We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on @math vertices, with a polynomial number of constraints, requires @math many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical formulations of common problems in operations research. In particular, it shows that for many classic vehicle routing problems and problems involving matchings, any compact mixed-integer linear description of such a problem requires a large number of integer variables. This provides a first non-trivial lower bound on the number of integer variables needed in such settings.
Kaibel and Weltge @cite_19 took a somewhat orthogonal approach in which they focus on the original variable space and assume integrality of all variables, with the goal of understanding how many linear inequalities are needed to describe a certain set of integer points. More precisely, they consider the set of integer points @math in some polyhedron @math , and then study how many facets a polytope @math needs to have such that @math . They show that even for natural polytopes that admit a compact extended formulation, like the spanning tree polytope, an exponential number of linear constraints is often necessary.
@cite_19
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Let @math X be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is @math X. This quantity, which we call the relaxation complexity of @math X, corresponds to the smallest number of linear inequalities of any integer program having @math X as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope. In addition to the material in the extended abstract Kaibel and Weltge (2014) we include omitted proofs, supporting figures, discussions about properties of coefficients in such formulations, and facts about the complexity of formulations in more general settings. We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . We show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any NP-complete combinatorial optimization problem unless NP = co-NP—a very unlikely event. We also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities. We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7 8-approximation for max-sat.