LOCAL SEARCH ALGORITHMS FOR THE MAXIMUM K-PLEX PROBLEM

Transcription

1 LOCAL SEARCH ALGORITHMS FOR THE MAXIMUM K-PLEX PROBLEM By ERIKA J. SHORT A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA

2 c 2008 Erika J. Short 2

3 To my parents, Tim and Jeanne, and my sisters, Alicia and Kayla. 3

4 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Panos M. Pardalos, for all of his encouragement and support. Also, I would like to acknowledge Dr. J. Cole Smith for serving on my committee and for all of his honest advice. Lastly, I would like to thank my family and friends for their never-ending moral support. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION THE MAXIMUM K-PLEX PROBLEM Foundations The Maximum k-plex Problem HEURISTIC APPROACH Development Heuristics Greedy Algorithm Tabu Search Method Computational results Exchange Algorithm Method Computational results CONCLUSIONS REFERENCES BIOGRAPHICAL SKETCH

6 Table LIST OF TABLES page 3-1 The tested DIMACS instances Tabu results for k = 1 and k = Exchange results for k = 1 and k = Result comparison for k = Result comparison for k =

7 Figure LIST OF FIGURES page 3-1 Greedy algorithm for k-plex Tabu algorithm for k-plex Exchange algorithm for k-plex

8 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science LOCAL SEARCH ALGORITHMS FOR THE MAXIMUM K-PLEX PROBLEM Chair: Panagote M. Pardalos Major: Industrial and Systems Engineering By Erika J. Short May 2008 The maximum k-plex problem is one of many ways to classify a cohesive subgroup in social network analysis. Considered a type of relaxed clique, the maximum k-plex problem is a degree based approach to identifying closely related vertices in a graph. The recent development of applying optimization to the maximum k-plex problem has the capability for growth. Local search algorithms were developed and applied to benchmark problems for the maximum k-plex problem. Computational experiments were performed, and new lower bounds have been successfully attained. 8

9 CHAPTER 1 INTRODUCTION The application of graph theory to the notion of social networks has been developed for many years. Unsurprisingly, the effects of globalization along with the an increase of interest in terrorism have propelled a new surge of interest in social network theory. Along with the study of social networks, certain mathematical models have been revived from the past. One such example is the maximum k-plex problem. The maximum k-plex problem has roots in the study of cohesive subgroups of a network. According to [25], the concept of cohesive subgroups comes from the close relationships between actors in a social network. By studying these groups, sociologists can form theories that explain the behavior of actors in social networks. The most famous of these cohesive subgroup models is known as the maximum clique problem. Although it is well-known, so are its difficulties. The restrictions associated with maximum clique have propelled the study and formation of clique relaxations. These relaxed approaches give way to new opportunities by focusing on specific properties of cohesive subgroups. In particular, the maximum k-plex problem focuses on the reachability and robustness of the graph, and relaxes the property of familiarity, as stated in [3]. This thesis will apply heuristic techniques to the maximum k-plex problem. Chapter 2 will cover the background of the problem. In chapter 3, heuristic approaches and results are addressed. Chapter 4 discusses conclusions and the future development of techniques that can be applied to the maximum k-plex problem. 9

10 CHAPTER 2 THE MAXIMUM K-PLEX PROBLEM 2.1 Foundations The maximum clique problem is well-known to be NP -hard. A clique can be represented on an undirected graph as a subset of nodes that are all pairwise adjacent. Attaining the maximum clique requires finding the largest clique that exists in the graph. With applications in many fields, the maximum clique problem has been extensively researched in the past [21]. Many sources have pointed out problems with the maximum clique problem, regarding its restrictiveness [25], modelling disadvantages [23] [10], and difficult approximation [5]. In light of these problems, researchers developed many different relaxations to the maximum clique problem. Since cliques can be described as cohesive subgroups of a social network, certain properties can be used to describe their nature: familiarity (each vertex has many neighbors), reachability (a low diameter), and robustness (high connectivity) [3]. Each of these properties has its own difficulties, resulting in different relaxation techniques. Some well-known relaxation models are k-clique [18], k-club [2], and k-clan [19]. This paper considers another model called the maximum k-plex problem. This particular relaxation is degree oriented and was developed by Seidman and Foster in 1978 [23]. 2.2 The Maximum k-plex Problem Let G = (V, E) be a simple, undirected graph, where V = {1, 2,..., n}, E = (i, j) such that i, j V, and the degree, deg G (v i ), is number of neighbors of v i in G. Given a subset, S V, let G[S] be the induced subgraph of S. If Equation 2 1 holds, the set S induces a k-plex. deg G[S] (v i ) S k v i S (2 1) 10

