New Lower Bounds on the Stability Number of a Graph

New Lower Bounds on the Stability Number of a Graph E. Alper Yıldırım June 27, 2007 Abstract Given a simple, undirected graph G, Motzkin and Straus [Canadian Journal of Mathematics, 17 (1965), 533 540] established that the reciprocal of the stability number of G (the size of the maximum stable set of G) is given by the minimum value of a certain quadratic function over the unit simplex. We propose two new lower bounds on the stability number of G based on this formulation. The first lower bound is obtained by minimizing the same objective function over the largest inscribed ball in the unit simplex. Using the fact that quadratic optimization over a full-dimensional ball admits a tight semidefinite programming relaxation, our lower bound can be computed to within any arbitrary precision in polynomial time. For regular graphs, we establish that this lower bound has a closed form solution and that it is tighter than some other existing lower bounds. The second lower bound improves upon the first lower bound and is obtained by a further refinement of the optimal solution that yields the first bound. We evaluate the new bounds and compare them with several other known lower bounds on the DIMACS collection of clique problems. Our computational results reveal that especially the improved lower bound is tighter than all other lower bounds on the majority of the instances. Key words: Maximum stable set, maximum clique, stability number, clique number, semidefinite programming. Department of Industrial Engineering, Bilkent University, 06800 Bilkent, Ankara, Turkey (yildirim@bilkent.edu.tr) 1

AMS Subject Classifications: 90C35, 90C22, 65K05, 90C20 1 Introduction Let G = (V, E) be a simple, undirected graph with a vertex set V = {1, 2,..., n} and an edge set E consisting of m edges, where each edge is identified with an unordered pair of its end vertices. A pair of vertices in V is said to be adjacent if they are connected by an edge in E. 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 α(g). A clique C V is a set of mutually adjacent vertices. Similarly, the clique number of G, denoted by ω(g), is the size of the maximum clique in G. For a graph G = (V, E), the complement graph G = (V, E) is obtained from G by removing the edges of G and connecting each pair of nonadjacent vertices of G by an edge. Clearly, S V is a stable set of G if and only if S is a clique of G. It follows that α(g) = ω(g). It is well-known that computing the stability number (equivalently, the clique number) of a graph is in general an NP-hard problem. The recent survey paper by Bomze at. al. [1] provides an account of the fairly rich literature including applications, formulations, exact algorithms, heuristics, and bounds and estimates. In fact, it is not only difficult to compute the exact stability number but no efficient algorithm can compute a good approximation to it. Hastad [5] proved that the stability number cannot be approximated within a factor of n 1/2 ɛ for any ɛ > 0 unless P = NP. Under a slightly stronger complexity assumption, the factor can be improved to n 1 ɛ. However, the stability number can be computed in polynomial time for certain classes of graphs such as perfect graphs and t-perfect graphs [4]. In this paper, we focus on computing lower bounds on the stability number of a given graph G. Similarly to the other known lower bounds [12, 2], our bounds rely on the continuous formulation of Motzkin and Straus [7], who established that the reciprocal of the stability number of a graph is given by the minimum value of a certain, usually nonconvex quadratic function over the unit simplex. We consider minimizing the same quadratic 2

objective function over the largest ball inscribed in the unit simplex. It is now well-known that quadratic optimization over a full-dimensional ball can be computed in polynomial time using a tight semidefinite programming relaxation (see, e.g., [9, 8, 10]). Since the optimal value of the latter optimization problem is an upper bound on the reciprocal of the stability number, the reciprocal of this optimal value provides our first lower bound on the stability number. For regular graphs, we establish that this lower bound has a closed form solution. Furthermore, for this class of graphs, the new bound is strictly better than some of the other known lower bounds on the stability number. The second lower bound is obtained by a further refinement of the optimal solution of the quadratic optimization problem that gives rise to the first lower bound. More specifically, the optimal solution lies inside the largest ball inscribed in the unit simplex. A family of feasible solutions in the unit simplex is constructed by extending this optimal solution appropriately until the boundary of the unit simplex. The improved lower bound is obtained by further minimizing the quadratic objective function over this family of feasible solutions which lies on a line segment. It follows that the latter lower bound is at least large as the former one. In an attempt to compare the new lower bounds with the other known bounds, we have performed computational experiments on the DIMACS collection of clique instances. Our experiments reveal that especially the improved lower bound provides a competitive alternative to the other known bounds. 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 [7] and reviews the known lower bounds on the stability number. Section 3 presents the first lower bound and establishes several properties. The improved lower bound is discussed in Section 4. The results of the computational experiments are presented in Section 5. Section 6 concludes the paper. 3

