Two Algorithms for Maximum and Minimum Weighted Bipartite ...

Two Algorithms for Maximumand Minimum Weighted

Bipartite Matching

Advisor: Prof. Yuh-Dauh Lyuu

Hung-Pin Shih

Department of Computer Science and Information EngineeringNational Taiwan University

Abstract

This thesis applies two algorithms to the maximum and minimum weightedbipartite matching problems. In such matching problems, the maximizationand minimization problems are essentially same in that one can be trans-formed into the other by replacing the weight on each edge with an inverseof the weight. Depending on the algorithms we used, we will choose the maxi-mization or minimization problems for illustrations. We apply the ant colonyoptimization (ACO) algorithm on a minimum weighted bipartite matchingproblem by transforming the problem to a traveling salesman problem (TSP).It may seem that makes the problem more complicated, but in reality it doesnot. We call this algorithm “ant-matching,” which can solve any weightedbipartite matching problems with or without a perfect matching. Besides, wealso apply the Metropolis algorithm to solve the maximum weighted bipartitematching problem. To analyze the performance on these two algorithms, wecompare the algorithms with the exact Hungarian algorithm, a well-knowncombinatorial optimization method for solving the weighted bipartite match-ing problem.

Keywords: Metropolis algorithm, Ant colony optimization, Maximumweighted bipartite matching, Minimum weighted bipartite matching

Contents

1 Introduction 4

2 Preliminaries 52.1 Maximum/Minimum Weighted Bipartite Matching . . . . . . 5

2.1.1 Weighted Bipartite Graph . . . . . . . . . . . . . . . . 52.1.2 Maximum/Minimum Weighted Bipartite Matching . . 5

2.2 Ant Colony Optimization . . . . . . . . . . . . . . . . . . . . . 62.3 Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . . . 7

3 Maximum/Minimum Weighted Bipartite Matching Prob-lems 93.1 ACO Algorithm for Minimum-Weighted Bipartite Matching . 9

3.1.1 Transforming Minimum-Weighted Bipartite Matchingto Traveling Salesman Problem . . . . . . . . . . . . . 10

3.1.2 Using ACO Algorithm to Solve Minimum-Weighted Bi-partite Matching . . . . . . . . . . . . . . . . . . . . . 12

3.1.3 The Complexity of Ant-Matching Algorithm . . . . . . 133.2 Metropolis Algorithm for Maximum-Weighted Bipartite

Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 The Complexity of Metropolis Algorithm . . . . . . . . 15

4 Experimental Results 174.1 Comparison of Complexity . . . . . . . . . . . . . . . . . . . . 174.2 Graph Sampling Algorithm . . . . . . . . . . . . . . . . . . . 184.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3.1 Results on Complete Bipartite Graphs . . . . . . . . . 194.3.2 Results on Sparse Bipartite Graphs . . . . . . . . . . . 204.3.3 Results on Ant-Matching Algorithm . . . . . . . . . . . 234.3.4 Results on Metropolis Algorithm . . . . . . . . . . . . 25

5 Conclusions 35

1

List of Figures

3.1 Transforming a bipartite matching problem to a TSP: (1) Theoriginal bipartite graph G. (2) Add u′

1 and its correspondingedges. (3) The final graph G′. . . . . . . . . . . . . . . . . . . 10

4.1 The running time of each algorithm on the complete bipartitegraph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 The running time of each algorithm on the bipartite graphwhere each vertex ui ∈ U has at most 2% ∗ |V | edges. . . . . . 21

4.3 The running time of each algorithm on the bipartite graphwhere each vertex ui ∈ U has at most 5 edges. . . . . . . . . . 22

4.4 The running times of the ant-matching algorithm on each dif-ferent bipartite graph. . . . . . . . . . . . . . . . . . . . . . . 24

4.5 The weights found by the ant-matching algorithm as propor-tions to the weights of the optimal solution on each differentbipartite graph. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6 The weight found by the Metropolis algorithm as proportionsto the weights of the optimal solution on each different bipar-tite graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2

List of Tables

4.1 Comparison of complexity . . . . . . . . . . . . . . . . . . . . 174.2 The weights as proportions to the weights of the optimal so-

lution on the complete bipartite graph. . . . . . . . . . . . . . 194.3 The weights as proportions to the weights of the optimal so-

lution on the bipartite graph where each vertex ui ∈ U has atmost 2% ∗ |V | edges. . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 The weights as proportions to the weight of the optimal solu-tion on the bipartite graph where each vertex ui ∈ U has atmost 5 edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.5 Results on each algorithm for complete bipartite graphs. . . . 264.6 Results on each algorithm for the bipartite graphs where each

vertex has at most 50% ∗ |V | edges. . . . . . . . . . . . . . . . 274.7 Results on each algorithm for the bipartite graphs where each



vertex has at most 5% ∗ |V | edges. . . . . . . . . . . . . . . . . 304.10 Results on each algorithm for the bipartite graphs where each

vertex has at most 2% ∗ |V | edges. . . . . . . . . . . . . . . . . 314.11 Results on each algorithm for the bipartite graphs where each

vertex has at most 5 edges. . . . . . . . . . . . . . . . . . . . . 324.12 Results on each algorithm for the bipartite graphs where each

vertex has at most 10 edges. . . . . . . . . . . . . . . . . . . . 334.13 Results on each algorithm for the bipartite graphs where each

vertex has at most 20 edges. . . . . . . . . . . . . . . . . . . . 34

3

Chapter 1

Introduction

The maximum/minimum weighted bipartite matching problem is a combina-torial optimization problem, and there exist many optimization algorithmsto solve the matching problem. But here we try some heuristic or sam-pling methods and expect them to generate reasonable sub-optimal solutionswithin reasonable time bounds. Specifically we use the ant colony opti-mization (ACO) and the Metropolis algorithm. In the ACO algorithm, wetransform this matching problem to a traveling salesman problem (TSP)and use the concepts of ACO algorithm as a basis for a new method to solvethe matching problem. We call the new method “ant-matching.” In theMetropolis algorithm, we randomly select a matching edge in each iterationto generate an approximate solution. Then we compare the performancesof these two algorithms with the exact Hungarian algorithm [2], which is aoptimization algorithm used to solve the weighted matching problems.

The structure of this thesis is organized as follows: In Chapter 1 andChapter 2, we define the problem of maximum/minimum weighted bipar-tite matching and introduce the background of the two above-mentionedalgorithms. In Chapter 3, we transform the weighted bipartite matchingproblem to a traveling salesman problem (TSP) and apply the concepts ofant colony optimization (ACO) algorithm as a basis for a new matching al-gorithm called “ant-matching.” In the latter part of Chapter 3, we applythe Metropolis algorithm to solve the maximum weighted bipartite matchingproblem. Finally, Chapter 4 compares these algorithms with other methodsin solving the weighted matching problems, and we conclude in Chapter 5.

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Chapter 2

Preliminaries

2.1 Maximum/Minimum Weighted Bipartite

Matching

2.1.1 Weighted Bipartite Graph

A bipartite graph G = (U, V,E) is a graph whose vertices can be dividedinto two disjoint sets U and V such that each edge (ui, vj) ∈ E connects avertex ui ∈ U and one vj ∈ V . If each edge in graph G has an associatedweight wij, the graph G is called a weighted bipartite graph.

