A Simpler Characterization of a Spectral Lower Bound on the Clique Number

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1 A Simpler Characterization of a Spectral Lower Bound on the Clique Number E. Alper Yıldırım April 4, 2009 Abstract Given a simple, undirected graph G, Budinich [Discrete Applied Mathematics, ), ] proposed a lower bound on the clique number of G by combining the quadratic programming formulation of the clique number due to Motzkin and Straus 965) with the spectral decomposition of the adjacency matrix of G. This lower bound improves the previously known spectral lower bounds on the clique number that rely on the Motzkin-Straus formulation. In this paper, we give a simpler, alternative characterization of this lower bound. For regular graphs, this simpler characterization allows us to obtain a simple, closed-form expression of this lower bound as a function of the positive eigenvalues of the adjacency matrix. Our computational results shed light on the quality of this lower bound in comparison with the other spectral lower bounds on the clique number. Key words: Maximum clique, maximum stable set, stability number, clique number, spectra of graphs. AMS Subject Classifications: 05C69, 90C35, 65K05, 90C20 Department of Industrial Engineering, Bilkent University, Bilkent, Ankara, Turkey yildirim@bilkent.edu.tr)

2 Introduction Let G = V, E) be a simple, undirected graph with a vertex set V = {, 2,..., n} and an edge set E consisting of m edges. A clique C V is a set of mutually adjacent vertices. The clique number of G, denoted by ωg), is the size of the maximum clique in G. A set S V is a stable set of G if each pair of vertices in S is mutually nonadjacent. The cardinality of the maximum stable set of G is called the stability number of G and is denoted by αg). The maximum clique problem is equivalent to the maximum stable set problem on G, the complement of G. We denote the n n adjacency matrix of G by A G. It is well-known that computing or even approximating) the clique number of a graph is in general an NP-hard problem [7]. The recent survey paper by Bomze et al. [2] provides an account of the fairly rich literature including applications, formulations, exact algorithms, heuristics, bounds, and estimates. However, the clique number and a maximum clique can be computed in polynomial time for certain classes of graphs such as perfect graphs and complements of t-perfect graphs [6, 3]. In the literature, several connections have been established between the clique or stability number of a graph and the spectral properties of the adjacency matrix or of the Laplacian matrix of G see, e.g., [4, 2, 3, 0, 8, 5, ]). In particular, Budinich [3] proposed a lower bound on the clique number of a given graph G by combining the quadratic programming formulation of the clique number due to Motzkin and Straus [9] and the spectral decomposition of the adjacency matrix A G. He established that this lower bound improves the previously known spectral lower bounds that rely on the Motzkin-Straus formulation. In this paper, we give a simpler, alternative characterization of this lower bound. In contrast with the characterization of the lower bound in [3], our characterization requires the solution of a much simpler quadratic linesearch problem. For regular graphs, this simpler characterization allows us to obtain a simple, closed-form expression of this lower bound as a function of the positive eigenvalues of A G. Our computational results shed light on the quality of the proposed lower bound in comparison with the other spectral lower bounds on the clique number. 2

3 This paper is organized as follows. In the remainder of this section, we define our notation. Section 2 discusses the continuous formulation of Motzkin and Straus [9] and reviews several known lower bounds on the clique number. Section 3 presents the simpler, alternative characterization of the lower bound due to Budinich [3]. The closed-form expression of this lower bound for the class of regular graphs is the topic of Section 4. The computational results are presented in Section 5. Section 6 concludes the paper.. Notation R n denotes the n-dimensional Euclidean space. For u R n, u i denotes the ith component of u. The complete graph on n vertices is denoted by K n. We reserve e to denote the vector of all ones in the appropriate dimension and e j to represent the unit vector whose jth component i. The n )-dimensional unit simplex in R n is denoted by n, i.e., n := {x R n : e T x =, x 0}. For a graph G = V, E) with V = {,...,n}, A G S n denotes the adjacency matrix of G. For a nonempty subset C V, we use χ C to denote the characteristic vector of C scaled by / C so that χ C n. 2 Formulation and Lower Bounds Given a simple, undirected graph G = V, E) with V = n, Motzkin and Straus [9] established that ωg) = max x n x T A G x, ) which is a continuous formulation of a combinatorial optimization problem. In addition to paving the way for the use of continuous optimization methods to solve the maximum clique problem, this formulation plays a central role in the derivation of lower bounds on the clique number of G. By ), ωg) µ x) := x T A G x, for all x n. 2) 3

