Maximum Cardinality Matchings on Trees by Randomized Local Search

Transcription

1 Maximum Cardinality Matchings on Trees by Randomized Local Search ABSTRACT Oliver Giel Fachbereich Informatik, Lehrstuhl 2 Universität Dortmund Dortmund, Germany oliver.giel@cs.uni-dortmund.de To understand the working rinciles of randomized search heuristics like evolutionary algorithms they are analyzed on otimization roblems whose structure is well-studied. The idea is to investigate when it is ossible to simulate clever otimization techniques for combinatorial otimization roblems by random search. The maximum matching roblem is well suited for this aroach since long augmenting aths do not allow immediate imrovements by local changes. It is known that randomized search heuristics like simulated annealing, the Metroolis algorithm, the (1+1) EA and randomized local search efficiently aroximate maximum matchings for any grah; however, there are grahs where they fail to find maximum matchings in olynomial time. In this aer, we examine randomized local search (RLS) for grahs whose structure is simle. We show that RLS finds maximum matchings on trees in exected olynomial time. Categories and Subject Descritors F.2.2 [Analysis of Algorithms and Problem Comlexity]: Nonnumerical Algorithms and Problems Comutations on discrete structures; G.3[Probability and Statistics]: Probabilistic Algorithms General Terms Algorithms, Theory Keywords Evolutionary algorithms, randomized local search, maximum cardinality matchings, runtime analysis suorted by the German Research Foundation (DFG) as a art of the collaborative research center Comutational Intelligence (SFB 531) Permission to make digital or hard coies of all or art of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies bear this notice and the full citation on the first age. To coy otherwise, to reublish, to ost on servers or to redistribute to lists, requires rior secific ermission and/or a fee. GECCO 06, July 8 12, 2006, Seattle, Washington, USA Coyright 2006 ACM /06/ $5.00. Ingo Wegener Fachbereich Informatik, Lehrstuhl 2 Universität Dortmund Dortmund, Germany ingo.wegener@cs.uni-dortmund.de 1. INTRODUCTION To understand how randomized search heuristics (RSHs) work we analyze their runtime. In general, we are interested in the question for what roblems simle RSHs including the Metroolis algorithm (MA), simulated annealing (SA), the (1+1) EA, and randomized local search (RLS) are suitable. One aroach is to analyze RSHs for otimization roblems whose structure is well-known. When a certain roblem turns out to be too difficult for a secific RSH, we ask for what instances of the roblem the RSH is successful. This aroach to understand the working rinciles of RSHs was started for MA and SA for the roblems grah bisection [8], clique [7], and also maximum matching [12, 11]. The question whether the right cooling schedule for SA is suerior to all choices of a constant temerature in MA for a quite natural roblem was answered in the affirmative [15]. The roof uses an instance of the combinatorial otimization roblem minimum sanning tree. A number of combinatorial otimization roblems have already been investigated in the field of evolutionary algorithms. Among them are sorting and single source shortest ath [13], maximum matching [4], minimum sanning tree [10], and also the NP-hard roblem artition [16]. The maximum matching roblem is well-suited for our urose. The roblem is not trivial. Only theoretical insight has led to olynomial time combinatorial algorithms. Exact otimization is difficult for simle RSHs because small changes of a good solution do not roduce better solutions in tyical situations. However, even very simle RSHs find aroximate solutions efficiently. But there are instances which require an exonential runtime for an otimal solution. Section 2 briefly summarizes known results regarding the maximum matching roblem and RSHs. For a better understanding, we first introduce the matching roblem and the heuristic RLS. A matching M is a subset of the edges of an undirected grah G such that no edges of M share a node. The maximum matching roblem is to find a matching of maximal cardinality for a given grah G. Edges not belonging to M are called free and nodes not belonging to an edge in M are also called free. If M is a matching, an M-augmenting ath is a simle ath in G connecting two free nodes such that matching edges and free edges alternate along the ath. Exchanging the state of matching edges and free edges on an augmenting ath imroves the matching by 1 edge. By Berge s theorem [1], every non-maximum matching has an augmenting ath. Consequently, maximum matchings can 539