11 Although the maximum k-plex problem was developed years ago, studies in optimization have neglected it until recently. In [3], the problem was reintroduced, considering its computational complexity and developing an integer programming formulation. In regards to complexity, when k is arbitrary, the maximum clique problem exists as a special case of maximum k-plex and is clearly NP -hard [3]. In addition, an NP -completeness proof for finding a k-plex of a fixed value of k in a graph was published [3]. Also, in [3] the first known computational results for the maximum k-plex problem was presented. Starting with an integer programming formulation, they constructed a branch-and-cut method to be applied to well-known data sets. A peeling procedure used to remove vertices with low degree was also used in preprocessing. Their benchmark solutions were obtained by implementing the method using ILOG CPLEX 9.0R[ILOG] software. No other published results are known to exist. 11

12 CHAPTER 3 HEURISTIC APPROACH 3.1 Development In optimization, the maximum k-plex problem has few computational results. A branch-and-cut algorithm that produced benchmark results for the problem was published in [3]. Branch-and-cut is considered an exact algorithm that uses cutting plane methods to increase the performance of a branch-and-bound algorithm. Although this method is exact, its performance is very slow. Therefore, approximation methods can be used to find solutions at a much faster rate. Since the maximum k-plex problem is relatively unexplored, heuristic methods have been developed to produce new benchmark solutions. Although the branch-and-cut algorithm was tested on several different instances, the experiments in this study are tested only against DIMACS instances [9]. These can be found in Table Greedy Algorithm 3.2 Heuristics The initial approach to approximating the maximum k-plex begins with a simple greedy algorithm for the maximum clique problem. This method involves selecting the vertex with the highest degree and adding it to the current solution set. Iteratively, neighboring vertices are added to set if they have the highest degree of all nodes that can be added to the clique. Vertices are continually added until no more can be added to the set. With little effort, this approach can be expanded to the maximum k-plex problem. This adaptation requires a slightly more detailed validity consideration. Instead of requiring all vertices to be connected to the candidate vertex, the new solutions must ensure that they satisfy the k-plex requirements. By definition, all vertices in the current set, G[S], and the candidate vertex, v i, have to satisfy Equation 2 1 to retain k-plex feasibility. In order to implement the algorithm, each vertex is classified when it is added to the current solution.vertices are classified as 12

13 one of two groups: those with positive potential and those with none. Potential, denoted pot(v i ), can be described as the amount of flexibility that the vertex has within the solution set. Potential is calculated by taking the difference between the k-plex degree requirement and the number of neighbors that the candidate vertex has within the set, see Equation 3 1. This property is useful in quickly determining the quality of candidate solutions. Similar to maximum clique, all vertices with zero potential must be neighbors to any vertex selected to be added to the k-plex in order to retain feasibility. pot(v i ) = deg G[S] (v i ) S + k v i S (3 1) Also, this approach requires the chosen vertex to have the maximum degree amongst all candidates outside of the set, deg G[ S] (v i ). In addition, if tie-breaking is necessary, the added vertex should be selected at random. The implementation of this algorithm is shown in Figure 3-1. Basic greedy algorithms are known to produce good results, but they are usually not significant enough to report. Alone, the greedy algorithm has a tendency to include vertices of high degree, although those vertices may not belong to the maximum k-plex. For more dependable results, the use of more advanced search techniques are needed. Therefore, this greedy approach simply provides a platform to continue development of a more in-depth heuristic Tabu Search Method In order to improve results, local search techniques were considered. Local search methods are characterized by searching the neighborhood of a current solution to find improvements. Local search heuristics have been known to produce good results when applied to difficult optimization problems. Within the classification of local search methods, tabu search is considered to be the most effective. Developed by Fred Glover, tabu search is a method that limits the number 13