4 Using the fact that x = /n)e n, it follows that ωg) n 2 n 2 2m, 3) where m = E is the number of edges of G. This bound matches the clique number for complete graphs K n and their complements. In addition to computing the clique number, the formulation ) can in some cases be used to identify a maximum clique. For any maximum clique C V, χ C is a global optimal solution of ). However, a drawback of the Motzkin-Straus formulation ) is the existence of global optimal solutions that do not correspond to characteristic vectors of maximum cliques of G. For instance, if G is the 4-cycle C 4, then χ V = /4)e is a global optimal solution of ) despite the fact that V is not a clique of G. In an attempt to circumvent this drawback of the Motzkin-Straus formulation, Bomze [] proposed the following regularized continuous formulation: 2ωG) = max x T A G x + ) x n 2 xt x. 4) In contrast with ), each local maximizer is strict and corresponds to the characteristic vector of a maximal clique maximal with respect to inclusion). Also, each global maximizer is strict and corresponds to the characteristic vector of a maximum clique. Similarly to ), this alternative formulation can be used to derive a lower bound on the clique number of G: ωg) β x) := 2 x T A G x /2) x T x), for all x n. 5) Despite the fact that the alternative formulation 4) has more appealing properties in terms of identifying cliques, the following lemma establishes that the Motzkin-Straus formulation actually yields better lower bounds. Lemma 2. Let G = V, E) be a simple, undirected graph such that V = n. For each x n, µ x) β x), i.e., the lower bound 2) obtained from the Motzkin-Straus formulation ) is at least as good as the lower bound 5) obtained from the Bomze formulation 4). 4

5 Proof. Let x n. Note that x T I + A G ) x x T ee T ) x = e T x) 2 =, where we used the fact that x 0 and I + A G is a matrix consisting only of zeroes and ones. Therefore, which completes the proof. β x) = 2 x T A G x ) 2 xt x, = 2 x T A G x x T I + A G ) x, x T A G x = µ x), By Lemma 2., we henceforth restrict our attention to the Motzkin-Straus formulation ). Most of the other lower bounds in the literature are obtained by combining the formulation ) with the spectral theory of graphs. Henceforth, we assume that G = V, E) is a connected graph without loss of generality, since the maximum clique problem can otherwise be decomposed into smaller problems on each connected component of G. We now collect some results about the spectra of such graphs. The reader is referred to [4] for further details. Theorem 2. Let G = V, E) be a connected graph with the adjacency matrix A G S n such that V = n 2 and let λ λ 2... λ n denote the spectrum of A G.. n i= λ i = λ n and λ λ n, i.e., λ is the spectral radius of A G. 3. A G is irreducible, which implies that there exists a positive eigenvector u R n, called the Perron eigenvector, corresponding to the simple eigenvalue λ, called the Perron root. 4. e R n is an eigenvector of A G corresponding to λ if and only if G is a regular graph. 5. A G has exactly one positive eigenvalue λ if and only if G is a complete multipartite graph. 5

6 Given a graph G, let λ > 0 and u R n denote the Perron root and the positive Perron eigenvector of A G, respectively. Using the feasible solution ˆx := / )u n of ), where := e T u, Wilf [2] established that ωg) µˆx) = s2 s 2 λ = λ s 2 λ +, 6) with equality if G is a complete graph. The lower bound 6) is an improvement over the lower bound 3). More recently, Budinich [3] proposed a new lower bound that makes use of all the eigenvectors of A G. In particular, if u, u 2,..., u n denotes the eigenvectors of A G of unit Euclidean norm corresponding to the eigenvalues λ λ 2... λ n, respectively, where u > 0, one can construct a family of unit vectors y i β) = β u i + β 2 u R n, i = 2, 3,..., n, 7) where β, ). Then, z i β) := /e T y i β))y i β) is a feasible solution of ) for i = 2, 3,..., n as long as β [l i, u i ], where Let l i := max j:u i j >0 and let g i := max β [l i,u i ] It follows from ) that u j u j )2 + u ij )2 < 0, u i := min j:u i j <0 z i β) T A G z i β) = max β [l i,u i ] u j u j )2 + u ij )2 > 0, i = 2, 3,..., n. 8) β 2 λ i + β 2 )λ βe T u i ) + i = 2, 3,..., n. 9) β 2 ) 2, g := max i=2,3,...,n g i. 0) ωg) g. ) Unless G is a complete multipartite graph, Budinich shows that ) strictly improves upon 6). A comparison of the three lower bounds reveals that 3) is the easiest to compute and is provably the weakest one. While 6) requires only the computation of the Perron root and the Perron eigenvector, one needs the full spectrum and the full set of eigenvectors to compute ). 6