2 u v v u v Figure 1: Fliing two bits may shorten or lengthen an augmenting ath by two edges. be constructed by iterated imrovements using augmenting aths. However, searching for augmenting aths efficiently is not trivial. Efficient algorithms like the blossom algorithm [3] are clever and their correctness roofs are difficult. The best known algorithms [9, 14] are quite comlicated. We work with a fitness function f : {0, 1} m Z for grahs with m edges. A search oint s {0, 1} m is interreted as the characteristic vector of the set of chosen edges. The idea is that the fitness equals the matching size so that the aim is to maximize f. Hence,f equals the number of chosen edges s s m if s describes a matching. There are two otions to handle non-matchings. One is to start the search with the emty set and never accet non-matchings. This aroach is usually chosen for SA and MA. The otion we choose here is to start the search with an arbitrary search oint and to define a enalty for nonmatchings. The enalty directs the search towards matchings. Let d(v, s) be the degree of a vertex v with resect to the edges chosen by s. The enalty (v,s) assigned to each vertex v equals r max{0,d(v, s) 1} for some fixed number r m + 1. Thus, all vertex enalties are 0 for matchings. Now, f(s) equals the number of chosen edges minus the sum of all vertex enalties, i. e., f(s) := s s m X (v, s). ( ) v The scaling factor r m + 1 ensures that the fitness of nonmatchings is strictly worse than the fitness of all matchings. The otima of f are maximum matchings and the local otima are maximal matchings. A maximal matching is a matching without augmenting aths of length 1. Here, we call such an augmenting ath, consisting of only one free edge, a selectable edge. If s reresents a matching without selectable edges, fliing only one bit of s either roduces a new search oint reresenting a smaller matching or a search oint reresenting a non-matching. Hence, randomized search heuristics with an elitist selection strategy need to fli at least two bits to overcome local otima. Indeed, it sufficestoflioneortwobitsineachstetoreachaglobally otimal solution. To see this, consider an augmenting ath P between free nodes u and v (Figure 1 to, dashed lines indicate free edges). If the first two edges or the last two edges of P fli, P shrinks by two edges. If this haens over and over again, P finally consists of only one selectable edge {u,v } (Figure 1 bottom). Now, fliing only the bit of {u,v } imroves the matching. Motivated by these observations, we consider the following randomized local search strategy RLS. u 1. Choose s {0, 1} m according to some robability distribution. 2. Reeat: Decide by a coin fli whether s is to be roduced by a 1-bit fli or 2-bit fli from s. Fli a uniformly chosen bit of s or a uniformly chosen air of bits accordingly. If f(s ) f(s t 1), set s := s. The aim is to analyze the number of fitness evaluations until RLS roduces a search oint s such that f(s) is maximal. This random number is called otimization time and we are interested in its exected value, the exected otimization time. We assume that the coin fli in the mutation ste is fair; however, this is not essential. It is only required that the robabilities of both outcomes of the coin fli are Ω(1). Then, RLS is essentially the (1+1) EA restricted to 1- and 2-bit flis. Indeed, the robability that a certain bit or a certain air of bits fli is Θ(1/m) res.θ(1/m 2 ) for both the (1+1) EA and RLS. We believe that the results resented in this aer also aly to the (1+1) EA but it is not obvious that the roofs can be adated. For an exerimental study see [2]. Next we summarize related results regarding the maximum matching roblem and RSHs. Then we study the effect of 1- and 2-bit flis in Section 3, which rovides some basic lemmas. In Section 4, we consider comlete trees and obtain a time bound of O(m 7/2 ) for RLS. For arbitrary trees, our bound of O(D 2 m 4 ) in Section 5 deends also on the diameter D of the tree. 2. RELATED WORK Sasaki and Hajek [12] show that simulated annealing (SA) and the Metroolis algorithm (MA) efficiently find aroximations for maximum matchings. These randomized search heuristics find a (1 + ε)-otimal solution (i. e., a matching such that maximum matchings are at most by a factor of (1+ε) larger) in exected olynomial time where the degree of the olynomial deends on ε>0. We rove the same for the (1+1) EA and RLS in [4]. The idea is that matchings which are at most (1 + ε)-otimal include an augmenting ath whose length is bounded by 2 1/ε 1 (comare [6]). The robability to fli the ath s edges in the desired order is then large enough. Remarkably, SA, MA, and the (1+1) EA utilize no knowledge about the maximum matching roblem; RLS utilizes no knowledge excet that 1- and 2-bit flis are sufficient. They are general search heuristics, i. e., they are not designed to otimize a articular class of fitness functions, and work in the black box scenario. This includes that access to the grah instance is limited to the fitness function, i. e., the grah is assumed to be hidden to the heuristics. He and Yao [5] come u with a similar aroximation result for a oulation based EA; however, their EA needs to access the grah instance and the variation oerators seem to be tailored to the maximum matching roblem to some extent. Although all mentioned heuristics can be viewed as olynomial time aroximation schemes for the maximum matching roblem, they may fail when it comes to exact otimization. Sasaki and Hajek [12] rove an exonential exected otimization time for MA and SA and a articular grahclasswhenthesearchstartswiththeemtymatching. Sasaki [11] roves with another technique that there exist starting oints with an exonential exected otimization time for these grahs. In [4], we rove an exonential exected otimization time for RLS and the (1+1) EA for any starting oint (excet the otimum). In fact, it can 540

