Computing the Stability Number of a Graph Via Linear and Semidefinite Programming

Abstract

We study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph. Our work is based on, and refines de Klerk and Pasechnik's approach to approximating the stability number via copositive programming (SIAM J. Optim. 12 (2002), 875-892). We provide a closed-form expression for the values computed by the linear programming approximations. We also show that the exact value of the stability number α(G) is attained by the semidefinite approximation of order α(G) − 1 as long as α(G) ≤ 6. Our results reveal some sharp differences between the linear and the semidefinite approximations. For instance, the value of the linear programming approximation of any order is strictly larger than α(G) whenever α(G) > 1.

Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

“…Peña et al [17] consider the set Q (r) n consisting of the matrices M for which such a decomposition (51) exists involving only the two highest order terms with |β| = r + 2, r. Therefore, Q (r) n is a subcone of the cone K (r) n with equality Q (r)…”

Copositive optimization – Recent developments and applications

“…Higher-order approximation alternatives, with a particular emphasis on SDPbased bounds on the clique number can be found in [48,109,86]. Similar copositive optimization approaches, among many others, were employed to obtain bounds on the (fractional) chromatic number of a graph [56].…”

“…Copositive approximation hierarchies [108,86,18,109,67,126,53] start with the zero-order approximation K (0) = P + N whose dual cone is the above discussed P ∩ N , and consist of an increasing sequence K (r) of cones satisfying cl ∪ r≥0 K (r) = C * . For instance, a higher-order approximation due to [108] uses squaring the variables to get rid of sign constraints: S ∈ C * if and only if y ⊤ Sy ≥ 0 for all y such that y i = x 2 i for some x ∈ R n , and this is guaranteed if the n-variable polynomial of degree 2(r + 2) in x,…”

Copositive Programming – a Survey

“…Refining these approaches, Peña et al [48] derive yet another hierarchy of cones approximating C. Adopting standard multiindex notation, where for a given multiindex β ∈ N n we have |β| := β 1 + · · · + β n and x β := x β1 1 · · · x βn n , they define the following set of polynomials…”