Consider a quadratic function q : Rn -> R given by: q(x) = xtQx, with Q E Rn×n. An unconstrained (−1, 1)-quadratic optimization problem can be expressed as follows: (QP) Z* = min{q(x) | x E {−1, 1}*}, where {−1, 1}n denotes the set of n-dimensional vectors with entries either equal to 1 or −1.We consider here that the matrix Q is symmetric and given by its spectrum, i.e. the set of its eigenvalues and associated unit pairwise orthogonal eigenvectors. Problem (QP) is a classical combinatorial optimization problem with many applications, e.g. in statistical physics and circuit design [2; 8; 10]. It is well-known that any (0,1)-quadratic problem expressed as: min{xt Ax + ctx | x E {0, 1}n}, A E Rn×n, c E Rn, can be formulated in the form of problem (QP) and conversely [9; 4]. The contribution of this work is 3-fold: (i) We slightly extend the known polynomially solvable cases of (QP) to when the matrix Q has fixed rank and the number of positive diagonal entries is O(log(n)). (ii) We introduce a new (to our knowledge) polynomial-time algorithmfor solving problem (QP) when it corresponds to such a polynomially solvable case. (iii) Preliminary experiments indicate that the proposed method may be computationally efficient