3 be shown that many starting oints (including the emty matching) lead to an exonential otimization time even with an overwhelming robability. For these starting oints, multi-start strategies of RLS and the (1+1) EA will also be unsuccessful. In site of carefully chosen worst-case examles, we believe that simle randomized search heuristics including the (1+1) EA and RLS are able to find maximum matchings for simle grahs efficiently. In [4], we resent the first ositive results on the otimization time of evolutionary algorithms for a class of grahs, namely ath grahs. Path grahs are trees consisting of m edges laid end to end such that the edges form a ath of length m. For every starting oint, the exected otimization time of the (1+1) EA and RLS is O(m 4 ). One can also rove that some second-best matchings, i. e., matchings which are by one edge smaller than maximum matchings, are starting oints with an exected otimization time of Ω(m 4 ). Hence, the final otimization ste from second-best to maximum matchings is essential. Our intuition is that simle randomized search heuristics are successful if the number of otential augmenting aths is olynomially bounded in most situations during the otimization rocess. In a tree with n nodes and m = n 1 edges, there is one and only one (simle) ath between any two nodes. Hence, there are at most `n 2 = O(m 2 )otential augmenting aths at any time. 3. PRELIMINARIES First we show that RLS finds some matching quickly. Lemma 1. For any initial search oint and any grah, RLS finds a matching in an exected time of O(m log m). Proof. Let = r k be the sum of the vertex enalties for the search oint s (see equation ( ) in Section 1). As r m+1, the fitness function rewards any decrease of and enalizes any increase of. The sum of all vertex degrees is 2m. Hence,k is less than 2m and there are at least k/2 m edges chosen by s whose elimination decreases k. The robability of a secific 1-bit fli equals Θ(1/m). Hence, the exected waiting time to decrease k is bounded by O(m/k). Summing u for 1 k<2m yields the claim. In the analyses, we distinguish situations with and without selectable edges. The next lemma considers a situation with a selectable edge and states that the next ste changing the situation will likely imrove the matching. Lemma 2. Given an arbitrary tree and a matching with at least one selectable edge. In exected time O(m), RLS has imroved the matching or there is no selectable edge but an augmenting ath of length 3. The matching is imroved with a robability of at least 1/2. Proof. We rove the lemma by roving the following two claims for all search oints describing matchings with a selectable edge: In the next ste, the robability of imroving the matching is at least 1/(2m), and the robability of destroying all selectable edges without imroving the matching is at most 1/(2m). The first claim is obvious since 1/(2m) is the robability of fliing exactly a secified selectable edge. Now we e e u v w (a) P z } { x 0 x 1 x 2 x l (b) Figure 2: (a) In an acceted ste, a free edge e incident uon a free node u and a matching edge e adjacent to e fli. (b) The augmenting ath P and its neighborhood. The figure only shows P and nodes and edges adjacent to P. rove the second claim. A ste fliing only matching edges is not acceted. A ste fliing only free edges is not acceted or it imroves the matching. Hence, we only have to consider mixed 2-bit flis choosing a free edge e and amatchingedgee. Since e must not become selectable, e has to be adjacent to e imlying that e is not selectable. To destroy a given selectable edge e, e must also be adjacent to e. In summary, the free edge e is adjacent to the selectable edge e at its free endoint, and at the other endoint, e is adjacent to the matching edge e. This shows that an augmenting ath of length 3 is created if e and e fli. By the matching roerty, the choice of e determines e, and, in a tree, e determines e between e and the given edge e. Hence, the ossible airs {e, e } are airwise disjoint imlying that their number is bounded above by (m 1)/2. The robability to fli any of these airs is at most ((m 1)/2) (1/2) `m 1 =1/(2m). 