7 3 A Simpler Characterization Given a simple, connected, undirected graph G = V, E) with V = n, recall the Motzkin- Strauss formulation: ωg) = max x n x T A G x. Let u, u 2,..., u n denote the eigenvectors of A G of unit Euclidean norm corresponding to the eigenvalues λ λ 2... λ n, respectively, where u > 0. For τ R, let us define the following family of solutions: w i τ) := ) u + τ u i si ) u ), i = 2, 3,..., n, 2) where s i := e T u i, i =, 2,..., n. 3) It is easy to verify that e T w i τ) =, i = 2, 3,..., n for all τ R. Furthermore, since u > 0, it follows that w i τ) n if and only if τ [τl i, τi u ], where τ i l := max j:d i j >0 u j d i j ) < 0, τi u u j := min > 0, i = 2, 3,..., n, 4) j:d i j <0 d i j ) where d i := u i si Therefore, we can define the following linesearch problems: ) u, i = 2, 3,..., n. 5) ν i := max w i τ) T A G w i τ), 6a) τ [τl i,τi u ] ) ) 2 si τ = max λ + λ i τ 2, 6b) τ [τl i,τi u ] { ) ) λ 2λ s i ) = max τ + λ i + λ ) } s i ) 2 τ 2, i = 2, 3,..., n. 6c) τ [τl i,τi u ] Let us define s 2 Since w i τ) n for τ [τl i, τi u ], i = 2, 3,...,n, it follows that s 2 s 2 ν := max i=2,3,...,n ν i. 7) ωg) ν. 8) 7

8 The next proposition establishes that the lower bound 8) is exactly the same as the improved bound ) due to Budinich. Proposition 3. Let G = V, E) be a simple, undirected, connected graph. The lower bounds 8) and ) agree. Proof. We prove the assertion by establishing a one-to-one correspondence between the feasible solutions of the linesearch problems 9) and 6). Let us fix i {2, 3,..., n}. Let β [l i, u i ], where l i and u i are given by 8). Therefore, z i β ) = e T y i β ) y i β ) = β s i ) + β u i + ) β ) 2 u, 9) β ) 2 where y i β ) is given by 7), is a feasible solution of the linesearch problem 9). We will show that z i β ) corresponds to a feasible solution of the line search problem 6) for i = i. Let us define τ := β β s i ) + β ) 2. 20) Consider the linesearch problem 6) corresponding to i = i. By 2), ) τ w i τ ) = τ s u i i ) + u, β = β s i ) + u i β ) + 2 β ) 2 β s i ) + u, β ) 2 = z i β ), where we used 20) in the penultimate line and 9) in the last one. Since z i β ) n, it follows that τ [τ i l, τ i u ], where τ i l Conversely, let ˆτ [τl i, τu i ]. By 2) and 4), ) ˆτsi ) w i ˆτ) = ˆτ u i + u and τ i u are given by 4). This implies that ν i g i. is a feasible solution of the linesearch problem 6) corresponding to i = i. Let us define ˆβ := ˆτ) 2 + ˆτ ˆτ si ) ) ). 2 /2 2) 8

