arXiv:1403.0529v1 [cs.CC] 3 Mar 2014 Limits to the scope of applicability of extended formulations for LP models of combinatorial optimization problems Moustapha Diaby OPIM Department; University of Connecticut; Storrs, CT 06268 moustapha.diaby@business.uconn.edu Mark H. Karwan Department of Industrial and Systems Engineering; SUNY at Buffalo; Amherst, NY 14260 mkarwan@buffalo.edu Abstract: We show that new definitions of the notion of “projection” on which some of the recent “extended formulations” works (such as Kaibel (2011); Fiorini et al. (2011; 2012); Kaibel and Walter (2013); Kaibel and Weltge (2013) for example) have been based can cause those works to over-reach in their conclusions in relating polytopes to one another when the sets of the descriptive variables for those polytopes are disjoint. Keywords: Linear Programming; Combinatorial Optimization; Computational Complexity, Extended Formulations. 1 Background definitions Definition 1 (“Standard EF Definition” (Yannakakis (1991); Conforti et al. (2010; 2013)) ) An “extended formulation” for a polytope X ⊆ R p is a polyhedron U = {(x, w) ∈ R p+q : Gx + Hw ≤ g} the projection, ϕ x (U ) := {x ∈ R p :(∃w ∈ R q :(x, w) ∈ U )}, of which onto x-space is equal to X (where G ∈ R m×p ,H ∈ R m×q , and g ∈ R m ). Definition 2 (“Alternate EF Definition #1” (Kaibel (2011); Fiorini et al. (2011; 2012)) ) A polyhedron U = {(x, w) ∈ R p+q : Gx + Hw ≤ g} is an “extended formulation” of a polytope X ⊆ R p if there exists a linear map π : R p+q −→ R p such that X is the image of U under π (i.e., X = π(U ); where G ∈ R m×p , H ∈ R m×q , and g ∈ R m ). Kaibel (2011), Kaibel and Walter (2013), and Kaibel and Weltge (2013) refer to π as a “projection.” Definition 3 (“Alternate EF Definition #2” (Fiorini et al. (2012)) ) An “extended formu- lation” of a polytope X ⊆ R p is a linear system U = {(x, w) ∈ R p+q : Gx + Hw ≤ g} such that x ∈ X if and only if there exists w ∈ R q such that (x, w) ∈ U. (In other words, U is an EF of X if (x ∈ X ⇐⇒ (∃ w ∈ R q :(x, w) ∈ U ))) (where G ∈ R m×p ,H ∈ R m×q , and g ∈ R m ). Remark 4 The purpose of this note is to point out that the scope of applicability of EF work based on the above definitions is limited to cases where U cannot be equivalent reformulated in terms of the w variables only. For simplicity of exposition, without loss of generality, we will say that G = 0 in the above definitions iff there exists a description of U which is in terms of the w-variables only and has the same or smaller complexity order of size. Or, equivalently, without loss of generality, we will say that G = 0 in the above definitions iff the x- and w-variables are 1