withi (1 ± ɛ) of its expecte value. We remi the reaer that the proof of Karger s theorem is base o a applicatio of uio bou over all cuts i the graph. I particular, it is show that the umber of cuts of size at most β times the miimum cut size is boue by β. O the other ha, for the samplig rate give i Theorem.3, we ca use Cheroff bous to claim that the probability that a cut of size β times the miimum cut eviates by (1 ± ɛ) from its expecte value is at most 1/ Ω(β). The theorem follows by combiig these two facts. Withi ay piece of the ecompositio, let c i be the umber of cuts of size i a let w i be the umber of miimal witess sets of size i. We kow by the correspoece argumet above that every Hall s theorem miimal witess set of size i correspos to a cut of size at most i, a at most two miimal witess sets (oe left a oe right) correspo to the same cut. Now, give a samplig probability p, the probability that oe of the eges i some miimal witess set are sample is at most i w i(1 p) i, which is at most i c i(1 p) i/. Therefore the probability that there is o matchig i this piece ca be at most twice the expressio use i Karger s theorem to bou the probability that there exists a cut from which o ege is sample whe the samplig rate is q, where 1 q = (1 p) 1/, or p =q q. Hece, it is sufficiet to use a samplig rate which is twice that require by Karger s samplig theorem to coclue that a perfect matchig survives with probability at least 1 1/ Ω(1). Puttig everythig together, the sample graph G will have a perfect matchig w.h.p. as log as we sample the eges with probability p > c l α for a sufficietly large costat c, thus completig the proof of theorem.1. We have mae o attempt to optimize the costats i this proof (a upper bou 1 l of α follows from the reasoig above). I fact, i a implemetatio, we ca use geometrically icreasig samplig rates util either the sample graph has a perfect matchig, or the samplig rate becomes so large that the expecte ruig time of Hopcroft a Karp [8] algorithm is Ω(m). 3 A Faster Algorithm for Perfect Matchigs i Regular Bipartite Graphs We ow show that the samplig theorem from the preceig sectio ca be use to obtai a faster raomize algorithm for fiig perfect matchigs i -regular bipartite graphs. Theorem 3.1. There exists a O(mi{m,.5 l }) expecte time algorithm for fiig a perfect matchig i a -regular bipartite graph with vertices a m = eges. Proof. Let G be a -regular bipartite graph with vertices a m = eges. If 3/4 l, we use the O(m) time algorithm of Cole, Ost, a Schirra [6] for fiig a perfect matchig i a -regular bipartite graph. It is easy to see that m.5 l i this case. Otherwise, we sample the eges i G at a rate of c l p = for some suitably large costat c (c = 48 suffices by the reasoig from the previous sectio), a by Theorem.1, the sample graph G cotais a perfect matchig w.h.p. The expecte umber of eges, say m, i the sample graph G is O( l ). We ca ow use the algorithm of Hopcroft a Karp [8] to fi a maximum matchig i the bipartite graph G i expecte time O(m ). The samplig is the repeate if o perfect matchig exists i G. This takes O(.5 l ) expecte ruig time. Hece, the algorithm takes O(mi{m,.5 l }) expecte time. Note that by abortig the computatio wheever the umber of sample eges is more tha twice the expecte value, the above algorithm ca be easily coverte to a Mote-Carlo algorithm with a worst-case ruig time of O(mi{m,.5 l }) a a probability of success = 1 o(1). Fially, it is easy to verify that the state ruig time ever excees O( 1.75 l ). 4 Uiform Samplig for Perfect Matchigs: A Lower Bou We ow preset a costructio that shows that the uiform samplig rate of Theorem.1 is optimal to withi a factor of O(l ). As before, for ay graph G the graph obtaie by samplig the eges of G uiformly with probability p is eote by G. Theorem 4.1. Let () be a o-ecreasig positive iteger value fuctio such that for some fixe iteger 0, it always satisfies oe of the followig two coitios for all 0 : (a) () / l, or (b) / l < () / l. The there exists a family of ()- regular bipartite graphs G with + o() vertices such that the probability that the graph G, obtaie by samplig eges of G with probability p, has a perfect matchig goes to zero faster tha ay iverse polyomial fuctio i if p = o(1) whe () satisfies coitio (a) above, a if ( p = o (()) l whe () satisfies coitio (b) above. )