2.1.2 Maximum/Minimum Weighted BipartiteMatching

In a bipartite graph G = (U, V,E), a matching M of graph G is a subsetof E such that no two edges in M share a common vertex. If the graphG is a weighted bipartite graph, the maximum/minimum weighted bipartitematching is a matching whose sum of the weights of the edges is maxi-mum/minimum. The maximum/minimum weighted bipartite matching canbe formulated as follows:

max / min∑

(ui,vj)∈E

wijxij

|U |∑i=1

xij = 1,∀j = 1, ..., |V |

|V |∑j=1

xij = 1,∀i = 1, ..., |U |

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xij ∈ {0, 1}

where xij is 1 denotes edge (ui, vj) is an edge in the maximum/minimumweighted bipartite matching.

2.2 Ant Colony Optimization

In the real world, ants are social insects and communicate with some inter-esting behavior. Each ant lays down pheromone on the path. When antsfind foods, they will go back to its nest. If some other ants encounter thepath, they will follow it based on the density of pheromone deposited. Thepheromone on the path that ants found initially will be reinforcing. Eventu-ally, ants will find the foods.

Over time, the pheromone on the path evaporates, thus reducing its at-tracting strength. The longer path an ant travels down and back, the moretime the pheromone has to evaporate. On a shorter path, an ant travelsdown the path and back faster, and the pheromone density remains highas the pheromone is laid on as fast as it can evaporate. Assume the antcolony converges to a path. If a shorter path is found by some ants, becausethe path is shorter than the original path that the ant colony converged to,the ants travel down the path and back faster. The density of pheromoneon the shorter path will increase gradually and the density on the originalpath will decrease. After a period of time, the ant colony will converge tothe shorter path. It will prevent the ant colony from converging to a localoptimal solution.

The ant colony optimization (ACO) algorithm comes from observing thebehavior of the ants. It simulates the ants and the pheromone evaporation onthe paths. In each iteration, the ants select the paths according to the densityof pheromone deposited. If the path has higher density, it will be selectedwith higher probability. After finding a path, the ants lay down pheromoneon the path. In the ACO algorithm, each path represents a solution. Besides,the pheromone evaporates gradually. The ACO algorithm is illustrated inAlgorithm 1.

Algorithm 1 ACO Algorithm for TSP

while termination condition not met dogenerate solutionspheromone update

end while

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The ACO algorithm has been used in solving the traveling salesman prob-lem (TSP) to find a nearly optimal solution [4]. In the TSP problem, thereare n cities. Each city has to be visited exactly once, and the tour endsat the starting city. The problem is to find a shortest tour to visit these ncities. Let dij be the distance between the city i and the city j and τij bethe pheromone on the edge connects i and j.

Each of the m ants decides independently on the city to be visited next.They base their decision on the density of pheromone τij and a heuristicfunction ηij. The probability of choosing the next city j from current city iat the tth iteration is

pkij =

[τij(t)]α(ηij)

β∑l∈Nk

l[τil(t)]α(ηij)β

,∀j ∈ Nki (2.1)

where ηij = 1dij

is a commonly used heuristic function, α and β are two

parameters to determine the relative influences of the pheromone and thedistance, and Nk

i is a list of cities that the kth ant has yet to visit.After all ants have constructed their complete tours, we update the

pheromone on each edge thus:

τij(t + 1) = (1− ρ)τij(t) +m∑

k=1

∆τ kij(t) (2.2)

where 0 < ρ ≤ 1 is the pheromone evaporation rate, ∆τ kij(t) is the pheromone

that the kth ant deposited at the tth iteration, which is defined as follows:

∆τ kij(t) =

{1/Lk(t) if(i, j) ∈ T k(t)0 if(i, j) /∈ T k(t)

(2.3)

where Lk(t) is the length of the kth ant’s tour at the tth iteration, T k(t) is thekth ant’s tour at the tth iteration. The shorter the tour the ants found, themore pheromone is deposited. After a number of iterations, the ant colonywill converge to a nearly shortest tour. Finally, select any starting city andfollow the edge with the highest density of pheromone to visit as the nextcity. After completing the tour, the tour is a solution of the TSP problem.

2.3 Metropolis Algorithm

The Metropolis algorithm is an algorithm to generate a sequence of samplesfrom a probability distribution that is hard to sample directly. It uses theMarkov chain Monte Carlo simulation to approximate the distribution [1].

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The Metropolis algorithm can generate samples from any unknown proba-bility distribution P (x) and requires only a function proportional to the den-sity of P (x) that can be calculated at x. The algorithm generates a Markovchain in which each state xt+1 depends only on the previous state xt. LetQ(xt, x′) denotes the candidate-generating density, where

∫Q(xt, x′)dx′ = 1.

The algorithm depends on the current state xt to generate a new sample x′

from Q(xt, ·)1. The sample x′ is accepted as the next state xt+1 if α ∈ U(0, 1)satisfies Eq. (2.4):

α <P (x′)Q(x′, xt)

P (xt)Q(xt, x′)(2.4)

If α satisfies Eq. (2.4), xt+1 = x′. Otherwise, the current state xt is retained,i.e., xt+1 = xt. The algorithm is illustrated in Algorithm 2.

Algorithm 2 Metropolis Algorithm

for t = 1, 2, ..., N dosample x′ from Q(xt, ·) and α from U(0, 1) {U(0, 1) is the uniform dis-tribution on(0, 1)}if α < P (x′)Q(x′,xt)

P (xt)Q(xt,x′)then

xt+1 = x′

elsext+1 = xt

end ifend forreturn {x(1), x(2), ..., x(N)}

1Lyuu: but if you do not know what Q is how can you generate x’????Hung-Pin: We don’t know Q at first. But we can observe the possible transitions betweenthe current state and the next state to find a possible Q. The possible Q may be corrector wrong, but it will describe the transitions. This is reasonable, if we don’t know theprobability distribution P (x), we observe the transitions between the current state andthe next state and infer a possible Q. We can use Q to sample the other next states andfind the unknown probability distribution P (x).

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Chapter 3

Maximum/Minimum WeightedBipartite Matching Problems

In a weighted bipartite graph G = (U, V,E), we want to find a maxi-mum/minimum weighted bipartite matching (maximum/minimum-weightedBM) whose sum of the weights of the edges is maximum/minimum. To solvethis matching problem, we present two methods. For illustrations, we willintroduce each method on a different matching problem. Specifically we ap-ply the ACO algorithm to solve the minimum-weighted BM problems andthe Metropolis algorithm to solve the maximum-weighted BM problems.

3.1 ACO Algorithm for Minimum-Weighted

Bipartite Matching

In a weighted bipartite graph G = (U, V,E), we want to find a matchingwhose sum of the weights of the edges is minimum. Here we apply theconcepts of ACO algorithm as a basis for a new method to solve the matchingproblem. First, we transform the matching problem to a TSP problem. Itmay seem that makes the matching problem more complicated, but in realityit does not. Second, we apply the ACO algorithm on the TSP problemand perform some simplifications to be explained later on solving the TSPproblem. The result is a new method for solving the minimum-weightedBM problem. In this chapter, the TSP refers to the problem of finding aminimum weighted complete tour on the complete graph with multiple edgesbetween vertices, and the number of edges between vertices will change asthe different tour selection.

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Figure 3.1: Transforming a bipartite matching problem to a TSP: (1) Theoriginal bipartite graph G. (2) Add u′

1 and its corresponding edges. (3) Thefinal graph G′.