1.1 Notation R n and S n denote the n-dimensional Euclidean space and the space of n n real symmetric matrices, respectively. For u R n, u i denotes the ith component of u and u represents its Euclidean norm. For a graph G = (V, E) with V = {1,..., n}, A G S n denotes the adjacency matrix of G. The complete graph on n vertices is denoted by K n. For X S n, we use A 0 (A 0) to indicate that A is positive semidefinite (positive definite). For X S n and Y S n, the trace inner product is denoted by X Y = n i=1 n j=1 X ijy ij. The identity matrix in S n is denoted by I 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 is 1. We use E i S n for the symmetric matrix e i (e i ) T, i = 1,..., n. The (n 1)-dimensional unit simplex in R n is denoted by n, i.e., n := {x R n : e T x = 1, x 0}. 2 Formulation and Lower Bounds Given a simple, undirected graph G = (V, E), Motzkin and Straus [7] established the following: Similarly, the clique number ω(g) satisfies 1 α(g) = min x n x T (I n + A G ) x. (1) 1 1 ω(g) = max x n x T A G x. (2) While solving (1) or (2) is in general NP-hard, each one provides a continuous formulation of a combinatorial optimization problem. Furthermore, these formulations play a central role in the derivation of several known lower bounds on the stability number or the clique number of G. Using the fact that x = (1/n)e n, it follows from (1) that α(g) n2 n + 2m 1, (3) where m = E is the number of edges of G. This bound matches the stability number for complete graphs K n and their complements. 4

The other lower bounds in the literature are based on combining the formulations (1) and (2) with the spectral theory of graphs. We now collect some results about the spectra of graphs. The reader is referred to [3] for further details. Theorem 2.1 Let G = (V, E) be a graph with the adjacency matrix A G S n and let λ 1 λ 2... λ n denote the spectrum of A G. 1. n i=1 λ i = 0. 2. If G contains no edges, then λ 1 =... = λ n = 0. 3. If G contains at least one edge, then 1 λ n n 1 and λ n λ 1 1, i.e., λ n is the spectral radius of A G. 4. λ n = n 1 if and only if G is a complete graph. 5. λ n = 1 if and only if the components of G consist of graphs K 2 and possibly K 1. 6. λ 1 = 1 if and only if the components of G are complete graphs. 7. λ 1 = λ n if and only if the component of G with the largest eigenvalue λ n is a bipartite graph. 8. A G is irreducible if and only if G is connected. In this case, there exists a positive eigenvector x P R n, called the Perron eigenvector, corresponding to λ P = λ n, called the Perron root. 9. e R n is an eigenvector of A G corresponding to λ n if and only if G is a regular graph. 10. A G has exactly one positive eigenvalue if and only if the nonisolated vertices of G form a complete multipartite graph. Under the assumption that G is a connected graph, it follows from part 8 of Theorem 2.1 that A G is an irreducible, symmetric, nonnegative matrix. Therefore, let λ P > 0 and x P R n 5

