Lower bounds on the size of semidefinite programming relaxations (1411.6317v1)

Abstract: 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 $2^{n^c}$, for some constant $c > 0$. 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-3-SAT.

• The paper establishes super-polynomial lower bounds on SDP relaxation sizes, proving limitations for efficient approximations of NP-hard problems.
• It introduces novel techniques linking positive semidefinite rank and sum-of-squares hierarchies to analyze polytopes such as CORR, cut, and TSP.
• The results imply that any polynomial-sized SDP cannot exceed a 7/8-approximation for MAX 3-SAT, guiding future research in optimization.

Overview of "Lower bounds on the size of semidefinite programming relaxations"

In the paper "Lower bounds on the size of semidefinite programming relaxations," Lee, Raghavendra, and Steurer propose novel methods for establishing lower bounds on the complexity of semidefinite programming (SDP) relaxations in the context of combinatorial optimization problems. The authors focus specifically on problems such as the cut, traveling salesman (TSP), and stable set polytopes on graphs, demonstrating that these cannot be effectively approximated by SDPs of certain bounded dimensions. The work introduces significant theoretical advancements by providing the first super-polynomial lower bounds on the semidefinite extension complexity for explicit polytope families.

One central contribution of this paper is the development of techniques to prove lower bounds on the positive semidefinite rank (psd rank) of a matrix, leveraging connections with the sum-of-squares (sos) SDP hierarchy. The authors show that for maximum constraint satisfaction problems (CSPs), polynomial-sized SDPs can be as effective as degree-O(1) sum-of-squares relaxations. Consequently, they conclude that any polynomial-sized SDP relaxation cannot surpass a 7/8-approximation for the MAX 3-SAT problem.

Key Results

The most critical results include the establishment of bounds on the psd rank for various polytopes:

• For the correlation polytope CORRn, with $n$ dimensions, the authors demonstrate a lower bound of $2^{\Omega(n^{2/13})}$ on its psd rank.
• The cut and TSP polytopes also exhibit lower bounds of a similar magnitude, supporting the conclusion that these polytopes resist characterization by low-dimensional SDPs.

The analysis relies on relating arbitrary SDP formulations to sum-of-squares formulations, demonstrating that for specific CSPs, polynomial-sized SDP relaxations align in effectiveness with the sos relaxations of a constant degree.

Implications and Future Research

The implications of this work extend across both theoretical and practical aspects of optimization theory and computational complexity. The presented bounds provide fundamental insights into the limitations of SDP relaxations, particularly in approximating NP-hard problems. These insights question the potential of small-sized SDPs in handling specific classes of combinatorial problems more effectively than previously thought possible.

The results urge further exploration into alternative relaxation hierarchies and their respective boundaries, heralding a deeper investigation into the utility of sum-of-squares methods in broader contexts. Future research directions may include refining these bounds or extending them to other combinatorial and optimization problems, potentially related to quantum computing or machine learning applications.

Moreover, the paper's methodologies and findings open avenues for exploring LP (Linear Programming) relaxations through analogous techniques, as the separation complexity constraints these approximations face could also manifest in other common optimization techniques. Exploring whether these SDP constraints indicate broader phenomena inherent in representing NP-hard problems within polyacceptable structures is a compelling open question.

In conclusion, the paper by Lee et al. highlights the complexities and limitations attached to sd-theoretical constructions and underscores an ongoing challenge in achieving efficient approximation algorithms for inherently difficult optimization problems.