2 Hence, on average, we have to create a selectable edge at most twice before the matching is imroved. To bound the exected time to imrove the matching it now suffices to bound the exected time to create a selectable edge. To this end, we study a situation without selectable edges and ick an augmenting ath. The aim is to bound the time until the selected ath finally consists of only one selectable edge. First, we consider only one ste, i. e., one iteration of the loo of RLS. Obviously, 1-bit flis will not be acceted. The next lemma describes the effect of 2-bit flis on an augmenting ath and claims that it is not too unlikely that the ath gets shorter. Lemma 3. Let P =(x 0,x 1,...,x l ) be an augmenting ath in a tree with resect to a matching without a selectable edge. An acceted mutation ste of RLS reserves the current matching size and either leaves P unchanged, lengthens P by two edges at one end, or shortens P by a multile of two edges at one end. If P is changed, the robability that P shrinks is at least 2/(deg(x 0)+deg(x l )). Proof. Because no edge is selectable, augmenting aths have at least three edges. Hence, only stes reserving the matching size are acceted. This imlies that only 2-bit flis choosing a free edge e and a matching edge e are acceted. Since e is free but not selectable, we obtain a new matching only if e is adjacent to e, i.e.,e = {u, v} and e = {v, w}, and u is free (see Figure 2(a)). We investigate acceted stes changing P. There are four ossibilities for the free edge e = {u, v} (see Figure 2(b)): 541

4 e is an outer edge of P, namely, e = {x 0,x 1} or e = {x l 1,x l }, e is an inner edge of P, e does not belong to P but its free endoint u does, neither e nor its free endoint u belong to P. In the first case, e is adjacent to e and lies on P. Fliing e and e shortens P by two edges. The second case is imossible because no endoint of e would be free. In the third case, either u = x 0 or u = x l. Fliing e and e lengthens P.Sincewbecomes a free node, the length increases by exactly 2. In the last case, P is only changed if the matching edge e connects two inner nodes of P. (To see this, observe that each edge incident uon a node of P is either free or it is a matching edge belonging to P.) The node w is then an inner node of P and becomes a free node. Either (x 0,...,w)or (w,...,x l ) becomes an augmenting ath. Since every augmenting ath has odd length, the new ath is by an even number of edges shorter than the old ath P. We have seen that the length of P can only increase if u = x 0 or u = x l. In such a ste, the fliing free edge e incident uon x 0 or x l determines which matching edge can fli. Therefore, there are at most deg(x 0)+deg(x l ) 2 2-bit flis increasing the length of P and at least 2 2-bit flis decreasing the length of P. This roves the claimed robability. In many situations, we will consider the number R of relevant stes rather than the total number of stes T. The definition of a relevant ste will deend on the situation. If an exected number of E(R) relevant stes is necessary to reach some target and in any situation the next ste is relevant with a robability of at least then the exected total number of stes E(T )isatmost 1 E(R). 4. COMPLETE k-ary TREES We investigate comlete k-ary trees, i. e., rooted trees where inner nodes have k successors and all leaves have the same distance to the root. (Paths grahs can be considered as comlete unary trees where k =1.) Fork 2, the diameter D of ath grahs is Θ(log k m)andatmostlog k m. Considering ath grahs and 2-bit fli again, there is one ossibility to shorten an augmenting ath at each end oint and one ossibility to lengthen it at each end oint in a tyical situation (see Figure 1 to). Thus, the robabilities of lengthening and shortening the ath are the same; they describe a fair game. The game gets unfair for k-ary trees and k 2. There are k ossibilities to lengthen an augmenting ath if the end oint is an inner node. Still there is only one ossibility to shorten it. On the one hand, 2-bit flis relevant to a secific augmenting ath now tend to lengthen the ath. One might exect that this leads to a larger otimization time. On the other hand, the length of all aths is now bounded logarithmically. The next lemma bounds the length of the shortest augmenting ath with resect to amatchingm. This gives rise to an uer bound which is in fact smaller than the lower bound for ath grahs. Lemma 4. Given a comlete k-ary tree with m edges and k 2. If M is a maximum matching and M amatching with m 1 edges less than M, there is an M-augmenting ath whose length is strictly less than L := 2 log k (2m/m ). 1 r l 1 l q s q q Figure 3: The Markov chain M in Lemma 5. Proof. Thenodesofthetreeindistanceh from the root belong to level h. Letd denote the deth of the tree imlying d log k m. Considering the symmetric difference M M, we obtain a set of m node-disjoint M-augmenting aths (see Theorem 1 in [6]). Let us assume that the length of a shortest augmenting ath in the set is l. A simle ath in a tree can contain at most two nodes from each level. This imlies that each augmenting ath in the set has an odd length of at least l<2d and contains at least one node on a level d d l/2. Hence, m is bounded above by the number of nodes on the levels 0,...,d l/2 imlying that m k k d l/2 = kd l/ k 1 < k k 1 kd l/2 2mk l/2. Solving for l yields the roosed bound l<l. The next lemma gives the hitting time of a random walk that is essential in all our analyses. The random walk is similar to the random walk describing the gambler s ruin roblem but has a reflecting barrier. Lemma 5. Given the homogeneous Markov chain M with state sace S = {0,...,l}, initial state l 2, andositive transition robabilities (0, 0) = 1, (1, 0) = r, (1, 2) = s, (i, i 1) = for i {2,...,l}, (i, i +1) = q for i {2,...,l 