denote the Perron root and the positive Perron eigenvector, respectively. Using the feasible solution (1/s P )x P n of (2), where s P := e T x P, Wilf [12] established that α(g) = ω(g) λ P s 2 P λ P + 1, (4) with equality if and only if G is a complete graph. The lower bound (4) is an improvement over the lower bound (3). More recently, Budinich [2] proposed a new lower bound that makes use of all the eigenvectors of A G. In particular, if {x 1, x 2,..., x n 1, x P } denotes the set of eigenvectors of A G, one can construct a family of unit vectors y j (µ) = µx j + 1 µ 2 x P R n, j = 1, 2,..., n 1. Then, z j (µ) := (1/e T y j (µ))y j (µ) is a feasible solution of (2) for j = 1, 2,..., n 1 as long as µ [l j, u j ], where l j 0 u j are chosen to ensure the nonnegativity of z j (µ). Let g j (µ) := z j (µ) T A G z j (µ) for j = 1,..., n 1 and let g := max j=1,...,n 1 max µ [lj,u j ] follows from (2) that g j (µ). It α(g) = ω(g) 1 1 g. (5) Unless G is a complete multipartite graph, Budinich shows that (5) strictly improves upon (4). A comparison of the three lower bounds reveals that (3) is the easiest to compute and is provably the weakest one. While (4) 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 (5). 3 A New Lower Bound In this section, we propose a new lower bound based on the continuous formulation (1). The new bound is obtained by minimizing the objective function in (1) over the largest ball inscribed in the unit simplex. We start with the characterization of the largest ball inscribed in the unit simplex. 6

Lemma 3.1 Let U := [u 1,..., u n 1 ] R n (n 1) be a matrix whose columns form an orthonormal basis for the orthogonal complement of e R n. The largest (n 1)-dimensional ball inscribed in n is given by { } B := x R n : x = 1 n e + Uw, w 1, w R n 1. (6) n(n 1) Proof. It is straightforward to show that the smallest (n 1)-dimensional ball enclosing n is given by B := { x R n : x = (1/n)e + Uz, z } n 1 n, z Rn 1. By the Löwner-John theorem [6], scaling the radius of B by a factor of 1/(n 1) yields an (n 1)-dimensional ball contained in n. The assertion follows from the fact that the scaling is tight for the unit simplex. Since B n by Lemma 3.1, we have from which it follows that 1 α(g) = min x n x T (I n + A G ) x min x B x T (I n + A G ) x =: ν, (7) α(g) 1/ν. (8) Therefore, (8) yields a lower bound on the stability number of G. where Note that ν = min x T (I n + A G ) x, x B { = min ((1/n)e + Uw) T (I n + A G ) ((1/n)e + Uw) : w 1/ } n(n 1) { = min w T Mw + 2v T w + γ : w 1/ } n(n 1), M := U T (I n + A G ) U = I n 1 + U T A G U S n 1, (9a) v := 1 n U T (I n + A G ) e = 1 n U T A G e R n 1, (9b) γ := n + 2m n 2. (9c) We first establish that the new lower bound (8) is at least as good as (3). 7

Lemma 3.2 For any graph G, we have α(g) 1 ν n2 n + 2m. (10) Proof. The first inequality follows from the definition of ν. The second inequality is a consequence of the fact that w = 0 R n 1 is a feasible solution of the problem yielding the new bound. We now discuss how to compute ν efficiently. It is now well-known that quadratic optimization over a full-dimensional ellipsoid admits a tight semidefinite programming (SDP) relaxation. The next proposition summarizes this result. Proposition 3.1 Given a graph G = (V, E), ν can be computed to within any arbitrary accuracy in polynomial time. Proof. We simply sketch the proof, which closely mimics the arguments in [13, Proposition 2.6]. Let F := M v T v S n, G := n(n 1)I n 1 0 S n. (11) γ 0 1 It follows from the results of [9, 8, 10] that quadratic optimization over a full-dimensional ball admits a tight SDP relaxation given by { ν = min w T Mw + 2v T w + γ : w 1/ } n(n 1), = min{f W : G W 0, E n W = 1, W 0}. Since any SDP problem can be solved to within any arbitrary accuracy in polynomial time using interior-point methods, the assertion follows. In addition to computing the optimal value ν, one can efficiently construct an optimal solution w R n 1 of the quadratic optimization formulation by transforming any optimal solution W S n of the SDP relaxation [10, Proposition 3]. We will use this result in the derivation of the improved lower bound in Section 4. 8