14 of future solutions based on a history of previous moves [12]. Unlike many other local search techniques, this method provides a way to escape local optima. With tabu search, a set of previously chosen moves are stored in a tabu list that disallows that move to be made again until a specified number of new moves have been taken. This limiting factor reduces the chance that the algorithm will continue to revisit the same local solution. Considering its past performance, tabu search was chosen as starting point for applying local search to the maximum k-plex problem. In order to implement tabu search, the qualities of the neighborhood structures need to be determined for efficient results [13]. For this method, the solutions that have been previously chosen using the greedy algorithm will be classified as tabu. These choices will be forbidden to be used again until a specified number of moves have been made. In such a greedy algorithm, the tabu solution will be overlooked, and the next best choice will be taken. The length of time that a move is classified as tabu is called its tabu tenure. Although there are many different ways of determining and adjusting the tabu tenure, most choices are based on empirical tests that begin with an initial set value for all problems [13]. For the results presented in this paper, the tabu tenure was chosen as a set value, with no adjustment within the algorithm itself. To implement the neighborhood search, each locally maximal solution needs to be changed. For this procedure, a local maximum is adjusted by removing a vertex from its current solution and adding new vertices based on the greedy heuristic. The selection of a deleted vertex is based on a partially random approach. Referencing to the values of potential for each vertex in the graph, vertices with zero potential are critical to the structure of the graph. Because these vertices are more restrictive to the solution, a random vertex from this subset is selected to be removed from the set. In such a case that a vertex cannot be selected with a potential of zero, the deleted vertex must be chosen from the rest of the set. This algorithm runs in a recursive manner, continuing until a stopping criteria is met. The pseudo code for the algorithm can be found in Figure

15 Computational results The tabu search method developed in this project was implemented using C programming language. Results were obtained using an AMD Opteron 2.2 GHz processor with 4 GB of RAM. Search parameters were equivalent for all experiments: tabu tenure = 10 iterations and run time = 1000 seconds. Although brief experiments were run for tabu tenure > 10, outputs provided insignificant results. The algorithm was applied to a variety of graphs from the benchmark clique instances of the Second DIMACS challenge [9]. Experiments on the graphs were done with values k = 1 (clique) and k = 2. Each experiment was performed for 20 independent runs. The results of the tests can be seen in Table 3-2, showing the average time to solution, maximum best solution, minimum best solution, and average best solution. These results are compared with later results and the benchmark solutions of [3] in Table 3-4 and Table 3-5. Comparison values indicate average time to solution and maximum best values. Values in square brackets are the bounds for non-optimal terminations in the benchmark instances from [3]. Values in parenthesis are new bounds. Experiments that did not obtain optimum (or current lower bound) are marked with an asterisk (*). The tabu search algorithm performed well on many problems, and new bounds were obtained for certain instances. Although 1-plexes are cliques, the algorithm provided improved results compared with previous k-plex results. According to DIMACS [9], the 1-plex bound attained for MANN 27.clq is the published size for maximum clique. Additionally, the algorithm found a new lower bound for hamming10-4.clq with k = 2. Although most results were matched or improved, some benchmark values were not attained Exchange Algorithm Method The proposed tabu heuristic uses a process of randomly deleting vertices and reevaluating the current solution set. New vertices are considered to be added to the 15

