We prove that any extended formulation that approximates the matching polytope on @math -vertex graphs up to a factor of @math for any @math must have at least @math defining inequalities where @math is an absolute constant. This is tight as exhibited by the @math approximating linear program obtained by dropping the odd set constraints of size larger than @math from the description of the matching polytope. Previously, a tight lower bound of @math was only known for @math [Rothvoss, STOC '14; Braun and Pokutta, IEEE Trans. Information Theory '15] whereas for @math , the best lower bound was @math [Rothvoss, STOC '14]. The key new ingredient in our proof is a close connection to the non-negative rank of a lopsided version of the unique disjointness matrix.
In a parallel line of work, Chan, Lee, Raghavendra and Steurer @cite_25 proved lower bounds on the size of linear programs for constraint satisfaction problems by relating it to known Sherali-Adams integrality gaps using techniques from Fourier analysis and convex optimization. For instance, they proved that any polynomial sized LP for Max-Cut on @math -node graphs cannot beat the trivial approximation factor of @math . Recently, Kothari, Meka and Raghavendra @cite_19 improved their lower bounds to @math for some constant @math . Lee, Raghavendra and Steurer @cite_2 proved that similar situation arises for SDP relaxations: when it comes to maximum constraint satisfaction problems, SDPs of size polynomial size cannot perform much better than sums of squares relaxations of constant degree. For instance, this implies that for @math - @math - @math , no SDP of polynomial size can beat the trivial approximation factor of @math .
@cite_19 @cite_25 @cite_2
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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. We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy. In particular, any polynomial-sized linear program for Max Cut has an integrality gap of 1 2 and any such linear program for Max 3-Sat has an integrality gap of 7 8. We show that for constraint satisfaction problems (CSPs), sub-exponential size linear programming relaxations are as powerful as nΩ(1)-rounds of the Sherali-Adams linear programming hierarchy. As a corollary, we obtain sub-exponential size lower bounds for linear programming relaxations that beat random guessing for many CSPs such as MAX-CUT and MAX-3SAT. This is a nearly-exponential improvement over previous results; previously, the best known lower bounds were quasi-polynomial in n (Chan, Lee, Raghavendra, Steurer 2013). Our bounds are obtained by exploiting and extending the recent progress in communication complexity for "lifting" query lower bounds to communication problems. The main ingredient in our results is a new structural result on "high-entropy rectangles" that may of independent interest in communication complexity. Arora, Rao and Vazirani [2] showed that the standard semi-definite programming (SDP) relaxation of the Sparsest Cut problem with the triangle inequality constraints has an integrality gap of O(√log n). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the ARV-Conjecture) by constructing an Ω(log log n) integrality gap instance. Khot and Vishnoi [16] had earlier disproved the non-uniform version of the ARV-Conjecture.A simple "stretching" of the integrality gap instance for the Sparsest Cut problem serves as an Ω(log log n) integrality gap instance for the SDP relaxation of the Minimum Linear Arrangement problem. This SDP relaxation was considered in [6, 11], where it was shown that its integrality gap is bounded from above by O(√log n log log n).