3.1 Regular Graphs Having established that the new lower bound can be computed efficiently, we now turn our attention into the special class of regular graphs. A graph G = (V, E) is said to be regular of degree k if each vertex in V has exactly k neighbors. For such graphs, it follows that 2m = nk, or k = 2m/n. Using the spectral properties of this class of graphs outlined in Theorem 2.1, we establish that the new lower bound (8) has a closed form solution. Furthermore, (8) is tighter than (3) on a large subset of this class of graphs. Proposition 3.2 Let G = (V, E) be a regular graph of degree k 1. Then, ν = n + 2m n 2 + 1 + λ 1 n(n 1) n + 2m n 2, (12) where λ 1 1 is the smallest eigenvalue of A G. This implies that α(g) 1 ν = 1 n+2m n 2 + 1+λ 1 n(n 1) n2 n + 2m. (13) Furthermore, unless the components of G are complete graphs, the new lower bound (8) is tighter than (3). Proof. Since G is a regular graph of degree k, we have A G e = ke, which implies that k R is an eigenvalue of A G with the corresponding eigenvector e R n. Therefore, the columns of the matrix U R n (n 1) in the statement of Lemma 3.1 can be chosen to consist of the remaining n 1 eigenvectors of A G. It follows from (9b) that v = (1/n)U T A G e = (k/n)u T e = 0 and M = U T (I n + A G )U = I n 1 + Σ, where Σ S n 1 is a diagonal matrix whose entries are the remaining n 1 eigenvalues of A G given by λ 1 λ 2... λ n 1. Therefore, { ν = min w T Mw + 2v T w + γ : w 1/ } n(n 1), { = min w T (I n 1 + Σ)w + γ : w 1/ } n(n 1), = 1 + λ 1 n(n 1) + n + 2m n 2, 9

where we used the fact that λ 1 1 for any graph with at least one edge (cf. part 3 of Theorem 2.1). This establishes (12) and hence (13). The last part of the assertion follows from parts 3 and 6 of Theorem 2.1. We next establish that the new lower bound is also tighter than the lower bound (4) for certain regular graphs. Proposition 3.3 Let G = (V, E) be a regular graph of degree k with k 1 such that the complement graph G is connected. Then, the lower bounds (3) and (4) coincide. If, in addition, G has at least one connected component that is not a complete graph, then each bound is strictly smaller than (8). Proof. Clearly, the complement graph G is also regular of degree n k 1. Therefore, x P = (1/ n)e R n is a Perron eigenvector of A G with the corresponding Perron root λ P = n k 1 and s P = e T x P = n (cf. parts 8 and 9 of Theorem 2.1). Therefore, the lower bound (4) is given by α(g) n k 1 n (n k 1) + 1 = n k + 1 = n2 n + 2m, where we used 2m = nk. This establishes the first part of the assertion. The second part follows from Proposition 3.2. The following result establishes that all four lower bounds coincide with the stability number on a certain class of regular graphs. Proposition 3.4 Let G = (V, E) be a regular graph of degree k with k 1 such that the complement graph G is a connected, complete multipartite graph. Then, each of the four lower bounds (3), (4), (5), and (8) coincides with α(g). Proof. By Proposition 3.3, the lower bounds (3) and (4) are equal to n 2 /(n+2m) = n/(k+1). Suppose that G has t connected components. By the hypothesis, each connected component is a complete graph on k + 1 vertices. Clearly, α(g) = n/(k + 1). Since each component 10

of G is a complete graph, it follows that the smallest eigenvalue of A G satisfies λ 1 = 1 (cf. part 6 of Theorem 2.1). By Proposition 3.2, the new lower bound (8) is also equal to n/(k + 1). Finally, since G is a regular, connected, complete multipartite graph, it follows from [2, Proposition 3] that the lower bounds (5) and (4) agree, which completes the proof. 3.2 Irregular Graphs We call a graph G irregular if it is not a regular graph. In contrast with regular graphs, a complete characterization of the spectra of such graphs does not exist. Therefore, the new lower bound (8) does not in general have a closed form solution for such graphs. Since the lower bound (5) requires the computation of the complete spectrum of A G, a complete characterization of graphs for which the new lower bound (8) is tighter than (5) is not straightforward. However, as the following simple example illustrates, there do exist graphs for which the new lower bound (8) is strictly better than (5). Example 3.1 Let G = (V, E), where V = {1, 2, 3, 4, 5}, E = {(1, 4), (2, 3), (2, 5), (3, 4), (3, 5)}, i.e., G is given by a path of length two connected to a vertex of a complete graph on three vertices. Clearly, α(g) = 2 and the lower bound (3) is equal to 25/15 = 1.6666. It is easy to verify that x = [1/4, 1/4, 0, 1/4, 1/4] T is an optimal solution of the quadratic optimization problem (1). Note that x (1/5)e = 1/ 20, which implies that x lies in the largest ball inscribed in 5 and hence is a feasible solution of the quadratic optimization problem defining ν (cf. (7)). Therefore, 1/ν = α(g) = 2, which implies that the lower bound (8) matches the stability number of G. The complement graph G is a connected, bipartite graph. The Perron root is given by λ P = 2.1358 and the Perron eigenvalue x P satisfies s P = e T x P = 2.1829. Therefore, the 11