1}, (l, l) =q, where0 <,r<1, q =1, and s =1 r (see Figure 3). The exected time to reach state 0 for the first time starting in state l is 8 2 l >< 2 + r 3 l 1 +2 if = q =1/2, r h l,0 = ` q! 1 1 l >: q q 1 + s q l 1 s l + 1 else. r r r In articular, for = q = r = s =1/2, h l,0 = l 2 + l. Proof. We claim that h 1,0 = 1 + q h2,1 and, for j 2, h j,j 1 = 8 >< 2(l j +1) if = q =1/2, ` q l j+1 1 >: if q. q Summing u the terms h j,j 1 for j {1,...,l} yields the lemma. By the law of total robability, h 1,0 =1+ 0+ q (h 2,1 + h 1,0). This imlies the first art of the claim. We rove the second art by induction on j. In accordance with the claim (for = q =1/2 andfor q), h l,l 1 =1/ since the transition (l, l 1) is the only transition leaving state l with ositive robability. For l 1 j 2, h j,j 1 =1+ 0+q(h j+1,j + h j,j 1) imlying that h j,j 1 = 1 (1 + q hj+1,j). 542

5 We aly the induction hyothesis and consider the cases = q =1/2 and q searately. In the first case, we obtain h j,j 1 = 1 1+q`2(l (j +1)+1) = 2(l j +1), and in the second case, ` h j,j 1 = 1 q l j! 1+q 1 q q = 1 ` q l j+1 ` ( q 1) + q q l j+1 1 =. q q This roves the claim and finishes the roof. Now we are reared to analyze RLS on totally balanced trees. Theorem 1. For any choice of the first search oint, the exected time until RLS finds a maximum matching on a comlete k-ary tree, k 2, isboundedbyo(m 7/2 ). Proof. By Lemma 1, the exected time to find a matching is small enough. Afterwards, we estimate the exected number of relevant stes to imrove the matching. In each ste,wechoosesomeshortestaugmentingathp.now,ina situation without selectable edges, a ste is called relevant if it is acceted and changes P. In a situation with at least one selectable edge, any ste imroving the matching or destroying all selectable edges is relevant. The exected number of all stes is then only by a factor of O(m 2 ) larger than the exected number of relevant stes because the robability of a relevant ste is Ω(1/m 2 ) in either situation. Assume we can guarantee that there is always some augmenting ath of length at most l. In situations with a selectable edge, we aly Lemma 2. In a ste changing the situation, the robability to imrove the matching is at least r =1/2, and the robability to create a ath of length three is at most 1/2. In situations without a selectable edge, we essimistically relace shortenings of the considered ath P with shortenings by only two edges. By Lemma 3, this leads to a robability of at least =1/(k + 1) of shortening this ath in relevant stes. We can now reresent the current length j of the ath by the state j/2 of the Markov chain in Lemma 5 with state sace {0,..., l/2 }. If we essimistically start with a ath of length l, the exected number of relevant stes until the matching is imroved (state 0 is reached) is at most k +1 k l/2 1 + k l/2 1 ( l/2 +1) +2 = O(k l/2 ). k 1 k 1 Until m 2m 1/2, Lemma 4 guarantees that there is some augmenting ath P of length at most l < log k m. This leadstoanexectednumberofo(m 1/2 ) relevant stes for an imrovement. Since M m, we aly this bound only O(m) times imlying that the exected number of relevant stes until m 2m 1/2 is O(m 3/2 ). Afterwards, we use the bound l 2log k m which is a trivial bound on the diameter D of the tree. Then we obtain an uer bound of O(m) on the number of relevant stes to imrove the matching. Since we aly the last bound only 2m 1/2 times, this leads to an additional exected number of O(m 3/2 ) relevant stes. This roves the theorem as argued at the beginning of this roof. 5. GENERAL TREES Path grahs have turned out to be the hardest k-ary trees for RLS. The uer bound of O(m 4 ) for ath grahs in [4] cannot be imroved if the search starts with an arbitrary search oint (see Section 2). Indeed, we conjecture that ath grahs are the hardest trees and the best ossible bound for RLS on trees is O(m 4 ). Here, we rove a weaker uer bound of O(D 2 m 4 ) on the exected otimization time where D denotes the diameter of the tree. We investigate an arbitrary search oint s describing a non-maximum matching. We are interested in the exected timetoobtainasearchointdescribingamatchingofthe same size with a selectable edge. We choose an arbitrary augmenting ath P connecting u and v. The endoints u and v of P move around. A ste is called u-relevant, if the endoint u moves. Lemma 6. Given a tree and a non-maximum matching without selectable edges. Let u denote an endoint of an arbitrary augmenting ath. RLS creates a selectable edge in an exected number of O(D 2 m) u-relevant stes. The roosed bound on the exected otimization time for trees follows from Lemma 6 and our results in Section 3. Theorem 2. For any choice of the first search oint, the exected time until RLS finds a maximum matching in a tree is bounded by O(D 2 m 4 ). Proof. Theexectedtimetocreateamatchingissmall enough (Lemma 1). Once we have a matching, it takes an exected number of O(D 2 m) u-relevant stes to roduce a selectable edge (Lemma 6) and each ste is u-relevant with a robability of Ω(1/m 2 ). Hence, the exected number of stes to roduce a selectable edge is O(D 2 m 3 ). When a selectable edge exists, a decision