Limitations of semidefinite programs for separable states and entangled games

Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $\omega(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al. These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published versio

Subsampling Mathematical Relaxations and Average-case Complexity

We initiate a study of when the value of mathematical relaxations such as linear and semidefinite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let $C$ be a family of CSPs such as 3SAT, Max-Cut, etc., and let $\Pi$ be a relaxation for $C$, in the sense that for every instance $P\in C$, $\Pi(P)$ is an upper bound the maximum fraction of satisfiable constraints of $P$. Loosely speaking, we say that subsampling holds for $C$ and $\Pi$ if for every sufficiently dense instance $P \in C$ and every $\epsilon>0$, if we let $P'$ be the instance obtained by restricting $P$ to a sufficiently large constant number of variables, then $\Pi(P') \in (1\pm \epsilon)\Pi(P)$. We say that weak subsampling holds if the above guarantee is replaced with $\Pi(P')=1-\Theta(\gamma)$ whenever $\Pi(P)=1-\gamma$. We show: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the relaxation considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every CSP under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semidefinite programs. Also there are unique CSPs for which subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semidefinite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value $1-\gamma$ have a cut value at most $1-\gamma/10$.Comment: Includes several more general results that subsume the previous version of the paper

Lift & Project Systems Performing on the Partial-Vertex-Cover Polytope

We study integrality gap (IG) lower bounds on strong LP and SDP relaxations derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover problem in which only t edges need to be covered. t-PVC admits a 2-approximation using various algorithmic techniques, all relying on a natural LP relaxation. Starting from this LP relaxation, our main results assert that for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P systems that have been used for positive algorithmic results (but the Lasserre hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices of the input graph. Our lower bounds are nearly tight. Our results show that restricted yet powerful models of computation derived by many L&P systems fail to witness c-approximate solutions to t-PVC for any constant c, and for t = O(n). This is one of the very few known examples of an intractable combinatorial optimization problem for which LP-based algorithms induce a constant approximation ratio, still lift-and-project LP and SDP tightenings of the same LP have unbounded IGs. We also show that the SDP that has given the best algorithm known for t-PVC has integrality gap n/t on instances that can be solved by the level-1 LP relaxation derived by the LS system. This constitutes another rare phenomenon where (even in specific instances) a static LP outperforms an SDP that has been used for the best approximation guarantee for the problem at hand. Finally, one of our main contributions is that we make explicit of a new and simple methodology of constructing solutions to LP relaxations that almost trivially satisfy constraints derived by all SDP L&P systems known to be useful for algorithmic positive results (except the La system).Comment: 26 page