lower bound (4) is given by α(g) 2.1358 (2.1829) 2 2.1358 + 1 = 1.8123 which is strictly smaller than α(g) = 2. The numerically computed lower bound (5) is equal to 1.9994, which implies that the lower bound (8) is the tightest among the four bounds. 4 An Improved Lower Bound In this section, we propose a new lower bound based on a refinement of the lower bound (8). Let us recall the quadratic optimization reformulation in (n 1)-dimensional space whose optimal value defines (8). ν = min {w T Mw + 2v T w + γ : w 1/ } n(n 1), w R n 1, (14) where M, v, and γ are defined as in (9). It is well-known that the optimal solution w R n 1 of (14) satisfies the following necessary and sufficient optimality conditions. (M + λ I n 1 ) w = v, (15a) λ (1/ ) n(n 1) w = 0, (15b) M + λ I n 1 0, (15c) λ 0, (15d) where λ R is the Lagrange multiplier corresponding to the inequality constraint. Let w R n 1 be an optimal solution of (14). By the reformulation in Section 3, it follows that x = (1/n)e + Uw B n, which implies that x R n is a feasible solution of the original quadratic optimization problem (1). In order to derive the improved lower bound, we first construct a family of feasible solutions of (1) given by x(θ) := (1/n)e + θuw, θ R. (16) 12

Clearly, x(1) = x n and e T x(θ) = 1 for all θ R. It follows then that x(θ) n for all θ [1, θ ], where θ := 1 min i:(uw ) i <0 n (Uw ) i 1. (17) Since e T Uw = 0, θ is well-defined and has a finite value unless w = 0, in which case, we define θ = +. The new lower bound is obtained by further minimizing the objective function of (1) on this family of feasible solutions x(θ) for θ [1, θ ]. In the reformulation (14) in (n 1)- dimensional space, this is equivalent to extending the solution w, i.e., replacing w by w(θ) := θw, until x(θ) = (1/n)e + Uw(θ) hits the boundary of the unit simplex n. Obviously, the new optimal value will be at least as small as ν therefore be at least as large as (8). To this end, let us define and its reciprocal will ν(θ) := x(θ) T (I + A G )x(θ) = θ 2 (w ) T Mw + 2θv T w + γ. (18) Let The new lower bound is given by The next proposition establishes that the improved lower bound (20) can easily be computed. ν := min ν(θ) 1 θ [1,θ ] α(g). (19) 1/ν α(g). (20) Proposition 4.1 Let w R n 1 be an optimal solution of (14) and suppose that (w, λ ) satisfies the optimality conditions (15). Then, ν = ν( θ), where 1, if λ = 0, θ := arg min ν(θ) = min{θ, vt w }, if (w ) T Mw > 0 and λ > 0, θ [1,θ ] (w ) T Mw θ, otherwise. Furthermore, the new lower bound (20) is tighter than (8) in the second and third cases above unless θ = 1. 13