16 set if they satisfy the k-plex feasibility conditions for the smaller set and do not belong to the tabu list. Once a local maximum is attained, the deleted vertex is chosen at random. Only after a vertex is deleted, can new vertices be considered for addition to the k-plex with the greedy approach. It has been considered that deleting a random node can have varying affects on the future growth of the k-plex. Once the algorithm reaches a locally optimal solution, it must reduce the k-plex size to move within the neighborhood. Because the vertex is deleted randomly, the solution may not reach its previous size if no other vertices can replace the deleted vertex. Therefore, a local search method is considered that will prohibit moves within the neighborhood that reduce the size of the k-plex. In this method, local search is conducted through exchanging vertices. When a local maximum is attained, a vertex within the set will be replaced by an exchange partner outside of the solution set. The exchange partner is a vertex chosen randomly and evaluated for possible exchange moves within the set. In order to decrease the complexity of the algorithm, the first move that satisfies conditions will be selected. Once feasibility is established, the two vertices can be exchanged. The algorithm then continues in a greedy fashion, adding new vertices to the set. Finding an exchange pair consists of evaluating simple conditions. After a vertex outside of the solution is chosen, it is randomly compared against the vertices within the set. When choosing an exchange move, two considerations need to be made: k-plex feasibility of the chosen vertex and stability of the current set. The first condition is for the chosen vertex to have k-plex feasibility after being added to the set. From the basic k-plex condition in Equation 2 1, the exchange conditions can be extended in terms of the relationship between the two chosen vertices. Because the feasibility of the new vertex is being evaluated before removing its exchange partner, the k-plex conditions are dependent on the relationship between the two. If the vertices are neighbors, the evaluated conditions must be offset by one, in order to compensate for the degree change when the current vertex is removed from the set. This situation occurs 16

17 because the degree of the new vertex will consider its exchange partner as still belonging to the set. The k-plex conditions for evaluating an exchange when the two vertices are not neighbors can be found in Equation 3 2. The conditions for when they are neighbors can be found in Equation 3 3, where S is the current solution set, and v new is the vertex being considered for addition to the set. N(v new ) S S k (3 2) N(v new ) S S k + 1 (3 3) Once the vertex is known to satisfy k-plex conditions, it must be compared with the current vertex to ensure that it does not break the feasibility conditions for the rest of the set. Similar to the requirements of the greedy algorithm, the potential values of the current solution are used to ensure the structure of the k-plex. Again, critical vertices are those with potential equal to zero, and any new vertex being added to the set must be connected to these vertices. With the conditions of exchange, potential values can only be increased or remain the same when a vertex is removed from the set. Therefore, only vertices with zero potential that are also neighbors to the vertex being considered for removal need to be checked for set stability. In order to prevent the neighboring vertices from attaining negative potential (breaking the k-plex), they all must also be neighbors of the exchange partner. This condition can be seen in Equation 3 4, where S is the current solution set, S BL is the subset S with potential equal to zero, v new is the vertex being added to the set, and v old is the vertex being removed from the set. N(v new ) S BL N(v old ) S BL (3 4) After all conditions are satisfied, the exchange can be made. A new solution set is formed, allowing the greedy algorithm to continue checking for new solutions. The 17

18 process continues to repeat until a stopping criteria is specified. The pseudo code for this algorithm can be found in Figure 3-3. In addition to this exchange algorithm, another method, using tabu search, was considered. This method incorporated the tabu search heuristic into the selection of exchange pairs. The tabu method would prevent exchanges with vertices that had been exchanged recently. Preliminary experiments were performed with tabu tenures of various sizes. After these tests, results showed a significant increase in process time for the exchange algorithm with tabu, with either no increase or a decrease in solution size. Therefore, no further experiments were performed on this combined algorithm Computational results The exchange local search algorithm developed in this project was implemented using C programming language. Results were obtained using an AMD Opteron 2.2 GHz processor with 4 GB of RAM. All experiments had the stopping criteria of run time = 1000 seconds. The algorithm was applied to the same set of benchmark clique instances as the tab algorithm, from the Second DIMACS challenge [9]. Again, experiments were performed with values k = 1 and k = 2. Each experiment was tested for 20 independent runs. Results of the experiments can be found in Table 3-3, showing average time to solution, maximum best solution, minimum best solution, and average best solution. These results have been compared with the tabu search results, along with the benchmark instances of [3] in Table 3-4 and Table 3-5. Comparison values indicate average time to solution and maximum best values. Values in square brackets are the bounds for non-optimal terminations in the branch-and-cut instances from [3]. Values in parenthesis are new bounds. Experiments that did not find optimal values (or current lower bounds) are marked with an asterisk (*). The exchange local search algorithm performed well on most problems. Although most search times were comparable with tabu search, some instances proved to require 18

