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A Hard Problem for Genetic Algorithms: Finding Cliques in Keller Graphs
Jamie Marconi and James A. Foster Laboratory for Applied Logic, Department of Computer Science, U. Idaho, Moscow, ID 83844-1010. E-mail: fmarc9442,
[email protected] Abstract | We present evidence that nding the maximum clique in Keller graphs is an example of a family of problems which are both natural and inherently dicult for genetic algorithms. Speci cally, we employ a hybrid genetic algorithm to nd the largest clique in Keller graphs. We present theoretical reasons why this problem is likely to be particularly hard for this family of graphs. Our results con rm this suspicion. We then discuss several characteristics of this graph family which confound genetic algorithms: its uniformity, edge density and small diameter. Keywords | Keller conjecture, Keller graphs, maximum clique, genetic algorithms, hardness, complexity
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I. Introduction
HIS paper presents our attempt to solve a very dicult combinatorial problem: approximating the maximum clique size in Keller graphs. This is part of an ongoing project in our laboratory to identify general problems and speci c instances which are hard for genetic algorithms (GAs). Our goal is to identify particular characteristics of a speci c problem instance which contribute to GA hardness. Knowledge about the complexity of GAs will allow future designers to decide how to approach their problem with GAs, and whether or not to use GAs at all. Eventually this work will lead to a more \natural" test bed for GAs that are more suited for the natural approach GAs use. So, even though are investigating limitations to using GAs as general purpose problem solvers, our objective is to eventually to recognize the particular problems for which they are unlikely to work. Theory predicts that approximating the maximumclique of an arbitrary graph is NP hard, but does not identify which families of problem instances will be intractable. Our empirical investigations indicate that the family of Keller graphs (to be described later) are particularly hard instances of the maximum clique problem for GAs. In this paper, we combine the best techniques from previous attempts to solve the maximum clique problem in order to explore the limitations of GAs on Keller graphs. We nd that this instance is indeed a very hard one. We then examine the Keller graphs analytically in order to determine what graph theoretic features are likely to contribute to this GA hardness. The remainder of this paper is organized as follows: Section I-A presents the Keller Conjecture, Section I-B discusses complexity theoretic limitations, and Section I-C describes other work to date related to this paper. Section II describes our GA implementation and the parameters we used in our experiments. Section III presents our empirical
results. Section IV describes the limitations any GA would face with the Keller Conjecture. A. The Keller Conjecture and Keller Graphs
Keller's Conjecture (KC) is that any tiling of ndimensional space with n dimensional cubes will contain at least two cubes which share a complete (n , 1)-dimensional face[1]. KC is known to hold for lattice tilings [2]. Perron proved that KC holds for dimension n 6 [3]. Lagarias and Shor proved that it fails for n 10 [4]. The conjecture remains open for n 2 f7; 8; 9g. Szabo [5] developed an algebraic approach to KC, which Corradi and Szabo [6] translated into a graph theoretic approach using Keller graphs (see their paper for an illustration): De nition 1: ,n = (N; E ) is the Keller Graph of dimension n, where the nodes are from N = f(a1; : : :; an) : ai 2 f0; 1; 2; 3gg and the edges are from E = f(vi ; vj ) : there are k and l, 1 k; l n and k 6= l with vi [k] 6= vj [k] and vi [l] , vj [l] 2 mod 4g. The structure of ,n is closely related to tilings of cubes in n dimensions. Let !(G) be the size of the largest complete subgraph, called a clique, in graph G. Corradi and Szabo showed that: Theorem 1: Keller's conjecture fails for dimension n if and only if !(,n ) 2n . In the same paper, Corradi and Szabo prove the following theorem. This theorem gives lower bounds on clique sizes by translating bounds on smaller graphs upwards. Theorem 2: For all n 1, !(,n+1 ) 2!(,n) They also present a 57 node clique for ,6 , which Shor [7] improved to 59. Shor [7] has also found a clique of size 123 for ,7, which implies that !(,n ) (123=64)2n for all n 7. One approach to KC, then, is to improve this bound on the clique size of ,6 , ,7 , ,8, and ,9 . Table I summarizes the current state of knowledge about the clique size of Keller graphs. TABLE I
State of the Art on the Clique Size of Keller Graphs !(,1 ) = 1 !(,2 ) = 2 !(,3 ) = 5 !(,4 ) = 12 !(,5 ) = 28
59 !(,6 ) < 64 123 !(,7 ) 128 246 !(,8 ) 256 492 !(,9 ) 512 210 !(,k ) for k 10
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B. Computational Complexity
II. The Hybrid GA Implementation
It is well known that the general problem of nding maximum cliques (MC) is dicult to even approximate. More precisely, it is NP hard to approximate !(G) for arbitrary G within a polynomial ratio of optimal [8], [9]. These limitations apply to stochastic algorithms, such as GAs, as well as to deterministic ones. However, even NP hard problems can have large families of input instances which are easy to solve. For example, it is easy to nd !(G) for trees, cycles, and even randomly constructed graphs. Soule and Foster [10] presents empirical evidence about the relative diculty of approximating !(G) for several natural graph families with GAs, and Soule, Foster, and Dickinson [11] present similar evidence for GPs.