Proof. First, suppose that λ = 0, which implies that M 0 by (15c). Therefore, w is the unconstrained minimizer of the convex quadratic objective function of (14) by (15a). Therefore, ν = ν(1) = min θ R ν(θ), which implies that ν = ν = 1/α(G) since the unconstrained global minimizer x(1) = (1/n)e + Uw is a feasible solution of (1). Suppose now that w = 1/ n(n 1) with λ > 0. By (15a), (w ) T Mw + λ n(n 1) = vt w. Note that ν (1) = 2((w ) T Mw + v T w ) = 2(λ /(n(n 1))) < 0, which implies that ν(θ) is strictly decreasing at θ = 1. Therefore, ν < ν if θ > 1. If (w ) T Mw > 0, then ν(θ) is a convex function and its global minimizer is given by θ = (v T w )/((w ) T Mw ) > 1 since ν (1) < 0. It follows that θ = min{θ, θ} in this case. Otherwise, ν(θ) is a concave, decreasing function and θ = θ, which completes the proof. 5 Computational Results In this section, we present our computational results on the DIMACS collection of clique problems (http://mat.gsia.cmu.edu/challenge.html). Each of the five lower bounds given by (3), (4), (5), (8), and (20) was computed on the complements of each of the sixty four instances. We used MATLAB to compute each of the five bounds on each of the instances. In particular, MATLAB s several built-in functions including eigs, eig, and fminbnd were employed to compute the bounds (4) and (5). We used the semidefinite programming (SDP) formulation given in the proof of Proposition 3.1 to compute the new lower bound (8). The resulting SDP problems were solved by the MATLAB-based interior-point solver SDPT3-3.02 [11] using the default parameters. In order to compute the improved lower bound, the optimal solution of the SDP formulation was transformed into an optimal solution of (14) using [10, Proposition 3]. Tables 1 and 2 present the results of the implementation on each of the sixty four instances. The first column presents the name of the instance. Note that the computations 14

GRAPH LOWER BOUNDS Instance V E α(g) (3) (4) (5) (8) (20) MANN-a9.co 45 72 16 10.7143 10.7345 12.1349 11.0434 12.0000 MANN-a27.co 378 702 126 80.1818 80.7926 116.9893 84.9989 117.0000 MANN-a45.co 1035 1980 345 214.4595 215.7641 329.9591 225.7705 330.0000 brock200-1.co 200 5066 21 3.8715 3.9064 4.0812 3.9347 4.1816 brock200-2.co 200 10024 12 1.9755 1.9927 2.0467 1.9939 2.0585 brock200-3.co 200 7852 15 2.5151 2.5367 2.6224 2.5441 2.6410 brock200-4.co 200 6811 17 2.8939 2.9203 3.0287 2.9325 3.0914 brock400-1.co 400 20077 27 3.9454 3.9637 4.0800 3.9783 4.1616 brock400-2.co 400 20014 29 3.9577 3.9748 4.0751 3.9895 4.1762 brock400-3.co 400 20119 31 3.9372 3.9548 4.0826 3.9694 4.1809 brock400-4.co 400 20035 33 3.9535 3.9715 4.0894 3.9862 4.1859 brock800-1.co 800 112095 23 2.8446 2.8510 2.8917 2.8494 2.9276 brock800-2.co 800 111434 24 2.8614 2.8687 2.9047 2.8713 2.9426 brock800-3.co 800 112267 25 2.8402 2.8474 2.8885 2.8500 2.9289 brock800-4.co 800 111957 26 2.8481 2.8551 2.8927 2.8577 2.9470 c-fat200-1.co 200 18366 12 1.0831 1.2280 1.2515 1.0847 1.2119 c-fat200-2.co 200 16665 24 1.1930 1.2412 1.3923 1.1947 1.8396 c-fat200-5.co 200 11427 58 1.7351 1.7386 2.4777 1.7420 2.3551 c-fat500-1.co 500 120291 14 1.0370 1.1725 1.1772 1.0373 1.1559 c-fat500-2.co 500 115611 26 1.0789 1.1618 1.1895 1.0793 1.1581 cfat-500-5.co 500 101559 64 1.2278 1.2414 1.4175 1.2286 1.6620 cfat-500-10.co 500 78123 126 1.5949 1.5960 2.2247 1.5970 2.2184 hamming6-2.co 64 192 32 9.1429 9.1429 31.9739 9.2477 32.0000 hamming6-4.co 64 1312 4 1.5238 1.5238 1.9998 1.5296 2.0000 hamming8-2.co 256 1024 128 28.4444 28.4444 127.8538 28.5315 128.0000 hamming8-4.co 256 11776 16 2.7527 2.7527 4.4129 2.7568 4.4138 hamming10-2.co 1024 5120 512 93.0909 93.0909 511.2482 93.0909 93.0909 hamming10-4.co 1024 89600 40 5.8182 5.8182 11.1271 5.8182 5.8182 johnson8-2-4.co 28 168 4 2.1538 2.1538 2.2899 2.1600 2.1972 johnson8-4-4.co 70 560 14 4.1176 4.1176 4.9466 4.1282 4.2452 johnson16-2-4.co 120 1680 8 4.1379 4.1379 4.1775 4.1391 4.1596 johnson32-2-4.co 496 14880 16 8.1311 8.1311 8.1884 8.1314 8.1344 keller4.co 171 5100 11 2.8195 2.8623 3.3098 2.8714 3.3563 keller5.co 776 74710 27 4.0093 4.0387 4.5947 4.0398 4.6867 p-hat300-1.co 300 33917 8 1.3209 1.4072 1.4292 1.3367 1.4611 p-hat300-2.co 300 22922 25 1.9504 2.3021 2.6251 2.0158 2.6934 p-hat300-3.co 300 11460 36 3.8760 4.1121 4.3121 4.0104 5.0851 p-hat500-1.co 500 93181 9 1.3379 1.4175 1.4415 1.3500 1.4887 p-hat500-2.co 500 61804 36 2.0144 2.3938 2.8129 2.0690 2.9523 p-hat500-3.co 500 30950 50 4.0064 4.2617 4.4060 4.1178 5.5334 p-hat700-1.co 700 183651 11 1.3315 1.4120 1.4375 1.3417 1.4930 p-hat700-2.co 700 122922 44 1.9875 2.3795 2.9213 2.0331 2.9392 p-hat700-3.co 700 61640 62 3.9523 4.2067 4.3439 4.0443 5.4686 p-hat1000-1.co 1000 377247 10 1.3236 1.3996 1.4531 1.3318 1.4745 p-hat1000-2.co 1000 254701 46 1.9592 2.3207 2.7616 1.9951 2.8790 p-hat1000-3.co 1000 127754 66 3.8985 4.1327 4.2218 3.9710 5.2792 Table 1: Computational Results 15