is made within an exected number of O(m) stes: With a robability of at least 1/2 the matching is imroved (Lemma 2). Using Markov s inequality we obtain that some O(D 2 m 3 ) stes imrove the matching with a robability of Ω(1); otherwise, we start over and wait for a new selectable edge. Hence, an exected number of O(D 2 m 3 ) stes imrove the matching. For a maximum matching, O(m) imrovements are sufficient. This leads to the claimed O(D 2 m 4 ) bound. It remains to rove Lemma 6. We focus on the distance between the endoints u and v. If this distance is only 1, the edge {u, v} is selectable but we may be lucky and roduce a selectable edge somewhere else earlier. To obtain recise statements we fix some node w as the root of the tree. For afixedrootw, T (r) is the subtree rooted at node r and T (r) = O(m) is its number of nodes. Claim 1. Assume there is no selectable edge, r w is the initial osition of u, andv/ T (r). The exected number of u-relevant stes until one of the following three events haens is bounded by T (r) : u leaves T (r), v enters T (r), a selectable edge is created. Proof. It suffices to bound the number of u-relevant stes until the first event u leaves T (r) haens assuming that the other two events do not haen. The roof investigates the random walk of u influenced by stes where an edge adjacent to u flis. Other stes moving u are only advantageous to us since u moves closer to r or leaves T (r) 543

6 r u r 1 r s T 1 T t Figure 4: The subtree T (r). Initially, the endoint u is at the root of T (r). r u... v T v (a)... u v T u T v (b) (see Lemma 3). The roof imlicitly uses the argument that a large degree of u imlies that not many subtrees of T (u) can be large. In the considered situation, the conditions of Lemma 3 hold and we may essimistically assume that the ath is never shortened by more than two edges in u-relevant stes. It is imortant to observe that u can only visit nodes in T (r) with an even distance to its initial osition, the root r of T (r). The roof is by induction on T (r). If T (r) =1,thenT (r) is a leaf and the exected number of u-relevant stes is at most 1. Now, let T (r) > 1andlets denote the degree of the root r. LetT 1,...,T t denote those subtrees of T (r) such that there are two edges between r and the root of each T i (Figure 4). Whenever u is at r, r is connected to s 1nodesr 1,...,r s 1 in T (r) andtoitsarent by free edges. Each node r j may be adjacent to several roots of the subtrees T 1,...,T t, but each r j is connected to at most one of these roots by a matching edge. Hence, at most s 1 roots of subtrees T i are reachable for u. When u leaves a subtree T i, it can only return to r or some node on the ath outside T (r). W. l. o. g. let T 1,...,T s 1 be s 1 largest subtrees in {T 1,...,T t}. For a subtree T of T (r) and u starting at the root of T,letE(T) denote the exected number of u-relevant stes until u leaves T (or a selectable edge is created). By the law of total robability, E(T (r)) 1 s 1+ 1 s `1+E(T1)+E(T (r)) s `1+E(Ts 1)+E(T (r)). Solving for E(T (r)) and using the induction hyothesis yields E(T (r)) s + T T s 1. BecauseT (r) includes all subtrees T i, the nodes r 1,...,r s 1, andr, the right-hand side of the last inequality is at most T (r). Claim 2. Assume there is no selectable edge, r is the initial osition of u, andv T (r). The exected number of u-relevant stes until one of the following three events occurs is bounded by O( T (r) ): u leaves T (r) and v remains in T (r), u and v are in the same roer subtree of T (r), a selectable edge is created. Proof. It is essential that the distance between u and r remains even. (Then the distance between v and r is odd.) In the following, T u and T v denote subtrees of T (r) rooted by a node on the level below r. Ifu is in a roer subtree of T (r), T u denotes the subtree of T (r) currently containing u; analogously for v (see Figure 5). First, we show that v cannot leave T (r) in the time we consider. If u is at r, thenv cannot leave T (r). If u leaves T (r), the second stoing criterion is fulfilled. If u moves to a node in T v (Figure 5(a)), Figure 5: The subtree T (r). (a) A secial situation and (b) the situation when u has moved into a subtree of T (r). the second stoing criterion is fulfilled. The endoint v might only leave T (r) ifu first moves to a subtree T u T v (Figure 5(b)). If this is the case, the ath P between u and v visits r. By Lemma 3, relevant stes can increase and decrease the length of P only by a multile of two edges at either end. Since u initially had an even distance to r, namely 0, u keesanevendistanceandv kees an odd distance to r. Now, if v leaves its subtree T v in a shortening ste, then v moves to some node on the ath P.Sincevcan- not visit r (odd distance), the new location of v could only be in T u and the last stoing criterion would be fulfilled first. Hence, v cannot leave T (r) before one of the stoing criteria is fulfilled. Now it is clear that we can uer bound the exected number of u-relevant stes until one of the stoing criteria is fulfilled by the exected number of u-relevant stes until one of the following events haens: Given