We began our study of the Keller graphs using Soule and Foster's algorithm. From this starting point we added two major optimizations: a graph coloring scheme, and an external MC estimator. This section starts with a more detailed description of the initial algorithm, then continues to describe our enhancements.
C. Evolutionary Solutions to MC
A.1 Problem Encoding We used a group representation for our graphs. Each chromosome is an integer vector where the set of indexes in the vector with the same value represents a subgraph. For example, if the ith element of a chromosome has the value j then the ith vertex is included in a separate subgraph consisting of all the other elements with value j (see Figure 1). The net result is that a single individual encodes multiple potential solutions which partition the input graph. This is similar to the grouping GA used successfully by Falkenauer [17].
A few researchers have investigated using genetic algorithms (GAs) for the general MC problem. For example, Carter and Park [12] found that a simple GA performed relatively poorly, but that simulated annealing worked well for random graphs. Park [13] argues that GAs in general will not perform as well as simulated annealing since crossover becomes less e ective as diversity declines. Bui and Eppley [14] built a hybrid GA/hill-climbing algorithm for MC which performed very well on several families of graphs. They incorporated local optimization in each generation. Their algorithm performed well on random graphs, but did poorly on Keller graphs. Soule and Foster [15] built another variation of the simple genetic algorithm without any hybrid features. Their design used group encoding on nodes and reordered alleles to keep t groups close together. Also, they used a time weighted tness function to reduce the likelihood of change as the GA approaches an optimal solution. This program found comparable solutions to those of Bui and Eppley. However, it also performed poorly on Keller graphs. Later work by Soule and Foster [10] con rms that their GA implementation solves MC well and quickly for several families of graphs, but that Keller graphs are not among them. They compute the diculty of several families of graphs from the DIMACS test suite [16], using epistasis invariance, distance correlation, and edge density. Their results con rm that, to the extent that these measures do in fact indicate GA hardness, it is particularly dicult to nd the maximum clique of Keller graphs with GAs. In this paper, we implement a hybrid GA. Our approach was to merge the best features of prior work, and to include input pre-processing. We use a reordered group encoding, with time weighted tness, do local hill-climbing with a standard heuristic, and impose a gradient on the nodes in the graph for the GA to follow. Our hope was that by combining these approaches we might nd larger cliques in Keller graphs than was possible with any of these approaches alone. We also hoped to improve the known bounds on the open cases of !(,n ). We succeeded in the rst goal, but failed in the second. We speculate about the particular properties of Keller graphs which make them so dicult later in this paper.
A. Original
The original algorithm was extensively tested on random graphs with embedded cliques. The algorithm performed very well on these graphs and when tested on the DIMACS test suite the algorithm performed on par with Bui and Eppley[14].