GRAPH LOWER BOUNDS Instance V E α(g) (3) (4) (5) (8) (20) p-hat1500-1.co 1500 839327 12 1.3392 1.4161 1.4809 1.3461 1.4865 p-hat1500-2.co 1500 555290 65 2.0232 2.4120 2.9499 2.0549 2.9333 p-hat1500-3.co 1500 277006 94 4.0503 4.3005 4.3723 4.1140 5.8240 san200-0.7-1.co 200 5970 30 3.2949 3.3197 6.1616 3.3398 3.7456 san200-0.7-2.co 200 5970 18 3.2949 3.7622 7.8417 3.4541 6.2527 san200-0.9-1.co 200 1990 70 9.5694 9.7858 16.0700 10.0117 12.4452 san200-0.9-2.co 200 1990 60 9.5694 9.6584 13.7211 9.8477 11.5973 san200-0.9-3.co 200 1990 44 9.5694 9.6333 11.5951 9.8021 10.4834 san400-0.5-1.co 400 39900 13 1.9950 2.1007 4.5864 2.0113 2.8396 san400-0.7-1.co 400 23940 40 3.3140 3.3248 6.6866 3.3345 3.4569 san400-0.7-2.co 400 23940 30 3.3140 3.3260 6.4173 3.3360 3.5700 san400-0.7-3.co 400 23940 22 3.3140 3.3480 6.3316 3.3483 3.9547 san400-0.9-1.co 400 7980 100 9.7800 9.8212 15.3951 9.9144 10.6407 san1000.co 1000 249000 15 2.0040 2.1069 5.1396 2.0126 2.8358 sanr200-0.7.co 200 6032 18 3.2616 3.2908 3.4253 3.3087 3.4826 sanr200-0.9.co 200 2037 42 9.3589 9.4562 10.0551 9.6411 10.8711 sanr400-0.5.co 400 39816 13 1.9992 2.0102 2.0438 2.0098 2.0583 sanr400-0.7.co 400 23931 21 3.3152 3.3328 3.4158 3.3414 3.5107 Table 2: Computational Results (continued) were performed on the complement graphs. The second group of columns reports the number of nodes V, the number of edges E, and the size of the maximum stable set α(g). The values with an asterisk correspond to the best known lower bounds on α(g). The lower bounds (3), (4), (5), (8), and (20) computed for each instance are presented in the third group of columns. The tightest lower bound for each instance is highlighted. A close examination of Tables 1 and 2 reveals that the tightest lower bound is either given by (5) or by (20). In particular, the new improved lower bound (20) was the tightest among all five lower bounds on thirty nine of the sixty four instances. The lower bound (5) was the tightest on the remaining twenty five instances. These results indicate that the improved lower bound (20) provides a competitive alternative to the other lower bounds. The lower bound (3), which is the easiest to compute, is always the weakest one among all five lower bounds. The lower bounds (4) and (8) usually yield similar values which are tighter than (3). As expected, the lower bounds (5) and (20) always outperform (4) and (8), respectively. Finally, while each of the lower bounds is usually significantly smaller than the stability 16