9 Then, ˆβ) 2 = ˆτ) 2 ) 2 ˆτ) 2 + ˆτ si ) /2 = ˆτs i ) ) ) ˆτ), 2 /2 2 + ˆτ si ) where we used the fact that ˆτ s i ) 0 since otherwise w i ˆτ) will necessarily have a negative component by 2). It follows that z i ˆβ) = = ˆβs i ) + ˆβ) 2 ˆτs i ) + = w i ˆτ). ˆτsi ) ) ) ˆβ u i + ˆβ) 2 u, ˆτu i + ˆτsi ) ) u ), Since w i ˆτ) n, it follows that ˆµ [l i, u i ], which implies that g i ν i. Since i is arbitrary, we have g i = ν i, i = 2, 3,..., n. Therefore, g = ν, which implies that / ν ) = / g ). Note that each linesearch problem 6) has a quadratic objective function and is therefore considerably simpler compared to its counterpart 9) required for the computation of ). Furthermore, this alternative characterization leads to a simple, closed-form expression for regular graphs, which is the topic of the next section. 4 Regular Graphs A graph G = V, E) is said to be k-regular if each vertex in V has exactly k neighbors. In this section, we turn our attention into the special class of regular graphs. We first establish certain properties of the lower bounds on such graphs. Then, we present a closed-form expression of the lower bound ) on this class of graphs using the simpler characterization outlined in Section 3. First, we establish that the lower bounds 3) and 6) coincide on this class of graphs. 9

10 Lemma 4. Let G = V, E) be a connected, k-regular graph with V = n vertices. Then, each of the lower bounds 3) and 6) is equal to n/n k). Proof. Since 2m = nk, the lower bound 3) is given by n 2 /n 2 2m) = n 2 /n 2 nk) = n/n k). Let us consider the lower bound 6). Since G is a regular graph of degree k, we have A G e = ke, which implies that k R is the Perron root with the corresponding Perron eigenvector u = / n)e. Since = e T u = n, it follows that the lower bound 6) is given by s 2 /s2 λ ) = n/n k), which establishes the assertion. Next, we establish that each of the three lower bounds 3), 6), and ) or, equivalently 8)) coincides with the clique number ωg) on connected, regular, complete multipartite graphs. Proposition 4. Let G = V, E) be a connected, k-regular, complete multipartite graph with V = n. Then, each of the three lower bounds 3), 6), and ) or, equivalently 8)) coincides with ωg). Proof. Note that the vertices of G can be partitioned into t subsets such that each subset contains exactly n/t mutually nonadjacent vertices. Since G is a k-regular, complete multipartite graph, we have k = t )n/t). Clearly, ωg) = t since each subset can contribute at most one vertex to any clique and there exists a clique of size t. By Lemma 4., the lower bounds 3) and 6) are equal to n/n k) = n/n t )n/t)) = t = ωg). Since G is a regular, connected, complete multipartite graph, it follows from [3, Proposition 3] that the lower bounds ) and 6) agree, which completes the proof. 4. Closed-Form Expression of Budinich s Lower Bound In this section, we present a simpler, closed-form expression of the lower bound ) for regular graphs relying on the alternative, simpler characterization of this lower bound given by 8). 0

11 Proposition 4.2 Let G = V, E) be a connected, k-regular graph with V = n. Let λ λ 2... λ n denote the eigenvalues of A G with the corresponding eigenvectors u, u 2,..., u n, where u > 0, and let P = {i {2, 3,..., n} : λ i > 0}. Then, the lower bound ) or, equivalently 8)) is given by { } n ωg) max n k, max i P k τ, 22) n i) 2 λ i where τ i := max { min min j:u i j <0 n u i j, j:u i j >0 nu i j }, i P. 23) Proof. Note that P = if and only if G is a complete, regular multipartite graph by Theorem 2.. In this case, the lower bound 8) is given by n/n k) and coincides with the clique number ωg) by Proposition 4.. Suppose that G is not a complete multipartite graph. Since G is a regular graph, it follows that u = / n)e is the Perron eigenvector of A G corresponding to the Perron root λ = k with = e T u = n. Therefore, s i = e T u i = 0, i = 2, 3,..., n, which implies that d i = u i, i = 2, 3,...,n by 5). By 4), τl i = max j:u i j >0 nu i j = min, τ i j:u i j >0 nu i u = min i = 2, 3,..., n. j j:u i j <0 n u i j, Therefore, we have, by 6), that ) k ν i = max τ [τl i,τi u ] n + τ2 )λ i, i = 2, 3,..., n. Clearly, ν i = k + τ n i) 2 λ i if i P, k n otherwise, where τ i is defined as in 23). The assertion follows from 8) and 7). We remark that Proposition 4.2 improves Wilf s lower bound for regular graphs given in [2, Theorem 3].