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A.2 The Fitness Function Our tness function is designed to give higher tness to chromosomes which select subgraphs with more edges, to give an additional bonus when those subgraphs are cliques, , and to adjust this advantage over time. Since there are v2 edges in the complete graph on v vertices, the density of , a graph G can be represented as e= v2 , where e and v are the number of edges and vertexes in G. To formalize this idea, let be a chromosome, and H 2 denote the fact that some group in represents subgraph H of G. We assume that H has e edges and v vertexes. Let C (H ) = 0 if H is not a clique, and C (H ) = 1 if H is a clique. Finally, let N be any upper bound on the maximum sub-clique size (for example, N could be the maximum degree of any vertex in the input graph G).
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The simple tness function, then, is: f () = max H 2
1 ,e=,v + C (H )v=N 2 2
becomes much more t and is correspondingly better represented in future generations. Very quickly the entire pop(1) ulation becomes multiple copies of the same individual.
For non-cliques, the rst term is always positive and less than one half while the second term is zero. Thus, for noncliques the tness function returns a value between zero and one half. This value is higher for non-cliques with more edges, so there is selective pressure among non-cliques to add high degree vertices. , A clique with v nodes has v2 edges, so for cliques, the rst term of the tness function is always one half. The second term is always positive and less than or equal to one half for cliques; it is maximized when the maximum clique has been found. Hence, for cliques the tness function returns a value between one half and one. Larger cliques will contain more vertices so they will have a higher tness than smaller cliques. So, this tness function computes a value between zero and one half for non-cliques, and between one half and one for cliques, and returns the value associated with the best group in the chromosome. Note that there is a subtle di erence between the two group case and the standard vertex encoding: in our representation, the tness is the maximum of both graphs selected by the groups, rather than that of the single subgraph selected by just the one bits in the chromosome. It is very important to decrease the likelihood of change as the GA approaches an optimal solution. So, to improve the rate of convergence the tness function was modi ed so that as the algorithm proceeded the better individuals would be given a relatively higher tness. This change was incorporated in the following weighted tness function: ft () = f ()(1+ (L=N ))
(2)
where L is the largest clique in the current population, N is an upper bound on the maximum clique size, and is the weighting parameter. This weighting is similar to the one used successfully by Carter and Park [12] under the name of simulated annealing. This weighted tness function takes the tness of an individual under the old tness function and raises it to a power which varies proportionally to the size of the largest clique found so far, generally increasing over time. As larger cliques are found more weight will be given to the more t individuals. Increasing the weighting parameter causes the algorithm to converge to the maximum clique faster. However, due to the exponential nature of the weighting function increasing too far leads to dramatic over-selection and a loss of the population diversity. With a low value for there is a nearly linear relationship between the size of the clique an individual encodes and the tness of that individual. However, with a large the relationship becomes sharply exponential: an individual with a larger clique size
B. Preprocessing: Imposing a Gradient
In order for a GA to work, there must be some tness gradient that it can explore. We realized that Keller graphs are very homogeneous in that local structure in the graph is invariant with respect to any particular node. In particular, all 4n nodes in ,n have the same degree, and it is rather high. Theorem 3: For any n, every node in ,n has degree d(n) = 4n , 3n , n. That is, each node in ,n is adjacent to exactly 4n , 3n , n other nodes. Proof: Fix a vertex v in ,n . There are two ways in which a di erent vertex u will fail to be adjacent to v. Case 1: If there is exactly one j such that v[j ] , u[j ] 2 mod 4, and for all other k, v[k] = u[k]. There are exactly n ways to choose such a j and u[j ] (since u[j ] = v[j ] + 2 mod 4 has only one possible value). Case 2: If there are exactly i di erent j such that v[j ] 6= u[j ], and for all of them v[j ] , u[j ] 6 2 mod 4. There are exactly,two possible values for such a u[j ], and n ways to select the j s. So, there are there arePexactly , i exactly ni=1 ni 2i ways to select us to meet this condition. Putting these together, since there are 4n , 1 nodes to which v may be adjacent, subtracting the sum of the two above cases gives the actual degree, of v. That is, the , P degree of v is 4n , 1 , n + ni=1 ni 2i = 4n , (n + 3n), as