19 more processing time. The resulting values from the experiments are comparable with the tabu search method. Most instances found the optimal solution or known lower bound. For some instances, the exchange algorithm found better solutions that the tabu search algorithm, including a new bound for hamming10-4.clq with k = 2. Comparing the overall results with tabu search, the exchange algorithm proves to be more stable. Maximum best solutions for both methods are similar, but average best solutions differ significantly. The exchange algorithm was consistently better at finding solutions for many instances, as can be seen in the difference of minimum best values. Overall, the exchange local search method is a well-performing heuristic for the maximum k-plex problem. 19

20 1 Greedy Algorithm 2 S 0 is the inital k-plex 3 S BL is the subset of S where N(v i ) S = S k for v i S 4 S S 0 5 while S is not maximal do 6 for i = 1 to n do 7 if N(v i ) S S k + 1 then 8 if N(v i ) S BL = S BL then 9 pot(v i ) N(v i ) S 10 if pot(v i ) > pot best then 11 pot best pot(v i ) 12 v best v 13 end if 14 end if 15 end if 16 end for 17 S S v best 18 end while Figure 3-1. Greedy algorithm for k-plex 1 Tabu Algorithm 2 S 0 is the inital k-plex 3 S S 0 4 while time < run time do 5 while S is not maximal do 6 Select v i S using the greedy algorithm; 7 if v i is not tabu then 8 S S v i 9 end if 10 end while 11 if S > S best then 12 S best S 13 end if 14 Select a random v i from S; 15 S S \ v i 16 end while Figure 3-2. Tabu algorithm for k-plex 20

21 1 Exchange Algorithm 2 S 0 is the inital k-plex 3 S S 0 4 while time < run time do 5 while S is not maximal do 6 Select v i S using the greedy algorithm; 7 S S v i 8 end while 9 Select a random v new from S; 10 while an exchange partner is not selected do 11 Select a random exchange partner v old from S; 12 if N(v new ) S BL N(v old ) S BL then 13 ifv old N(v new ) then 14 if N(v new ) S S k + 1 then Select v old as exchange partner; S vold 17 S v new 18 end if 19 end if 20 else 21 if N(v new ) S S k then Select v old as exchange partner; S vold 24 S v new 25 end if 26 end else 27 end if 28 end while 29 end while Figure 3-3. Exchange algorithm for k-plex 21

22 Table 3-1. The tested DIMACS instances Graph V E c-fat200-1.clq c-fat200-2.clq c-fat200-5.clq c-fat500-1.clq c-fat500-2.clq c-fat500-5.clq c-fat clq hamming6-2.clq hamming6-4.clq hamming8-2.clq hamming8-4.clq hamming10-2.clq hamming10-4.clq johnson8-2-4.clq johnson8-4-4.clq MANN a9.clq MANN a27.clq MANN a45.clq san clq keller4.clq

23 Table 3-2. Tabu results for k = 1 and k = 2 k = 1 k = 2 Graph Time(s) Max Min Avg Time(s) Max Min Avg c-fat200-1.clq c-fat200-2.clq c-fat200-5.clq c-fat500-1.clq c-fat500-2.clq c-fat500-5.clq c-fat clq hamming6-2.clq hamming6-4.clq hamming8-2.clq hamming8-4.clq hamming10-2.clq hamming10-4.clq johnson8-2-4.clq johnson8-4-4.clq MANN a9.clq MANN a27.clq MANN a45.clq san clq keller4.clq

