An effective modeling and solution approach for the generalized independent set problem

Abstract

The generalized independent set (GIS) problem was first introduced by Hochbaum and Pathria (Forest Sci 43(4), [544][545][546][547][548][549][550][551][552][553][554] 1997) and independently explored in greater detail by Hochbaum (Manage Sci 50(6), 2004). This problem, with applications in forest management and a variety of related areas, is a generalization of the classical maximum independent set problem. In this paper we highlight a natural, nonlinear formulation for the problem that is an attractive alternative to the linear model found in the literature. The effectiveness of this alternative formulation is demonstrated by computational experience on test problems of varying size and density, disclosing a dramatic reduction in the time to obtain optimal and near optimal solutions and an ability to solve much larger problems.

The unconstrained binary quadratic programming problem: a survey

“…Wang, Alidaee, Glover, and Kochenberger (2006) discuss the problem of grouping machines and parts together in a flexible manufacturing system in a manner that facilitates economies in time and cost. Kochenberger, Alidaee, Glover, and Wang (2007) discuss the use of UBQP as a tool for modeling and solving the generalized independent set (GIS) problem. In each case the original model was re-cast into the form of UBQP and successfully solved in this new form.…”

The generalized vertex cover problem and some variations

“…GVC2-HP in [15] was presented as a maximization problem with c i ≥ 0, ∀i ∈ V and q 2 ij ≤ 0, ∀(i, j) ∈ E. Clearly, this is equivalent to our definition of GVC2-HP. Kochenberger et al [21] gave a nonlinear formulation for GVC2-HP and compare the computational effectiveness of this non-linear integer programming formulation with an available linear integer programming formulation.…”

Solving the maximum vertex weight clique problem via binary quadratic programming

“…During the past decade, binary quadratic programming (BQP) has emerged as a unified model for numerous combinatorial optimization problems, such as max-cut [20,37], set partitioning [24], set packing [2], generalized independent set [19] and maximum edge weight clique [1]. A review of the additional applications and the reformulation procedures can be found in [18,21].…”