claimed. Even more interesting, every node is in some maximum clique. Theorem 4: For any Keller graph ,n , and for any node v, if there is a clique of size c in ,n then there is a clique of size c which contains v. Proof: Let u be a node in a clique of size c, and let v be another node in ,n which is not in this clique (otherwise the result is trivial). We construct a one to one transformation from arbitrary nodes in the rst clique into nodes adjacent to v. Choose an arbitrary node w, adjacent to u, in the given clique. Let d = v , u mod 4. That is, for all 1 i n, d[i] = v[i],u[i] mod 4. Then we claim that z = w+d mod 4 is adjacent to v. To see this, assume that i and j are such that u[i] 6= w[i] and u[j ] , w[j ] 2 mod 4. Then z [i] = w[i] + d[i] mod 4 = w[i] + v[i] , u[i] mod 4 and so z [i] 6= v[i] since w[i],u[i] 6 0 mod 4. Similarly, v[j ],z [j ] v[j ] , (w[j ]+ d[j ]) v[j ] , (w[j ]+ v[j ] , u[j ]) u[j ] , w[j ] 2 mod 4. Therefore, v is adjacent to z . Clearly, the transformation v 7! v + d is one to one. So, every edge in the clique containing u maps to a single edge adjacent to v. Consequently, there is a clique of size c which contains v, as was to be shown. We attempted to force some structure onto ,n which the GA could then explore. This was inspired by Holland's suggestion (in a presentation at the Int. Conf. on
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Genetic Programming, 1996) that evolutionary techniques
(yi , y^i )2 (3) (yi , y)2 Here, y^i is the ith value of the linear regression line and weights y is the mean of all yi . We wanted a value that would increase as the weights in a group became closer together and decrease as they spread apart. By making y^i = 1 the r2 value give the value we are looking for, because the function is calculating the variance from the horizontal line where all weights would be the same. Scaling the original tness function (1) with r2 we get: 1 ,e=,v + C (H )v=N + r2 (4) f () = max 2 H 2 3 r2 = 1 ,
C. External Estimator
Next, we hybridized the GA to run a local optimization program every 50 generations. The program we used was nmclique, which is available from DIMACS[16]. nmclique uses a randomized greedy heuristic to search for large cliques (see the DIMACS Challenge Steering Committee documentation [16] for details). The GA would insert the clique that nmclique found into the population (replacing one existing chromosome) and continue running. The clique found by nmclique is placed in the chromosome in one group and the remaining nodes are placed in another. D. Parameters
We use the following parameters on all trials: Population Size (P) 600; Generations 900; Crossover Rate 0.6. The mutation rate was approximately one expected mutation per chromosome every twelve generations, which is .07 percent per bit for ,4, and was chosen to maximize performance. The crossover rate is the same as suggested by Goldberg [18]. The data plotted in each gure were obtained by averaging over ve trials. The maximum clique found is normalized relative to the maximum clique in the graph. III. Results
A. Empirical Problems
Figure 2 shows output from running the GA on ,4. This plot shows some interesting phenomena. The area marked by (A) shows how the GA typically starts and how the old GA usually ran to completion. At this point the GA has just created its population and is slowly increasing the clique size. This increase is often very slow and often loses
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can take advantage even of random gradients. Speci cally, we labeled each node with integer weights. We selected an arbitrary node v and labeled it 1, then labeled each node in ,n with the distance of that node from v. We then modi ed our tness function according to the r2 value of the group. r2 is the coecient of determination, interpreted as the proportion of observed variation in our weights that can be explained by the simple linear regression model. r2 is calculated as follows:
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ground as is shown with the local negative slopes of the line. The points in Figure 2 labeled (B) are cliques inserted by nmclique on ,4. These are optimal cliques. These points are particularly interesting because of the wide range of possible outcomes of such a insertion. The integer weighting of the nodes can make an insertion have little e ect on the population. Because Keller Graphs have many maximal cliques throughout the graph, and the node weighting has a localization e ect on the search space nmclique can insert a very large clique with a relatively small tness. This new chromosome is alone in the population and is likely to be destroyed by crossover. This e ect is seen in the last (B) point in Figure 2 where the population does not increase at all after insertion. Another e ect that the insertion can have is to increase the population tness as a whole. In this instance the new clique is in the area the GA is searching, and crossover is likely to help. On the plot this is shown as a sharp increase followed by a level but slowly increasing region, e ectively