3.1.1 Transforming Minimum-Weighted BipartiteMatching to Traveling Salesman Problem

In a weighted bipartite graph G = (U, V,E), we choose either U or V as thebase vertices and transform the minimum-weighted BM problem to a TSPproblem. We choose the vertices set U as our base vertices for illustrations.First, for each ui ∈ U , we create a corresponding virtual vertex u′

i ∈ U ′,where U ′ is the set of vertices correspond to U and |U ′| = |U |. Then, foreach (ui, vj) ∈ E where ui ∈ U, vj ∈ V , we create a corresponding back edge(u′

i, vj) ∈ E ′ with an associated weight 0, where E ′ contains the edges wecreate. Furthermore, we create another back edge (ui, u

′i) ∈ E ′ with a large

associated weight wmax. Here we define this large associated weight wmax as20 ∗maxi∈|U |,j∈|V | wij. Finally, for each virtual vertex u′

i ∈ U ′, we create theedges (u′

i, uk) ∈ E ′ to connect each vertex uk ∈ U, uk 6= ui, with an associatedweight wik = 0. Thus, we have a new graph G′ = (U, V,E, U ′, E ′). The abovesteps are illustrated in Algorithm 3 and Figure 3.1. In Figure 3.1, each backedge (u′

i, vj) ∈ E ′ is in green and (ui, u′i) ∈ E ′ is in red. Besides, the edges

(u′i, uk) connect each vertex u′

i to all the other vertices in U are in grey.In the new graph G′ = (U, V,E, U ′, E ′), on each base vertex ui ∈ U , we

define that if we choose the edge (ui, vj) ∈ E, then we will follow the edge(vj, u

′i) ∈ E ′ back to vertex u′

i ∈ U ′. Thus, for each base vertex ui ∈ U , thereare some paths ui−vj −u′

i where (ui, vj) ∈ E and (vj, u′i) ∈ E ′, and the path

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Algorithm 3 Transform Graph G to G′

Input: Weighted bipartite graph G = (U, V,E)and each edge (ui, vj) ∈ E has a weight wij.

E ′ = ∅for ui ∈ U do

create vertex u′i ∈ U ′

for (ui, vj) ∈ E doadd back edge (u′

i, vj) to E ′ with weight 0end foradd back edge (ui, u

′i) to E ′ with weight wmax

for uk ∈ U doadd edge (u′

i, uk) to E ′ with weight 0end for

end forreturn graph G′ = {U, V,E, U ′, E ′}

ui−u′i where (ui, u

′i) ∈ E ′. So there exist a edge from ui to its corresponding

virtual vertex u′i ∈ U ′. Furthermore, each virtual vertex u′

i is connected toall the other base vertices uk ∈ U . Thus, if we start from any base vertexui ∈ U on the graph G′, we can visit each base vertex in U exactly once andend at ui (a Hamiltonian cycle). If we want to find a matching M in thebipartite graph G, we can find it with a Hamiltonian cycle in the graph G′

as follows.Let Availi(V ) ⊆ V be the vertex set adjacent to the vertex ui ∈ U such

that for each vertex vj ∈ Availi(V ), each edge (ul, vj) ∈ E is not yet anmatching edge in the matching M . On each vertex ui ∈ U , the ant mayselect12 the edge (ui, vj) ∈ E followed by (vj, u

′i) ∈ E ′ where vj ∈ Availi(V ),

or select the edge (ui, u′i) ∈ E ′. If the ant select the edge (ui, vj) ∈ E followed

by (vj, u′i) ∈ E ′, we add a new matching edge (ui, vj) to M . If the ant select

the edge (ui, u′i) ∈ E ′, we do not change M . After visiting all the base vertex

ui ∈ U exactly once, we will find the matching M .

1Lyuu: what do you mean by select? what if both are possible, which one do youselect?Hung-Pin: We choose edge based on the pheromone. It will be explained later. Here wejust explain the reduction.

2Lyuu: this is strange. if you are only describing a reduction, why bring up ”select”?Also if you use ”select”, it is better to write ”the ant may select”?Hung-Pin: I replace reduction with transformation. Moreover, because there are multi-paths between ui and u′

i, the different selection has different means in the matching M . Iuse the ”select” to explain the differences and the relations between the matching problemand the TSP problem.

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Thus, if we find a tour to visit each base vertex ui ∈ U exactly onceand then return to the starting base vertex, and the total weight of thistour is minimum, we have a minimum weighted bipartite matching M onthe weighted bipartite graph G. Now, we have successfully transformed theminimum-weighted BM problem to the TSP problem.

3.1.2 Using ACO Algorithm to Solve Minimum-Weighted Bipartite Matching

After transforming the weighted bipartite graph G = (U, V,E) to a newgraph G′ = (U, V,E, U ′, E ′), we can use the ACO algorithm to solve theTSP problem on G′ and find the solutions of the original matching problem.But here we can perform some simplifications on solving the TSP problem.In the graph G′, the algorithm needs to visit all vertices in the set U andcorresponding set U ′ of U to do matching. With the simplification, becausewe define that if the ant select the base vertex ui ∈ U , then the ant will followthe paths ui − vj − u′

i or the path ui − u′i back to the vertex u′

i, we can seeU and U ′ same because if the algorithm visits the base vertex ui ∈ U , it willvisit its corresponding virtual vertex u′

i ∈ U ′ next.3 Therefore, the algorithmcan just visit each ui ∈ U and select a matching edge (ui, vj) ∈ E. It willhelp this method more efficient and applicable for the minimum-weightedBM problems. We call this new method “ant-matching.”

In the ACO algorithm, the ants choose the next vertex independentlybased their decision on the density of pheromone and a heuristic function.In the ant-matching algorithm, the ants choose the next vertex based on thesame mechanisms. In each iteration t, the ants choose the next base vertexui ∈ Availk(U)(t) based on the density of pheromone τp(i)i(t) as Eq. (3.1),

where Availk(U)(t) is the set of available base vertices that the kth ant hasyet to visit at the tth iteration and up(i) ∈ U is the predecessor of the vertexui. Let Nk

i (t) ⊆ V be the set of the kth ant’s available adjacent vertices ofui at the tth iteration. If |Nk

i (t)| > 0, each of the m ants chooses one vertexvj ∈ Nk

i (t) to do matching based on the density of pheromone τij(t) anda heuristic function ηij(t) as Eq. (2.1), then adds the weight wij to Lk(t),where Lk(t) is the length of the kth ant’s tour in iteration t. Otherwise, the

3Lyuu: I do not think this is true, as we discussed last Thursday. You want it to betrue, but it may not be true for a general TSP solver. you need to say YOUR TSP solverwill impose this conditionHung-Pin: I add a sentence before this sentence to explain the condition. Besides, atthe end of the section 3.1, I redefine the TSP problem in the transformation. It is not ageneral TSP. It likes a related TSP problem which generalized TSP deals with ”states”.Thus, I redefine the TSP problem.