number on most of the instances, our computational results indicate that especially the lower bounds (5) and (20) either match the stability number or provide a very good approximation to it on some of the instances. These results indicate that the progress on lower bounds may have significant implications for the computation of the stability number, which does not admit any efficient, nontrivial approximation. 6 Concluding Remarks In this paper, we proposed two lower bounds on the stability number of a given graph G. Both of our bounds rely on the continuous formulation (1) and can be efficiently computed. Our computational results indicate that especially the improved lower bound (20) has a promising performance in comparison with the other lower bounds. Given the hardness of even approximating the stability number, the construction of improved bounds may have significant implications since the maximum stable set problem has many applications in diverse areas. In the near future, we intend to continue our work on obtaining upper and lower bounds by considering various tractable inner and outer approximations to the continuous formulation (1) of the stability number. References [1] I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization (Supplement Volume A), pages 1 74. Kluwer Academic, Boston, Massachusetts, U.S.A., 1999. [2] M. Budinich. Exact bounds on the order of the maximum clique of a graph. Discrete Applied Mathematics, 127:535 543, 2003. [3] D. M. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs. Pure and Applied Mathematics. Academic Press, Inc., New York, 1979. 17

[4] M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer, New York, 1988. [5] J. Hastad. Clique is hard to approximate within n 1 ɛ. Acta Mathematica, 182(1):105 142, 1999. [6] F. John. Extremum problems with inequalities as subsidiary conditions. In Studies and Essays, presented to R. Courant on his 60th birthday January 8, 1948, pages 187 204. Interscience, New York, 1948. Reprinted in: Fritz John, Collected Papers Volume 2 (J. Moser, ed), Birkhäuser, Boston, 1985, pp. 543 560. [7] T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of Turán. Canadian Journal of Mathematics, 17:533 540, 1965. [8] F. Rendl and H. Wolkowicz. A semidefinite framework for trust region subproblems with applications to large scale minimization. Mathematical Programming, 77(2):273 299, 1997. [9] R. J. Stern and H. Wolkowicz. Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM Journal on Optimization, 5(2):286 313, 1995. [10] J. F. Sturm and S. Z. Zhang. On cones of nonnegative quadratic functions. Mathematics of Operations Research, 28(2):246 267, 2003. [11] R. H. Tütüncü, K. C. Toh, and M. J. Todd. Solving semidefinite-quadratic-linear programs using SDPT3. Mathematical Programming, 95:189 217, 2003. [12] H. S. Wilf. Spectral bounds for the clique and independence numbers of graphs. Journal of Combinatorial Theory, Series B, 40:113 117, 1986. [13] E. A. Yıldırım. On the minimum volume covering ellipsoid of ellipsoids. SIAM Journal on Optimization, 17(3):621 641, 2006. 18