moving the line up. This e ect was seen often in Figure 2 in regions (A), (C), and (D). The most common e ect we saw with the insertions, especially with ,4 , was termination because of optimal insertion. The GA was made to stop running if the population contained the maximum known clique after crossover. The GA would run nmclique, insert the output, crossover the population, and then check to see if it was done. If the new chromosome survived the crossover it was the end. This effect is shown by the end of Figure 2 at (C) the GA ends with a quick hop to the top. nmclique is much faster than our GA: nmclique found a clique in ,5 of size 25 in less than a minute, while the GA would take about as long to do a single generation. The runtime for nmclique can be adjusted by increasing the TRIALNUM parameter, where more trials will usually nd larger cliques. After increasing TRIALNUM on ,5 nmclique would start to consistently nd the MC as it did for ,4. For ease of analysis, we kept this number low to see what e ects non-MC output had on the GA. Figure 3 shows the e ects of sub-optimal solutions entered into the population. As the gure shows there is no noticeable dif-
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IV. Discussion
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ference between e ects of optimal solutions entered and sub-optimal solutions. For instance in Figure 3 at (A) a nearly optimal solution is entered and causes little change in the population, and at (B) a lower-optimal solution is entered but has a greater e ect on the population as a whole. Just the opposite e ects occurs at (C). The GA slows because of the time weighted tness function, in that after it has a large clique it takes more generations to increase the size of that clique. We call this slowing e ect plateauing because of the nearly horizontal line on clique size versus generation (Figure 3 (C)). We believe our hybrid GA is plateauing quickly because the nmclique heuristic introduces a large clique that drasticly changes the the size of the largest clique in the population. If the clique survives crossover, change is very slow because of the weighted tness function. The large fraction of the population that does not have a high tness have little chance to change, even though this might be bene cial to the population. This does suggest the possibility of using nmclique to make the initial populations, however. We attempted using nmclique to generate initial populations and our results were not helpful. We used the same technique for creating the chromosome as we did during insertion: insert nmclique output in group zero and the remaining nodes in group one. Using this method we expected all members of the population to grow at about the same rate. We found that nmclique was too good|no matter how low we set the options after the initial crossover the population would contain the MC and the GA would terminate. This occurred in tests on ,4 and ,5 . By forcing the GA to continue we found that the tness of the population is very low, for the same reasons as ineffective nmclique insertions. Because Keller Graphs have many maximal cliques throughout the graph, and the node weighting has a localization e ect on the search space the population will have many very large cliques without high tnesses. Because of the low tness and many possible optimal solutions, the population members do not crossover well. In fact, the maximum clique in the population begins to shrink quickly. When nmclique is run to insert in this new type of population it has the same kinds of e ects we saw earlier.
We suspect that there are some inherent reasons why it is particularly dicult for genetic algorithms (or any gradient search heuristic) to estimate the maximum clique size of Keller graphs. The main problem seems to be the homogeneity of the graphs. As we pointed out in Theorems 3 and 4, each node is essentially interchangeable with respect to large cliques. Theorem 4 in particular indicates that there are several potential optima, and that any suboptimal solution will be found by a local search from any node. It would be interesting to compute the maximum size of sun owers, collections of vertex sets which share a common intersection, for Keller graphs. We suspect that such structures mislead gradient searches on graphs. It also appears to be inherently dicult to apply a gradient to the nodes in the graph. This is in part because all the nodes are relatively close together. Theorem 5: For any n, ,n has a diameter of at most two. Proof: We assume n > 1, since the results are trivial otherwise. Let u and v be two arbitrary, distinct, nonadjacent nodes in ,n. We build a node w such that both u and v are adjacent to w. We describe what the values of two components on the w vector are. The other values are arbitrary. There are three cases. Case 1: There is exactly one i such that u[i] , v[i] 2 mod 4 and for all i 6= i, u[j ] = v[j ]. In this case, for such i and j , let w[j ] = u[j ] + 2 mod 4 and let w[i] be any value other than u[i] and v[i]. By construction, w is adjacent to