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ant does nothing but adds a large associated weight wmax to Lk(t). Aftereach of m ants visiting all the vertices in the set U , update the pheromoneon each edge and start the next iteration. The pheromone deposited on eachedge (ui, uk) ∈ E ′ represents an order relation such that if one ant choosesthe vertex ui, it will choose the vertex uk next time. And a large amount ofpheromone on the edge (ui, vj) represents that the edge (ui, vj) is a preferredmatching edge. Thus, we have the order relations to do matching on thevertex set U and the preferred matching edges on each vertex ui ∈ U . Wecan find the solutions of the matching problems according to the density ofpheromone. The algorithm is listed in Algorithm 4, and Algorithm 5 is thedetail.

pkp(i)i =

[τp(i)i(t)]α∑

l∈Availk(U)(t)[τp(i)l(t)]α∀i ∈ Availk(U)(t) (3.1)

Algorithm 4 Ant-Matching Algorithm

for t = 1 to tmax dofor k = 1 to NumberOfAnts do

choose ui ∈ Availk(U)(t) based on τp(i)i(t)choose vertex vj ∈ Nk

i (t) based on τij(t) and ηij(t)end forupdate pheromone

end for

3.1.3 The Complexity of Ant-Matching Algorithm

In the ant-matching algorithm, each ant visits each vertex ui in the vertexset U and searchs each edge adjacent to ui exactly once. Thus, generatinga solution costs O(U + E). Moreover, it costs O(U2 + E) to update thedensity of pheromone on each edge (ui, uk) where ui, uk ∈ U and on each edge(ui, vj) ∈ E. If there are m ants and tmax iterations. The total complexity ofthe ant-matching algorithm is O(tmax(m(U + E) + U2 + E)).

3.2 Metropolis Algorithm for Maximum-

Weighted Bipartite Matching

In a weighted bipartite graph G = (U, V,E), we can choose the vertex set U orV arbitrarily to do matching. We choose the vertex set U for illustrations. We

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Algorithm 5 Ant-Matching Algorithm in Detail


MatchingListBest = []Weightmin = ∞for t = 1 to tmax do{Generate Solution}for k = 1 to NumberOfAnts do

Availk(U)(t) = U ;MatchingListk(t) = []Weightk(t) = 0while |Availk(U)(t)| > 0 do

choose available ui ∈ Availk(U)(t) based on τij(t)

remove ui in Availk(U)(t)if ui has available adjacent vertices Nk

i (t) ⊆ V thenchoose vj ∈ Nk

i (t) based on τij(t) and ηij

add (ui, vj) into MatchingListk(t)Weightk(t) = Weightk(t) + wij

elseadd (ui, φ) into MatchingListk(t)Weightk(t) = Weightk(t) + wmax

end ifend whileif Weightk(t) ≤ Weightmin(t) thenMatchingListBest = MatchingListk(t)

end ifend for{Update pheromone}for k = 1 to NumberOfAnts do

for l = 1 to |MatchingListk(t)| and (xl, yl) ∈ MatchingListk(t) do∆τ k

xl−1,xl(t) = 1/Weightk(t)

if yl 6= φ then∆τ k

xl,yl(t) = 1/Weightk(t)

else∆τ k

xl,xl(t) = 1/Weightk(t)

end ifend for

end forfor (ui, vj) ∈ E do

τij(t + 1) = (1− ρ)τij(t) +∑m

k=1 ∆τ kij(t)

end forfor ui, uj ∈ U, ui 6= uj do

τij(t + 1) = (1− ρ)τij(t) +∑m

k=1 ∆τ kij(t)

end fort = t + 1;

end for

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apply the Metropolis algorithm on the maximum weighted bipartite matching(maximum-weighted BM) problem. Let x be the current search state. x ∈{0, 1}|E| describes the set of matchings (ui, vj), where xij = 1 denotes amatching edge, xij = 0 denotes not.

In each iteration, choose (ui, vj) ∈ E at random and flip xij. Thus, inthe current state x, choose a new search state x′ = {x′

00, ..., x′|U ||V |} where

x′ij = 1− xij, and x′

gh = xgh, if the edges (ug, vh) 6= (ui, vj).After choosing a new state x′, using a fitness function f to decide if we

choose the new search state x′ as the next state. If f(x′) ≥ f(x), select x′. If

f(x′) < f(x), select x′ with the probability ef(x′)−f(x)

K , where K is a constant,otherwise, select x. The algorithm is illustrated in Algorithm 6.

Algorithm 6 Apply Metropolis Algorithm to Maximum-Weighted BM


loopchoose (ui, vj) ∈ E at random and flip xij

sample α from U(0, 1)if f(x′) ≥ f(x) then

select x′

elseif α ≤ e

f(x′)−f(x)K then

select x′

end ifend if

end loop

Here we choose 0m as the starting state x for the maximization prob-lems and choose a fitness function f , where f(x) = 0 for the state xwhich there exist two edges in the matching sharing a common vertex, andf(x) =

∑ui∈U,vj∈V xijwij for the state x, not which has two edges that share

the same vertex. This fitness function f will guarantee the matching is rea-sonable in each iteration.

3.2.1 The Complexity of Metropolis Algorithm

In the Metropolis algorithm, generating a new search state costs a little. Togenerate a new search state we choose a edge (ui, vj) ∈ E at random and flipxij. It will cost O(1) to generate a new search state. Then, it costs O(1) tocalculate the weight of the new search state and costs O(1) to verify if the

15

new state is reasonable. If the new search state is better than the best state,record the new search state as the best state and it will cost O(U). Thus, ifthere are c iterations, the complexity of the Metropolis algorithm is O(cU).

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Chapter 4

Experimental Results

After introducing the two methods on solving the weighted bipartite match-ing problems, we write them in programs to do some experiments and com-pare them with the Hungarian algorithm, a well-known algorithm for solvingthe weighted bipartite matching problems. In a weighted bipartite graphG = (U, V,E) and each edge (ui, vj) ∈ E has an associated weight wij.We replace each edge weight wij with 1

wijto transform a maximum weighted

bipartite matching problem to a minimum weighted bipartite matching prob-lem. Thus, we can use these algorithms to solve the maximum or minimumweighted bipartite matching problem.

4.1 Comparison of Complexity

Before the experiments, we compare the complexity first. The complexity ofeach algorithm is listed as follows:

Table 4.1: Comparison of complexity

Algorithm ComplexityAnt-Matching O(tmax(m(U + E) + U2 + E))Metropolis Algorithm O(cU)Hungarian Algorithm O(V 3)

In Table 4.1, we can easily find that the complexity of the MetropolisAlgorithm is the smallest. But we can’t easily compare the complexity of theant-matching algorithm with the Hungarian algorithm. To verify that if the

17

ant-matching algorithm is faster than the Hungarian algorithm or not, wewill do some experiments. Moreover, we also test if the Metropolis algorithmand the ant-matching algorithm can generate a reasonable solution for theweighted bipartite matching problem within a smaller time and discuss theirusability.

4.2 Graph Sampling Algorithm

Before the experiments, we build some graph generating algorithms to gen-erate the weighted bipartite graphs. There are two graph generating algo-rithms:

• Generating algorithm of complete bipartite graphs.

The generating algorithm is used to generate the complete weightedbipartite graph, which ∀ui ∈ U,∀vj ∈ V, ∃(ui, vj) ∈ E.

• Generating algorithm of sparse bipartite graphs.

The graph generating algorithm generate a weighted bipartite graphG = (U, V,E) randomly and each vertex ui ∈ U has at most a specifiednumber of adjacent vertices. It does not ensure that the generatedgraphs has a perfect matching.

We will use these algorithms to generate the graph samples to do theexperiments.

4.3 Results

In the experiments, we configure some variables at first. In the ant-matchingalgorithm, there are 200 iterations and 20 ants in each iteration and we makeα = 2 and β = 1 to make the relative influence of the pheromone doublesthat of the distance. In the Metropolis algorithm, we initialize 1, 000, 000, 000cycles to generate the new search states. After configuring the variables, westart the experiments, and the results are listed as follows. We will analyzethe results in different aspects:

• TimeThe cost of time in solving the weighted bipartite matching problems.