12 5 Computational Results Recently, using an upper bound on the smallest eigenvalue λ n of A G, Nikiforov [] established that 2m ωg) + ) n 2m 2m λ 24) n n n), with equality if and only if G is a complete regular multipartite graph. In an attempt to assess and compare the quality of the four spectral lower bounds 3), 6), 8) or, equivalently )), and 24) we evaluated each bound on each of the sixty six instances in the DIMACS collection of clique problems using MATLAB. The major work in computing each of the lower bounds 6), 8), and 24) was the computation of the eigenvalue decomposition of the adjacency matrix A G. Rather than presenting the bounds for each instance, we report our computational results in terms of several statistics since they provide much more insight about the quality of each lower bound. As shown in [3], the lower bound 8) or, equivalently )) is an improvement over each of the lower bounds 3) and 6), which is also confirmed by our computational results. The lower bound 3), which is the easiest to compute, is always the weakest one among all four lower bounds. The lower bound 6) is tighter than 3). The lower bounds 8) and 24) always outperform 3) and 6). Our results reveal that the lower bounds 8) and 24) are generally incomparable. Among the four lower bounds, the lower bound 24) was the sharpest on thirty nine instances while Budinich s bound 8) outperformed the other bounds in the remaining twenty seven instances. To assess the quality of each lower bound in further detail, we computed, for each instance, the ratio of each lower bound to the corresponding clique number. This ratio can be viewed as an approximation factor. We restricted our analysis to the fifty five instances in the DIMACS collection whose clique numbers are known. For each lower bound 3), 6), 8), and 24), we report several statistics related to these approximation ratios in Table. Table reveals that Budinich s lower bound 8) achieves the best average approximation ratio among all the lower bounds despite the fact that Nikiforov s lower bound 24) was 2

13 3) 6) 8) 24) Average Standard Deviation Maximum Minimum Table : Several Statistics About the Average Approximation Ratios the sharpest on the majority of the instances. Furthermore, while Budinich s lower bound agrees with the clique number on the DIMACS instances hamming6-2, hamming8-2, and hamming0-2, the lower bound 24) is at most within about 68% of the clique number among all instances. Figure : Distribution of Approximation Ratios The distribution of the approximation ratios for each lower bound is depicted in Figure. The horizontal axis represents the approximation ratios in ten equal intervals given by 0, 0.], 0., 0.2],..., 0.9, 0] and the vertical axis denotes the number of DIMACS in- 3

14 stances whose approximation ratio falls into the corresponding interval. Each of the four lower bounds is denoted by a different color as stated in the figure. A close examination of Figure reveals that Budinich s lower bound 8) achieves larger approximation ratios on more instances in comparison with the other lower bounds. We remark that the lower bound 8) is within 90% of the clique number on five instances whereas none of the other lower bounds achieves an approximation ratio of more than 0.7 on any of the instances. In our computational experiments, we used each of the two characterizations ) and 8) to compute Budinich s lower bound. While both bounds agree in theory, our computational results revealed that the original characterization ) always yields slightly smaller values in comparison with the simpler characterization 8) presented in this paper. This might be due to the fact that the original characterization is more prone to numerical errors. It follows that the simpler characterization seems to be more appealing also from a computational point of view. 6 Concluding Remarks In this paper, we presented a simpler, alternative characterization of a spectral lower bound on the clique number due to Budinich [3]. Our characterization leads to a closed-form expression of this lower on regular graphs. Our computational results shed light on the quality of this lower bound in comparison with other spectral lower bounds on the clique number. Given the hardness of any nontrivial approximation of the clique number, the construction of efficiently computable lower bounds may have significant implications towards the computation of clique number on larger graphs. For instance, good lower bounds can lead to a considerable reduction in the running time of search algorithms such as branch-and-bound. In the near future, we intend to continue our work on obtaining efficient upper and lower bounds by considering various tractable inner and outer approximations to the continuous formulation ) of the clique number. 4

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