both v and u. Case 2: There is at least one i such that v[i] 6= u[i] but v[i] , u[i] 6 2 mod 4, and there is at least one j such that v[j ] = u[j ]. Let w[i] = v[i] + 2 mod 4 and w[j ] = v[j ] + 2 mod 4 for such i and j . Clearly, w is adjacent to both v and u. Case 3: For all 1 i n, v[i] 6= u[i] and v[i] , u[i] 6 2 mod 4. Let w[1] = v[1] + 2 mod 4 and w[2] = u[2] + 2 mod 4. Again, w is adjacent to both v and u. This indicates that our distance based gradient would never give a distance coloring greater than two to nodes in the graph. Another dicult with Keller graphs is their high degree, as pointed out in Theorem 3. It is interesting to note that the degree of ,n can be re-written as 4n (1 , (3=4)n) , n, which approaches 4n as n approaches in nity. This means that the number of edges in a Keller graph approaches the maximum possible number of edges for large n. Smith and Grant [19] report empirical studies of constraint satisfaction problems in which parameters of the problems display a critical threshold at which they apparently become very dicult to solve. Perhaps the problem complexity of MC for Keller graphs is sensitive to the degree of ,n in a similar way. In any
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case, this makes it likely that any edge-based search will be quite complex and time consuming. This also implies that pre-processing by removing all nodes not adjacent to some xed node (which is justi ed by Theorem 4) will not help very much in the limit, though it will help slightly with the smaller dimensions. Finally, we do know that MC is an inherently intractable problem in the sense that no general purpose randomized algorithm can approximate the maximum clique for arbitrary input graphs within a polynomial ratio of optimal, unless surprising theoretical collapses occur (namely, NP = co , RP) [20]. The hybrid GA we used was still a general purpose algorithm. Perhaps we need a special approach for Keller graphs. Our results insofar as they apply to solving MC for Keller graphs are negative|we did not improve any current bounds on MC size. However, these negative results contribute something to our understanding of what makes a problem hard for GAs, without relying exclusively on traditional computational complexity theory, an ad hoc test suite, or simplistic empirical comparisons. This paper begins with a family of problems which are known to be hard asymptotically and hard to approximate. We found a particular family of instances of this problem which appears to be genuinely hard for GAs. We then analyzed this family in order to determine what features might create this \GA hardness", so that practitioners may be able to recognize these features in other problems. We were looking for features which can be adjusted so as to give rise to a family of problems of scalable GA hardness. This approach is similar to that used when studying deceptiveness, except that our problems are natural ones. Our tests with the Keller graphs did show that Keller graphs have properties that make an excellent test instance for GAs. Good test problems should be: Computationally challenging Scalable with respect to complexity Natural A suitable problem must be provably dicult. It is not sucient that a problem be NP hard, as has been claimed in the literature for results on SAT, TSP, binpacking, and even MC on random graphs. Such claims ignore the actual nature of NP-completeness, which predicts only asymptotic complexity on some instance of the problem, rather than universal or even average case complexity. In fact, SAT, TSP, binpacking, and MC are all either easy to solve for random instances (the very ones we tend to rely on) or are so easy to approximate that exact solutions are unimportant. Using such problems as empirical test cases gives a false impression of the power of evolutionary computation. By contrast, the general MC problem is provably dicult and the immense size and subtle structure of Keller Graphs have confounded our GA, and will highlight limitations of other GAs. Test suite problems should also have adjustable diculty. The ability to change the input size of the problem without changing the structure of the input is also important. This will have the e ect of changing the calculation time that
a GA might take, but not the problem diculty. For the Keller graphs, only the size of ,n changes as n increases, not the graph structure of the graph. Finally, contrived test problems, like the DeJong test suite and fully deceptive problems, may help illuminate some aspects of particular GA searches. But they tend to tell us little about the inherent complexity theoretic limitations of GAs. It is better to use problems with known inherent complexity limitations, such as MC. It is even better if our problems have practical motivations, so that their resolution will be of interest outside the GA community. Keller graphs meet this requirement, since their maximum clique size is a long-standing open problem in combinatorics. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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