• WeightThe maximum weight found by each algorithm. We solve the maximumweighted bipartite matching problems in our experiments.

18

Size Ant-Matching Metropolis Size Ant-Matching Metropolis100x100 95.13% 60.87% 1100x1100 91.20% 52.07%200x200 94.21% 56.73% 1200x1200 91.10% 52.29%300x300 92.94% 55.75% 1300x1300 91.52% 51.98%400x400 92.93% 54.93% 1400x1400 90.96% 51.59%500x500 92.94% 53.81% 1500x1500 90.93% 51.64%600x600 92.67% 54.49% 1600x1600 90.95% 51.72%700x700 92.36% 52.79% 1700x1700 90.95% 50.92%800x800 91.66% 52.40% 1800x1800 90.75% 51.25%900x900 92.16% 52.84% 1900x1900 90.58% 51.59%1000x1000 91.73% 52.54% 2000x2000 90.62% 51.71%

Table 4.2: The weights as proportions to the weights of the optimal solutionon the complete bipartite graph.

4.3.1 Results on Complete Bipartite Graphs

We use the ant-matching algorithm, the Metropolis algorithm, and the Hun-garian algorithm to solve the maximum weighted bipartite matching prob-lems on each complete bipartite graph with different problem size. Table 4.5is the numeral result in solving the matching problems. In Table 4.5, the re-sult found by the Hungarian algorithm is optimal. Table 4.2 lists the weightsfound by the ant-matching algorithm and the Metropolis algorithm as pro-portions to the weights of the optimal solution. In Table 4.5, we can see thatthe weights found by the ant-matching algorithm are close to the weights ofthe optimal solution. In Figure 4.1, the running time of the ant-matchingalgorithm increases gradually depending on the sizes of matching problems.Even the running time of the ant-matching algorithm is larger than the run-ning time of the Hungarian algorithm, but the growth rate of running time ofthe ant-matching algorithm is smaller than the Hungarian algorithm. Thus,if the problem size is very large, the ant-matching algorithm will be fasterthan the Hungarian algorithm.

The weights found by the Metropolis algorithm are close to the half ofthe weights of the optimal solution. But the running time of the Metropolisalgorithm is very small. In practice, we can find the result with less numberof cycles in the Metropolis algorithm. The initialized 1, 000, 000, 000 cycles isused to raise the running time, it will help to draw the transitions on differentproblem sizes in Figure 4.1.

19

Figure 4.1: The running time of each algorithm on the complete bipartitegraph.

4.3.2 Results on Sparse Bipartite Graphs

In the sparse bipartite graph G = (U, V,E) which each vertex ui ∈ U has atmost 2% ∗ |V | edges, the numeral result of this matching problem is listedin Table 4.10. Table 4.3 is the result of weights found by the ant-matchingalgorithm and the Metropolis algorithm as proportions to the weights ofthe optimal solution. In Table 4.3, the weights found by the ant-matchingalgorithm are close to the weights of the optimal solution. And in Figure 4.2,the running time of the ant-matching algorithm is much less than that of theHungarian algorithm as the problem size increases. Thus, in a large scalebipartite matching problem, we can use the ant-matching algorithm to finda solution in a smaller time, and the result is close to the optimal solution.

In Table 4.3, we can see the weights found by the Metropolis algorithmare close to half of the weights of the optimal solution. In Figure 4.2, evenwe initializes 1, 000, 000, 000 cycles in the Metropolis algorithm, the runningtime is still small. Thus, if we want to find a matching in a rapid time anddon’t care the weight, we can use the Metropolis algorithm to find a solution.

20

Figure 4.2: The running time of each algorithm on the bipartite graph whereeach vertex ui ∈ U has at most 2% ∗ |V | edges.


Table 4.3: The weights as proportions to the weights of the optimal solutionon the bipartite graph where each vertex ui ∈ U has at most 2% ∗ |V | edges.

21

Figure 4.3: The running time of each algorithm on the bipartite graph whereeach vertex ui ∈ U has at most 5 edges.

Here is another sparse bipartite graph G = (U, V,E) where each vertexui ∈ U has at most 5 edges, and the numeral result of this matching problemis listed in Table 4.11. Figure 4.3 is the graphic illustration of the time costsby each algorithm. We can see that the running times of the ant-matchingalgorithm and the Metropolis algorithm are still much less than the runningtimes of the Hungarian algorithm. In Table 4.4, the weights found by theant-matching algorithm are close to the optimal solutions. Thus, in a largescale and very sparse graph, we can use the ant-matching algorithm to finda matching in a smaller time and the weight of the matching is close to theweight of the optimal solution.

22


Table 4.4: The weights as proportions to the weight of the optimal solutionon the bipartite graph where each vertex ui ∈ U has at most 5 edges.

4.3.3 Results on Ant-Matching Algorithm

In addition to the above-mentioned bipartite matching problems, we alsosolve the matching problems on different bipartite graphs in our experiments.The numeral results are listed in Tables 4.5–4.13. We will analyze the resultsand indicate the features of each algorithm.

In the ant-matching algorithm, Figure 4.4 is the graphic illustration forthe running times on each different bipartite matching problem. In Figure4.4, we can easily see that the running time of the ant-matching algorithmdepends on number of edges. The less the number of edges in the bipartitegraph, the less time the ant-matching algorithm costs on solving the matchingproblem.

Figure 4.5 illustrates the transitions of the weights on each different graphon different problem sizes. In Figure 4.5, as the problem size increases, theweights found by the ant-matching algorithm on each different graph as pro-portions to the weights of the optimal solution decrease slowly. Besides, theweights as proportions to the optimal solution are greater than 90%. Withthe advantages of the smaller running time, we can use the ant-matchingalgorithm to find a matching in a smaller time, which the weight is close tothe optimal solution.

23

Figure 4.4: The running times of the ant-matching algorithm on each differ-ent bipartite graph.

Figure 4.5: The weights found by the ant-matching algorithm as proportionsto the weights of the optimal solution on each different bipartite graph.

24

Figure 4.6: The weight found by the Metropolis algorithm as proportions tothe weights of the optimal solution on each different bipartite graph.

4.3.4 Results on Metropolis Algorithm

The running time of the Metropolis algorithm is much smaller than the othertwo algorithms. Even the cost times have slight change on different prob-lem sizes, but they are still smaller than the other two algorithms. But theweights found by the Metropolis algorithm are not as impressive. In Figure4.6, as the problem size increases, we can see that the weights found by theMetropolis algorithm as proportions to the optimal solution lie in between45% and 55%. Thus, if we don’t care the weights found by the Metropolis al-gorithm, we can use the Metropolis algorithm to solve the bipartite matchingproblem within a small time.

25

Tab

le4.

5:R

esults

onea

chal

gorith

mfo

rco

mple

tebip

ar-

tite

grap

hs.

Tim

e(m

s)W

eigh

tSiz

eA

nt

Met

ropol

isH

unga

rian

Ant

Met

ropol

isH

unga

rian

100x

100

1242

212

7454

7893

.762

1278

459

.991

3797

598

.562

8713

820

0x20

028

328

1261

4010

9418

6.75

3371

311

2.45

1180

919

8.23

4685

930

0x30

067

969

1441

7241

2527

7.28

7194

116

6.33

6095

729

8.34

1794

940

0x40

011

9875

1986

2510

438

370.

1943

328

218.

8004

151

398.