6 Proof. Note that the theorem asserts that essetially o samplig ca be oe whe () / l. We shall omit the epeece o i () to simplify otatio. Defie H (k) =(U, V, E), 0 k, as a bipartite graph with U = V = such that k vertices i U (respectively V ) have egree ( 1) a the remaiig vertices have egree. We will call the vertices of egree ( 1) eficiet. Clearly, for ay 0 k, the graph H (k) exists: startig with a -regular bipartite graph o vertices, we ca remove a arbitrary subset of k eges that belog to a perfect matchig i the graph. I the followig costructio, we will use copies of H (k) as builig blocks to create our fial istace. I oig so, oly the set of eficiet vertices i a copy of H (k) will be coecte to (eficiet) vertices i other copies i our costructio. We ow efie a -regular bipartite graph G. Let γ = l (ote that γ sice / l ). We choose W =, k j = γ for 1 j < W, a γ k W = γ(w 1) γ. We also efie K() = l if () / l a K() = otherwise. The graph G cosists of K() W copies of H (k) that we iex as {H i,j } 1 i K(),1 j W. The subgraph H i,j is a copy of H (kj), where k j is as efie above. Note that the sum of the umber of eficiet vertices over each of the parts of H i,j, 1 j W, equals for all fixe i. Moreover, the umber of eficiet vertices i H i,j is the same for all i whe j is hel fixe. We ow itrouce two istiguishe vertices u a v a a aitioal eges as follows: 1. For every 1 i<k() a for every 1 j W, all eficiet vertices i part V of H i,j are matche to the eficiet vertices i part U of H i+1,j (that is, we isert a arbitrary matchig betwee these two sets of vertices);. All eficiet vertices i part U of H 1,j for 1 j W are coecte to u; 3. All eficiet vertices i part V of H K(),j for 1 j W are coecte to v. Essetially, we are coectig the graphs H i,j for fixe j i series via their eficiet vertices, a the coectig the left es of these chais to the istiguishe vertex u a the right es of the chais to the istiguishe vertex v. We ote that the graph G costructe as escribe above is a -regular bipartite graph with K()W + = + o() vertices. Cosier the sample graph G. Suppose G has a perfect matchig M. I the matchig M, if u is matche to a vertex i part U of H 1,j for some 1 j W, the there must be a vertex i part V of H 1,j that is matche to a vertex i part U of H,j. Proceeig i the same way, oe coclues that for every i, 1 i<k() there must be a vertex i part V of H i,j that is matche to a vertex i part U of H i+1,j. Fially, vertex v must be matche to a vertex i part V of H K(),j. This implies that the sample graph G ca have a perfect matchig oly if at least oe ege survives i G betwee every pair of ajacet elemets i the sequece below: u H 1,j H,j... H K() 1,j H K(),j v. Now suppose that we sample eges uiformly with probability p. It follows from the costructio of G that for ay fixe j, the probability that at least oe ege survives betwee every pair of ajacet elemets i the sequece u H 1,j H,j... H K() 1,j H K(),j v is equal to ( 1 (1 p) kj ) K()+1 (pkj ) K()+1. Hece, the probability that at least oe such path survives i G is at most W ( ) K()+1 p max k j 1 j W by the uio bou. Whe () / l, we have γ = 1, W =, k j = 1 a K() = /. So the bou trasforms to (4.1) Wp K()+1 = p / +1, which goes to zero faster tha ay iverse polyomial fuctio i whe p = o(1) sice K() = / = Ω(l ). Whe / l, we have k j γ where γ = l, W = bou becomes (4.) γ W (pγ) K()+1 = a K() = l. (pγ) l +1, γ Hece, the which goes to zero faster tha ay iverse polyomial fuctio i whe p = o ( l ). This completes the proof of the theorem. The costructio give i Theorem 4.1 shows that the samplig upper bou for preservig a perfect matchig prove i Theorem.1 is tight up to a factor of O(l ). Ackowlegmets We thak Rajat Bhattacharjee for may helpful iscussios i the early stages of this work.

7 Refereces [1] G. Aggarwal, R. Motwai, D. Shah, a A. Zhu. Switch scheulig via raomize ege colorig. FOCS, 003. [] A. Beczur. Cut structures a raomize algorithms i ege-coectivity problems. PhD Thesis, [3] G. Birkhoff. Tres observacioes sobre el algebra lieal. Uiv. Nac. Tucumá Rev. Ser. A, 5: , [4] B. Bollobas. Moer graph theory. Spriger, [5] R. Cole a J.E. Hopcroft. O ege colorig bipartite graphs. SIAM J. Comput., 11(3): , 198. [6] R. Cole, K. Ost, a S. Schirra. Ege-colorig bipartite multigraphs i O(E log D) time. Combiatorica, 1(1):5 1, 001. [7] H.N. Gabow a O. Kariv. Algorithms for ege colorig bipartite graphs a multigraphs. SIAM J. Comput., 11(1):117 19, 198. [8] J.E. Hopcroft a R.M. Karp. A 5/ algorithm for maximum matchigs i bipartite graphs. SIAM J. Comput., (4):5 31, [9] D. Karger. Raom samplig i cut, flow, a etwork esig problems. Proceeigs of the twety-sixth aual ACM symposium o Theory of computig, pages , [10] D. Karger. Usig raomize sparsificatio to approximate miimum cuts. Proceeigs of the fifth aual ACM-SIAM symposium o Discrete algorithms, pages 44 43, [11] D. Köig. Uber graphe u ihre aweug auf etermietetheorie u megelehre. Math. Aale, 77:453465, [1] R. Motwai a P. Raghava. Raomize Algorithms. Cambrige Uiversity Press, [13] A. Schrijver. Bipartite ege colorig i O( m) time. SIAM J. o Comput., 8:841846, [14] D.A. Spielma a N. Srivastava. Graph sparsificatio by effective resistaces. STOC, pages , 008. [15] D.A. Spielma a S.-H. Teg. Nearly-liear time algorithms for graph partitioig, graph sparsificatio, a solvig liear systems. STOC, pages 81 90, 004. [16] J. vo Neuma. A certai zero-sum two-perso game equivalet to the optimal assigmet problem. Cotributios to the optimal assigmet problem to the Theory of Games, :5 1, 1953.