Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Abstract

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size.We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, travelling salesman polytope or dominant of the odd-cut polytope, needs Ω(n/ log n) many integer variables, where n is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only O(n) or O(n log n) (for the traveling salesman polytope) many integer variables.Our results build upon a new decomposition technique that, for any convex set C, allows for approximating any mixed-integer description of C by the intersection of C with the union of a small number of affine subspaces.

Computational aspects of relaxation complexity: possibilities and limitations

“…Independently of which of the two cases occurs, one always has a i, j − a i,k + a i, ≥ 0. We have thus found the point y:=e j − e k + e ∈ Y that satisfies Ay ≥ 0, which is a contradiction to (7). This shows m ≥ log 2 log 2 d and concludes the proof.…”

“…Since P might have exponentially many facets, many researchers have investigated extended formulations of P, which are polyhedra Q in a higher-dimensional space that linearly project onto P. Such extended formulations might have significantly less facets; see the surveys by Conforti et al [9] and Kaibel [23] and the references therein. Alternatively, one can consider extended relaxations of X , i.e., mixed-integer programming models that allow to use auxiliary (integer) variables; see, e.g., Bader et al [4], Cevallos et al [7], or Weltge [35,Ch. 7.1].…”

Extended Formulations for Stable Set Polytopes of Graphs Without Two Disjoint Odd Cycles

“…Note that this does not follow from the main result of [1]. As noted in [5,Thm. 5.4], integer hulls of bimodular integer programs can have exponential extension complexity.…”

“…Indeed, for every vector y = (y 0 , y 1 ) ∈ σ(R(G)) there exists a vector y H (the image of (x 01 , x H ) under σ H ) with (y 0 , y H ) ∈ σ(STAB(G ′ 0 )) and (y 1 , y H ) ∈ σ(STAB(G ′ 0 )). This implies that the inequalities in (5) and (6) are satisfied. Now (4) follows since it is the sum of these two inequalities.…”

“…This would imply in particular that the stable set problem on graphs with ocp(G) ⩽ k is polynomial for every fixed k. Conforti, Fiorini, Huynh, Joret, and Weltge [7] recently proved this is true under the additional assumption that G has bounded (Euler) genus. 5 Furthermore, by itself the bimodular algorithm does not imply any linear description of the stable set polytope of graphs G with ocp(G) = 1. It turns out that for such graphs, STAB(G) may have many facets with high coefficients that do not seem to allow a "nice" combinatorial description in the original space.…”