3521

658

500x

500

1868

9128

1625

2021

946

3.15

6447

526

8.15

7171

349

8.32

2094

560

0x60

026

8281

3264

2234

484

554.

4458

956

326.

0030

6259

8.29

3561

700x

700

3630

0033

7172

6478

164

4.91

3209

436

8.58

9427

869

8.26

6603

880

0x80

047

4672

3459

2210

5953

731.

7736

169

418.

3070

854

798.

3324

054

900x

900

6024

6937

1109

1462

0482

7.94

8320

247

4.67

2823

689

8.37

4043

110

00x10

0074

2375

3768

2820

4000

915.

8608

818

524.

5644

648

998.

4223

776

1100

x11

0090

2640

3725

3130

0031

1001

.691

501

571.

8960

081

1098

.358

4512

00x12

0010

6860

941

7531

3951

2510

91.7

3615

762

6.56

6850

211

98.3

5680

213

00x13

0012

5360

938

5781

5370

3211

88.2

4128

167

4.89

3523

712

98.3

4793

414

00x14

0014

4840

640

0343

7088

9012

72.0

0647

472

1.39

6174

413

98.3

7025

415

00x15

0016

6812

539

5672

9073

7513

62.4

5832

477

3.79

7069

614

98.4

1646

916

00x16

0018

9523

439

9672

1098

031

1453

.659

121

826.

6484

5415

98.3

6725

517

00x17

0021

5092

241

9453

1424

421

1544

.753

336

864.

8778

489

1698

.379

739

1800

x18

0024

6317

240

9375

1785

843

1631

.940

541

921.

6115

056

1798

.372

2319

00x19

0026

9170

342

8485

2110

999

1719

.485

088

979.

4262

441

1898

.372

438

2000

x20

0029

7348

542

6828

2662

853

1810

.807

4510

33.3

9225

919

98.3

5114

3

26

Tab

le4.

6:R

esult

son

each

algo

rith

mfo

rth

ebip

artite

grap

hs

wher

eea

chve

rtex

has

atm

ost

50%∗|V|e

dge

s.

Tim

e(m

s)W

eigh

tSiz

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nt

Met

ropol

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unga

rian

Ant

Met

ropol

isH

unga

rian

100x

100

4922

1275

3194

92.7

2780

627

60.3

6169

385

96.8

6275

204

200x

200

1932

813

1110

1047

187.

1510

867

113.

0348

6719

6.86

5259

530

0x30

043

485

1331

8839

5327

6.33

2672

516

3.41

2203

929

6.58

2848

440

0x40

077

406

1422

9788

1336

6.02

1748

921

6.40

9315

396.

7368

031

500x

500

1210

0020

2375

2017

245

6.86

3981

526

4.64

2575

149

6.74

1958

600x

600

1741

2525

7937

3410

954

9.52

7526

831

2.29

1804

859

6.68

1299

770

0x70

023

6500

2910

9356

641

641.

3882

497

366.

1369

673

696.

8035

924

800x

800

3089

2231

8203

9221

973

4.11

2888

541

7.24

3250

879

6.73

2962

590

0x90

039

0672

3456

4013

9641

818.

4287

195

467.

2815

802

896.

6588

299

1000

x10

0048

2782

3613

6019

9453

913.

0191

076

522.

0801

954

996.

9157

8311

00x11

0058

8250

3734

2131

2516

1002

.044

8656

6.10

8963

510

96.6

4038

912

00x12

0070

1359

3894

2240

3218

1098

.590

562

614.

3345

073

1196

.734

613

00x13

0082

6625

3808

7556

9453

1184

.955

507

668.

5840

256

1296

.792

552

1400

x14

0096

1250

3857

6670

7172

1273

.914

744

720.

9959

4613

96.7

6358

515

00x15

0011

0164

139

3406

9009

8513

67.0

7456

277

9.05

6539

114

96.7

1836

416

00x16

0012

5404

739

2016

1133

766

1453

.961

3681

7.48

3947

115

96.7

1902

117

00x17

0014

1400

041

0844

1365

297

1545

.651

154

863.

5369

943

1696

.774

299

1800

x18

0015

8639

141

3610

1691

687

1630

.413

165

917.

1276

354

1796

.742

078

1900

x19

0017

6645

340

4891

2063

985

1724

.556

971

974.

1730

304

1896

.714

927

2000

x20

0019

5975

042

1547

2492

984

1813

.187

7110

28.6

3545

519

96.7

6367

5

27

Tab

le4.

7:R

esult

son

each

algo

rith

mfo

rth

ebip

artite

grap

hs

wher

eea

chve

rtex

has

atm

ost

25%∗|V|e

dge

s.

Tim

e(m

s)W

eigh

tSiz

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nt

Met

ropol

isH

unga

rian

Ant

Met

ropol

isH

unga

rian

100x

100

5875

1210

4763

91.0

8770

013

55.6

1176

776

93.1

4795

471

200x

200

1337

512

3406

640

185.

5426

607

108.

4255

932

193.

4444

159

300x

300

3153

112

5875

2494

277.

7223

822

159.

6677

912

293.

3814

713

400x

400

5606

213

9734

6453

359.

3233

843

214.

4873

6439

3.57

3760

350

0x50

087

578

1356

5614

573

454.

6455

209

256.

6291

784

493.

3040

891

600x

600

1252

3414

3218

2760

354

2.86

8112

931

2.70

9116

559

3.67

2341

870

0x70

017

0110

1755

4749

844

640.

9559

432

364.

1409

9369

3.62

9180

780

0x80

022

1953

2171

2579

956

726.

0061

843

411.

0983

714

793.

5412

079

900x

900

2799

0624

4953

1261

6882

4.97

3926

446

1.20

3447

689

3.42

4683

910

00x10

0034

4454

2715

0017

6792

914.

2384

497

509.

7117

555

993.

6410

455

1100

x11

0041

6157

2880

0024

9002

1002

.637

891

560.

4437

2410

93.3

7439

212

00x12

0049

5171

3064

6940

6296

1092

.740

705

611.

8370

349

1193

.319

361

1300

x13

0058

0250

3241

4156

7642

1188

.161

513

659.

0457

447

1293

.535

325

1400

x14

0069

0297

3305

6271

1439

1284

.472

726

716.

8991

148

1393

.315

201

1500

x15

0077

2047

3396

0989

3031

1365

.775

754

765.

3170

4914

93.5

2115

416

00x16

0087

8171

3605

6311

7493

514

59.4

6000

280

9.93

3155

915

93.2

746

1700

x17

0099

2062

3594

5313

5833

615

53.8

4162

886

5.64

4716

816

93.5

4095

1800

x18

0011

1137

536

5704

1757

088

1641

.120

297

913.

3638

176

1793

.344

5819

00x19

0012

3661

037

6813

1951

383

1727

.499

884

964.

0143

746

1893

.537

905

2000

x20

0013

7064

138

4891

2623

716

1822

.796

555

1009

.587

833

1993

.525

222

28

Tab

le4.

8:R

esult

son

each

algo

rith

mfo

rth

ebip

artite

grap

hs

wher

eea

chve

rtex

has

atm

ost

10%∗|V|e

dge

s.

Tim

e(m

s)W

eigh

tSiz

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nt

Met

ropol

isH

unga

rian

Ant

Met

ropol

isH

unga

rian

100x

100

4859

1269

5378

81.9

8107

088

52.7

4386

7384

.895

1305

320

0x20

010

531

1262

5065

517

7.58

3311

110

3.46

2862

184.

