Fifty-Plus Years of Combinatorial Integer Programming

Throughout the history of integer programming, the field has been guided by research into solution approaches to combinatorial problems. We discuss some of the highlights and defining moments of this area. 1 Combinatorial integer programming Integer-programming models arise naturally in optimization problems over combinatorial structures, most notably in problems on graphs and general set systems. The translation from combinatorics to the language of integer programming is often straightforward, but the new rendering typically suggests direct lines of attack via linear programming. As an example, consider the stable-set problem in graphs. Given a graph G = (V,E) with vertices V and edges E, a stable set of G is a subset S ⊆ V such that no two vertices in S are joined by an edge. The stable-set problem is to find a maximum-cardinality stable set. To formulate this as an integer-programming (IP) problem, consider a vector of variables x = (xv : v ∈ V ) and identify a set U ⊆ V with its characteristic vector x̄, defined as x̄v = 1 if v ∈ U and x̄v = 0 otherwise. For e ∈ E write e = (u, v), where u and v are the ends of the edge. The stable-set problem is equivalent to the IP model max ∑ (xv : v ∈ V ) (1) xu + xv ≤ 1, ∀ e = (u, v) ∈ E, xv ≥ 0, ∀ v ∈ V, xv integer, ∀ v ∈ V. To express this model in matrix notation, let A denote the edge-vertex incidence matrix of G, that is, A has rows indexed by E, columns indexed by V , and for each e ∈ E and v ∈ V , entry Aev = 1 if v is an end of e and Aev = 0 otherwise. Letting 0 and 1 denote the vectors of all zeros and all ones, respectively, problem (1) can be written as max(1x : Ax ≤ 1, x ≥ 0, x integer). (2) In a similar fashion, the vertex-cover problem can be modeled as min(1x : Ax ≥ 1, x ≥ 0, x integer). (3) This later problem asks for a minimum-cardinality set C ⊆ V such that every edge in E has at least one of its ends in C. By dropping the integrality constraints on the variables, we obtain linear-programming (LP) relaxations for the IP models. From these relaxations we get the LP dual min(y 1 : yA ≥ 1 , y ≥ 0) (4)