24 Table 3-3. Exchange results for k = 1 and k = 2 k = 1 k = 2 Graph Time(s) Max Min Avg Time(s) Max Min Avg c-fat200-1.clq c-fat200-2.clq c-fat200-5.clq c-fat500-1.clq c-fat500-2.clq c-fat500-5.clq c-fat clq hamming6-2.clq hamming6-4.clq hamming8-2.clq hamming8-4.clq hamming10-2.clq hamming10-4.clq johnson8-2-4.clq johnson8-4-4.clq MANN a9.clq MANN a27.clq MANN a45.clq san clq keller4.clq

25 Table 3-4. Result comparison for k = 1 Graph Tabu Time(s) Exg Time(s) BC Time(s) c-fat200-1.clq c-fat200-2.clq c-fat200-5.clq c-fat500-1.clq c-fat500-2.clq c-fat500-5.clq c-fat clq hamming6-2.clq hamming6-4.clq hamming8-2.clq hamming8-4.clq hamming10-2.clq hamming10-4.clq (40) (40) [38,379] johnson8-2-4.clq johnson8-4-4.clq MANN a9.clq MANN a27.clq (126) (126) [125,148] MANN a45.clq 334* * [342,422] san clq 12* 0.00 (18) [17,27] keller4.clq 9*

26 Table 3-5. Result comparison for k = 2 Graph Tabu Time(s) Exg Time(s) BC Time(s) c-fat200-1.clq c-fat200-2.clq c-fat200-5.clq c-fat500-1.clq c-fat500-2.clq c-fat500-5.clq c-fat clq hamming6-2.clq hamming6-4.clq hamming8-2.clq [128,130] hamming8-4.clq [16,80] hamming10-2.clq [512,534] hamming10-4.clq (47) (48) [45,458] johnson8-2-4.clq johnson8-4-4.clq [14,16] MANN a9.clq MANN a27.clq 235* * [236,260] MANN a45.clq 660* * [662,739] san clq 24* * 0.00 [62,86] keller4.clq 15* * 0.00 [40,45]

27 CHAPTER 4 CONCLUSIONS This study developed new methods for applying heuristics to the maximum k-plex problem. The first proposed algorithm used tabu search, along with a greedy technique, to implement a local search procedure. The tabu algorithm was tested on a set of problems provided by DIMACS [9]. It showed good results for most problems, finding optimal solutions for 15 of 16 known solutions for k = 1 and 11 of 11 known solutions for k = 2. For benchmarks with unknown optimal solutions, the algorithm found new bounds for two of four solutions with k = 1 and one of nine solutions with k = 2. A small number of instances did not break lower bounds. The exchange algorithm was experimented on the same set of DIMACS instances as tabu search [9]. In comparison with tabu search, the exchange algorithm performed more reliably; terminating solutions were consistently higher. For values of both k = 1 and k = 2, all known optimal solutions were found. For instances with bounded solutions, the exchange algorithm met or surpassed lower bounds for three of four solutions with k = 1 and five of nine solutions with k = 2. Again, a small number of instances did not meet lower bounds, and these examples of poor performance are primarily shown in high density graphs. It is often necessary to find solutions to optimization problems in a limited period of time. Development of approximation methods to estimate solutions quickly is necessary for the quick evaluation of large and complex problems. These heuristics provide new method that can be applied to finding k-plexes in a graph. Extending the research of local search methods for the maximum k-plex problem is very promising. With additional tuning, the tabu search algorithm could improve dramatically from its current performance. Incorporating techniques that can be adjusted during the search procedure for both the tabu and exchange algorithms is expected to 27

28 have improved results. In addition to building upon current search techniques, other methods, such as simulated annealing, may have successful results as with many other applications. Further research regarding the maximum k-plex problem will have positive effects on the field of optimization and social sciences. Applications of finding large k-plex values in graphs are expected to have an impact on determining cohesive subgroups in many real life situations. Other than social network analysis, the maximum k-plex problem has the opportunity and possibility of being implemented in many other fields of study. 28

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31 BIOGRAPHICAL SKETCH The author, Erika Short, is a master s student at the University of Florida. She studies in the department of Industrial and Systems Engineering, with her primary focus in operations research. 31