that v stays in its subtree T v either u leaves T (r) oru moves into T v.this can be done in the same way as in the roof of Claim 1. The only difference is that the terms for subtrees contained in T v can now be bounded by 0. This can only decrease the uer boundo( T (r) ) on the exected number of u-relevant stes. Note that the case r = w where u cannot leave T (r) is included. In the following let r be the node where the ath from u to the root w of the tree and the ath from v to w meet, i. e., r is the root of the smallest subtree containing both u and v. Let d(r) be the deth of r, i. e., the distance between r and the root w. We consider the random variable d(r)with resect to time. A selectable edge is created if the distance between u and v equals 1. This is certainly the case if r is the root of a subtree with only two levels. This haens if d(r) D 1. The idea is to rove that d(r) tends to grow. The situation deicted in Figure 5(a) is of articular interest in our analysis. There is no selectable edge, one endoint of P is located at the root r of T (r), and the other endoint is contained in T (r). We call this a secial situation. W.l.o.g. the endoint at r is named u and the other v. Claim 3. In a secial situation it is ossible to define disjoint events E 1,...,E 4 such that E 1 is the event that d(r) decreases (and it can decrease by at most 2). E 2 imlies that d(r) increases by at least 2, 544

7 E 3 imlies that d(r) increases by at least 1, and E 4 is the event that a selectable edge is created. The exected number of u-relevant stes for one of these events to haen is O(m). The robability that E 1 haens first is less than or equal to the robability that E 2 haens first. When one of the events E 1,...,E 3 has haened, the exected number of u-relevant stes until a secial situation is reached again (or E 4 haens) is O(m). Inthistime,d(r) cannot decrease. Proof. The initial situation is assumed to be a secial situation. Hence, the ath P connecting u and v has a minium length of 3 edges and T (r) includes at least 4 levels. We consider only situations without selectable edges. Otherwise, E 4 has haened and we can sto. Furthermore, u (initially located at r) kees an even distance to r until a new secial situation is reached. By Claim 2, after an exected number of O(m) u-relevant stes, both endoints are in the same roer subtree of T (r), or u has left T (r). First assume that v finishes such a hase, i. e., v leaves T v (see Figure 5(b) and the roof of Claim 2 for the definition of T u and T v). From the roof of Claim 2, we also know that v cannot reach r (odd distance) but a node in T u on the ath P. If this haens (event E 3), the deth d(r) increases by at least 1 and the new situation is a secial situation. Now assume that u finishes the hase and consider the last ste. If the length of P decreases by more than 2 in the last ste then u enters T v and reaches a node on the ath P (event E 2). The deth d(r) increases by at least 2 and a secial situation is reached. Otherwise, u visits r before the last move. Given that u finishes the hase in the next ste, it either moves to T v with a robability of at least 1/2 or leaves T (r) with a robability of at most 1/2 (Figure 5(a)). (If r = w, evente 1 is not ossible.) In the first case (event E 2 ), d(r) increases by 2 and a secial situation is reached. In the second case (event E 1), d(r) decreases by at most 2: The endoint u may move to its re-redecessor in the tree (Figure 6(a)). Then a secial situation is reached. Another otion is that u moves to some sibling of r, say r (Figure 6(b)). Then the new situation is not a secial situation. In the following, we again assume that no selectable edge is created. By Claim 1, after an additional waiting time of an exected number of O( T (r ) ) =O(m) u-relevant stes, u has left T (r ) and is back at r (oratsomenodeonthe ath inside T (r)) such that a secial situation is reached, or v is also in T (r ). In the last case, v must have visited the arent of r and r (or a node on the ath inside T (r )) first such that a secial situation has been reached. Hence, there is no additional decrease of the deth d(r) in the additional waiting time for a new secial situation. The events E 2 and E 1 can only occur in the same situation (where u finishes the hase and u visits r before the last move). Now, given that one of these two events haens in the next ste, Prob(E 2 ) 1/2. Hence, the robability that the event E 2 := E 2 E 2 finishes the hase is only larger than the robability that event E 1 finishes the hase. Moreover, we have shown that after an event E 1, E 2, or E 3 has haened, a secial situation is reached within an exected number of O(m) u-relevant stes where d(r) isnot decreased. Now we can combine our results and rove Lemma 6. u r r u v T (r) v (a) r r r u r u v T (r) v (b) Figure 6: If d(r) decreases, u moves to the reredecessor of r (a) or to a sibling r of r (b). Proof of Lemma 6. Initially, we choose w := u to be the root of the tree. Then d(r) = 0 and we start in a secial situation. This imlies that T (r) includes at least 4 levels. When a situation is reached where T (r) includes only