1782

001

300x

300

2520

312

7531

2509

270.

0122

5815

2.43

3590

228

4.04

4278

140

0x40

045

078

1280

4763

1236

8.40

3189

419

8.92

4660

438

3.33

2709

750

0x50

070

110

1347

3413

108

459.

6486

038

249.

8956

0148

3.22

5838

600x

600

1007

9713

2078

2936

455

3.28

167

295.

4547

724

582.

7038

311

700x

700

1366

4113

5094

5015

664

8.52

8165

934

4.96

9619

682.

8400

679

800x

800

1775

7813

7641

8563

073

4.94

4871

395.

7375

045

782.

5284

336

900x

900

2248

7514

1422

1103

3381

3.92

1163

544

5.66

5788

788

2.89

5943

910

00x10

0027

9047

1590

0018

3946

898.

1066

518

494.

2512

265

982.

8046

155

1100

x11

0033

6750

1839

2230

1091

1001

.708

216

547.

6307

167

1082

.368

178

1200

x12

0040

1532

2038

9138

9089

1078

.901

513

592.

5325

897

1182

.699

767

1300

x13

0046

6312

2270

7955

5828

1168

.468

644

642.

0305

438

1282

.202

102

1400

x14

0053

8859

2461

7269

5479

1263

.152

943

690.

8885

926

1382

.615

354

1500

x15

0062

7906

2582

9788

8542

1356

.610

098

738.

1014

563

1482

.409

127

1600

x16

0071

7609

2921

0911

1835

714

40.8

1085

478

9.83

3652

715

82.3

2484

117

00x17

0079

6484

2853

4414

1042

415

52.5

1965

483

4.94

5819

816

82.5

6552

818

00x18

0088

9594

3201

1017

3230

616

28.8

5150

688

7.68

5158

617

82.2

0673

519

00x19

0098

9969

3341

7222

0918

717

21.1

8842

593

2.32

2166

218

82.1

3431

620

00x20

0010

9737

531

2610

2609

406

1817

.020

365

983.

5757

335

1982

.354

23

29

Tab

le4.

9:R

esult

son

each

algo

rith

mfo

rth

ebip

artite

grap

hs

wher

eea

chve

rtex

has

atm

ost

5%∗|V|e

dge

s.

Tim

e(m

s)W

eigh

tSiz

eA

nt

Met

ropol

isH

unga

rian

Ant

Met

ropol

isH

unga

rian

100x

100

4766

1653

4463

72.5

5963

638

45.2

6483

225

74.3

5403

938

200x

200

1051

615

7875

953

166.

5589

723

96.9

0158

747

171.

1226

1830

0x30

024

313

1449

2237

6626

1.30

2440

513

9.07

6186

268.

4547

5940

0x40

043

203

1527

9791

4135

3.86

6419

318

4.21

1072

836

8.54

7148

250

0x50

067

344

1525

9417

594

451.

1410

476

231.

1197

816

468.

8982

772

600x

600

9646

815

3016

3446

854

2.86

2225

527

7.07

6739

356

9.05

7167

670

0x70

013

1047

1547

5061

953

638.

3631

942

328.

4142

032

667.

9605

229

800x

800

1706

5615

4547

9429

772

7.69

5069

237

3.37

8100

776

7.87

0686

190

0x90

021

5015

1679

8413

8031

820.

9100

048

424.

8350

4286

9.32

1103

510

00x10

0026

5735

1585

9320

4500

914.

2964

7246

9.80

6912

196

7.68

7227

211

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1593

1637

1927

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941

519.

3492

103

1068

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1200

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567.

2015

287

1167

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663

1300

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0044

2172

1777

3454

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1190

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8862

1.70

2371

512

67.8

4560

514

00x14

0053

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1814

0770

9016

1241

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366

0.98

2468

1369

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415

1500

x15

0058

9953

1886

2592

7703

1343

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651

711.

0699

621

1467

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3816

00x16

0067

6547

1941

7211

4225

014

40.3

7264

777

0.59

0262

815

68.2

2760

817

00x17

0076

7094

1892

6613

8893

815

27.2

2796

808.

5954

361

1667

.407

893

1800

x18

0084

7500

1950

1515

7448

516

20.8

5849

885

5.38

6488

117

67.3

9563

719

00x19

0094

9563

2065

6220

5587

517

10.7

9986

890

8.37

5572

818

67.8

1361

920

00x20

0010

4714

121

6719

2510

532

1780

.259

9795

4.86

7843

419

67.0

5880

2

30

Tab

le4.

10:

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each

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Ant

Met

ropol

isH

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rian

100x

100

4500

1778

9063

55.1

8013

862

41.1

9267

619

57.0

0068

792

200x

200

9641

1710

4795

313

8.71

0132

279

.542

3864

214

0.93

1125

830

0x30

022

422

1660

0042

0322

4.02

9447

612

1.26

5374

227.

7981

038

400x

400

3926

616

2171

8484

320.

7804

736

167.

8679

841

329.

4465

666

500x

500

6100

016

0546

1893

841

4.58

7583

920

8.02

6716

442

6.02

9811

860

0x60

087

531

1614

6936

734

509.

6318

166

256.

2433

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4.80

1663

970

0x70

011

8922

1610

9458

453

604.

8175

264

298.

9940

862

625.

8276

844

800x

800

1546

7216

1297

9237

570

1.31

0001

235

1.32

063

724.

0302

865

900x

900

1952

3416

2375

1364

6979

1.28

1692

139

8.26

7256

682

3.44

6362

910

00x10

0024

0234

1668

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480.

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20.2

9556

1200

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0034

5016

1698

4438

9687

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453

4.25

6685

211

23.2

5550

113

00x13

0040

2484

1704

8451

9985

1167

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4457

4.13

9165

112

22.3

1780

514

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1726

2570

3188

1253

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631.

8811

8913

21.5

2103

815

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7734

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866

668.

2464

149

1419

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406

1600

x16

0061

0609

1716

0911

4004

714

45.3

3847

972

1.35

0684

115

20.9

5315

517

00x17

0069

1313

1753

4414

8257

815

32.5

6083

976

1.55

5357

216

20.4

5389

818

00x18

0077

2937

1761

8817

0409

416

27.4

2030

481

2.94

1664

117

20.4

6442

919

00x19

0085

7343

1800

4720

5701

617

20.9

4202

685

5.19

3381

218

19.9

6171

320

00x20

0095

5719

1723

1224

6809

418

09.9

0783

790

8.01

1921

719

19.6

5387

31

Tab

le4.

11:

Res

ult

son

each

algo

rith

mfo

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ropol

isH

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100x

100

2610

1441

2578

71.0

1632

804

47.7

6754

179

72.2

9785

256

200x

200

9547

1479

5410

7814

4.32

6146

788

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6114

147.

6499

849

300x

300

2071

914

2969

3625

209.

4333

408

123.

6406

937

218.

1550

131

400x

400

3610

914

5453

9172

274.

9515

693

160.

1249

942

284.

7093

798

500x

500

5578

214

5828

1889

035

3.15

5080

619

5.75

8127

136

0.60

9790

860

0x60

080

657

1469

2236

235

418.

2425

001

228.

5038

435

432.

3359

955

700x

700

1092

5015

5937

6348

449

1.71

2577

326

6.47

9316

350

5.53

7890

680

0x80

014

1719

1499

6899

828

566.

7097

6730

1.73

2119

658

2.12

9979

900x

900

1769

5315

1015

1553

2864

2.08

9699

334

0.17

7652

865

9.11

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510

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4310

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x15

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541.