two levels, the considered ath P is a selectable edge since its length must be odd. In a situation where T (r) includes only three levels, the augmenting ath is a selectable edge or the the next P -relevant ste creates a selectable edge. To see this, observe that the length of P is then 1 or 3. There is nothing to show if the length is 1. If the length is 3, neither u nor v can be at the root of T (r) andtheath could be lengthened by at most one edge at u or v. Hence, by Lemma 3, the next P -relevant ste can only shorten P. This imlies that the exected number of u-relevant stes to create a selectable edge is by at most 1 larger than the exected number of u-relevant stes for the event d(w) D 2. By the O(m) bounds in Claim 3, it is sufficient to rove an uer bound of O(D 2 ) on the exected number of the events E 1, E 2,andE 3 until the event d(w) D 2 occurs. We slow down this stochastic rocess by assuming that E 1 decreases d(r) byexactly2,e 2 increases d(r) byexactly2, and Prob(E 1 haens first) = Prob(E 2 haens first). We rove that the robability of the event d(r) D 2ina hase including D 2 /2 +2D of the events E 1,...,E 4 is at least 1/2. The claim then follows since we may start over and choose a new root w if the hase were unsuccessful. If the number of E 3-events in the hase is at least D 2, the result follows since, with robability 1/2, event E 2 haens at least as often as event E 1. Otherwise, there are at least D 2 /2+D events E 1 and E 2, and we disregard the hel of E 3-events. Now the events E 1 and E 2 describe a symmetric random walk as analyzed in Lemma 5 for = q = r = s = 1/2. Starting there in state l = (D 2)/2 (standing for d(r) = 0), the exected number of stes to reach the state 0 545

8 is less than D 2 /4+D/2. By Markov s inequality, D 2 /2+D stes are successful with a robability of at least 1/2. 6. CONCLUSIONS Randomized search heuristics can be fooled by carefully constructed grahs but RLS finds maximum matchings for simle grahs (trees) in exected olynomial time. The results indicate that randomized search heuristics can use algorithmic ideas not known to the designer of the algorithm. Our results contribute to the understanding of how simle randomized heuristics work. It would be interesting to see how efficient related heuristics like SA and MA erform on trees. 7. REFERENCES [1] C. Berge. Two theorems in grah theory. Proceedings of the National Academy of Sciences of the United States of America, 43: , [2] P. Briest, D. Brockhoff, B. Degener, M. Englert, C. Gunia, O. Heering, T. Jansen, M. Leifhelm, K. Plociennik, H. Röglin, A. Schweer, D. Sudholt, S. Tannenbaum, and I. Wegener. Exerimental sulements to the theoretical analysis of EAs on roblems from combinatorial otimization. In Proceedings of the 8th Conference on Parallel Problem Solving from Nature (PPSN 04), volume 3242 of Lecture Notes in Comuter Science, ages Sringer, [3] J. Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17: , [4] O. Giel and I. Wegener. Evolutionary algorithms and the maximum matching roblem. In Proceedings of the 20th Annual Symosium on Theoretical Asects of Comuter Science (STACS 03), volume 2607 of LNCS, ages Sringer, [5] J. He and X. Yao. Time comlexity analysis of an evolutionary algorithm for finding nearly maximum cardinality matching. Journal of Comuter Science and Technology, 19(4): , [6] J. E. Hocroft and R. M. Kar. An n 5/2 algorithm for maximum matchings in biartite grahs. SIAM Journal on Comuting, 2(4): , [7] M. Jerrum. Large cliques elude the Metroolis rocess. Random Structures and Algorithms, 3(4): , [8] M. Jerrum and G. B. Sorkin. The Metroolis algorithm for grah bisection. Discrete Alied Mathematics, 82: , [9] S. Micali and V. V. Vazirani. An O( V E ) algorithm for finding maximum matching in general grahs. In Proceedings of the 21st Annual Symosium on Foundations of Comuter Science (FOCS 80), ages IEEE, [10] F. Neumann and I. Wegener. Randomized local search, evolutionary algorithms, and the minimum sanning tree roblem. In Proceedings of the 6th Genetic and Evolutionary Comutation Conference (GECCO 04), volume 3102 of LNCS, ages Sringer, [11] G. Sasaki. The effect of the density of states on the Metroolis algorithm. Information Processing Letters, 37(3): , [12] G. H. Sasaki and B. Hajek. The time comlexity of maximum matching by simulated annealing. Journal of the ACM, 35: , [13] J. Scharnow, K. Tinnefeld, and I. Wegener. Fitness landscaes based on sorting and shortest aths roblems. In Proceedings of the 7th Conference on Parallel Problem Solving from Nature (PPSN 02), volume 2439 of LNCS, ages Sringer, [14] V. V. Vazirani. A theory of alternating aths and blossoms for roving correctness of the O( VE) maximum matching algorithm. Combinatorica, 14:71 109, [15] I. Wegener. Simulated annealing beats Metroolis in combinatorial otimization. In Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP 05), volume 3580 of LNCS, ages , [16] C. Witt. Worst-case and average-case aroximations by simle randomized search heuristics. In Proceedings of the 22nd Annual Symosium on Theoretical Asects of Comuter Science (STACS 05), volume 3404 of LNCS, ages Sringer,