3211

637

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186

1600

x16

0057

9688

1527

6611

8665

611

27.4

5140

156

8.69

1967

1157

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919

1700

x17

0065

3219

1559

5415

8126

611

91.9

1572

160

4.58

5513

512

28.6

7696

718

00x18

0072

6531

1534

0718

5415

612

70.5

7783

264

0.53

8688

613

06.9

4436

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00x19

0080

3360

1546

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9723

413

40.3

5130

667

2.03

5546

513

81.9

8123

2000

x20

0088

3531

1549

3827

7382

814

14.7

3028

709.

1355

942

1457

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701

32

Tab

le4.

12:

Res

ult

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each

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rith

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hs

wher

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Met

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100x

100

2906

1362

0378

83.4

4144

831

52.7

4386

7384

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1305

320

0x20

010

110

1388

2810

3116

6.08

0900

594

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6199

416

8.62

7658

430

0x30

021

907

1343

2933

9024

9.85

2134

913

6.84

3487

225

5.06

1219

840

0x40

038

250

1358

7590

0033

1.81

2769

318

0.87

7369

734

0.69

7786

550

0x50

058

516

1375

3118

329

412.

3106

6121

8.24

1374

642

4.53

8064

460

0x60

082

735

1386

7233

031

498.

8163

551

256.

5263

599

508.

9098

678

700x

700

1124

3713

9640

5545

357

3.20

1359

229

5.60

8140

759

2.91

7256

680

0x80

014

5750

1393

5990

516

661.

1020

851

337.

5970

951

678.

5129

0890

0x90

018

3062

1395

4712

9547

745.

2667

431

381.

3007

525

766.

6953

569

1000

x10

0022

6360

1399

0619

2563

824.

9628

987

410.

9344

0885

0.36

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1100

x11

0027

7265

1414

6927

9922

905.

3093

731

448.

8447

992

932.

8235

5212

00x12

0033

0922

1409

2235

8219

984.

2998

233

488.

0119

238

1015

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472

1300

x13

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4031

1416

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3188

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441

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5911

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3529

1400

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1422

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971

567.

5875

857

1185

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969

1500

x15

0051

0875

1428

2886

0047

1231

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836

614.

0386

2412

69.0

6846

716

00x16

0057

6782

1437

9710

5696

813

08.3

388

641.

7723

808

1350

.805

353

1700

x17

0064

8484

1526

2513

8350

013

80.1

1882

167

9.10

0884

114

30.6

6642

818

00x18

0072

6531

1610

4716

8826

614

68.2

9940

671

8.42

2965

915

18.5

5681

319

00x19

0073

1750

1550

4720

0012

515

53.8

7212

376

0.14

8536

916

05.3

8131

820

00x20

0079

9125

1562

6623

8110

916

28.2

8393

979

6.71

1340

116

87.4

1599

2

33

Tab

le4.

13:

Res

ult

son

each

algo

rith

mfo

rth

ebip

artite

grap

hs

wher

eea

chve

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atm

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20ed

ges.

Tim

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Met

ropol

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Ant

Met

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100x

100

3344

1327

0393

88.9

9404

4555

.415

9606

691

.473

1706

520

0x20

011

078

1345

3111

2517

7.50

1070

610

3.46

2862

184.

1782

001

300x

300

2345

312

9156

3703

268.

6427

545

147.

1410

087

275.

9809

938

400x

400

4040

613

0719

9844

357.

4541

1519

6.09

4746

536

8.38

6148

950

0x50

061

750

1337

9719

610

442.

6658

518

234.

5835

032

459.

3350

209

600x

600

8698

513

6078

3273

452

8.64

6773

227

6.50

8276

955

1.40

3598

670

0x70

011

7078

1328

2859

937

617.

6368

435

319.

0734

327

642.

3899

665

800x

800

1510

3113

3563

9211

070

3.52

2271

436

1.94

6939

973

4.85

6971

390

0x90

018

9203

1338

7514

9500

796.

7183

423

405.

7479

686

827.

3289

196

1000

x10

0023

1671

1352

9620

3485

883.

5310

944

4.44

9194

791

8.72

2924

511

00x11

0028

1328

1367

9628

9250

974.

4097

692

488.

6040

136

1013

.999

974

1200

x12

0033

4265

1394

3839

7609

1055

.555

762

535.

0272

726

1104

.694

687

1300

x13

0039

2110

1390

6355

1984

1149

.346

358

577.

1818

476

1197

.436

389

1400

x14

0045

4032

1399

8465

2531

1232

.319

395

613.

9420

703

1285

.780

484

1500

x15

0052

0484

1426

5787

6735

1324

.146

313

653.

2243

0813

78.5

8299

316

00x16

0058

9375

1397

0411

0184

414

07.4

3937

868

9.68

4826

414

69.7

4472

817

00x17

0066

2219

1408

4413

4882

814

94.9

5388

973

6.84

7802

515

60.6

4509

118

00x18

0074

0328

1412

8215

5965

615

85.7

1184

478

2.90

3499

116

54.7

4217

319

00x19

0082

2156

1417

0320

1576

616

61.6

1008

181

4.08

8682

917

45.0

1535

120

00x20

0090

8079

1436

7223

3507

817

54.3

5467

586

1.03

6132

1836

.726

347

34

Chapter 5

Conclusions

This thesis presents two methods for the maximum and minimum weightedbipartite matching problems. In the ant-matching algorithm, we can find amatching which the weight is close to the optimal solution. And if in a largescale weighted bipartite matching problem, using the ant-matching algorithmto find a matching will be faster than using the Hungarian algorithm. In theMetropolis algorithm, we can use the algorithm to find a matching in a veryshort time, but the weight of the matching as proportions to the optimalsolution is not good. If we want to find a matching in a large weightedbipartite graph within a rapid time, the Metropolis algorithm is the bestchoice.

Besides, with the property of the ant colony optimization algorithm whichthe pheromone on the path changes gradually, the ant-matching algorithmwill be useful in solving the dynamic weighted bipartite matching problemswhere the vertices and edges in the bipartite graph change with the time. Inthe Metropolis algorithm, in each iteration the algorithm randomly choosesan edge as a matching edge or not. It will not be influenced by the change ofthe bipartite graph. Thus, it is applicable on solving the dynamic weightedmatching problems.

35

Bibliography

[1] S. Chib and E. Greenberg. (1995) Understanding the Metropolis-Hasting Algorithm. The American Statistician, Vol. 49, No. 4 (Novem-ber 1995), pp. 327–335.

[2] J. Munkres. (1957) Algorithms for the Assignment and Transporta-tion Problems. Journal of the Society of Industrial and Applied Math-ematics, Vol. 5, No. 1 (March 1957), pp. 32–38.

[3] Y. Nakamichi and T. Arita. (2001) Diversity Control in AntColony Optimization. Proceedings of the Inaugural Workshop on Arti-ficial Life (AL’01), pp. 70–78, Adelaide, Australia, December 2001.

[4] T. Stutzle, M. Dorigo. (1999) ACO Algorithms for the Travel-ing Salesman Problem. In K. Miettinen, M. Makela, P. Neittaanmaki,and J. Periaux, editors, Evolutionary Algorithms in Engineering andComputer Science. Wiley, 1999.

[5] I, Wegener. (2005) Simulated Annealing Beats Metropolis in Com-binatorial Optimization. ICALP 2005. LNCS 3580, pp. 589–601.

36

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