Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut

Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively

Faster SDP hierarchy solvers for local rounding algorithms

Convex relaxations based on different hierarchies of linear/semi-definite programs have been used recently to devise approximation algorithms for various optimization problems. The approximation guarantee of these algorithms improves with the number of {\em rounds} $r$ in the hierarchy, though the complexity of solving (or even writing down the solution for) the $r$'th level program grows as $n^{\Omega(r)}$ where $n$ is the input size. In this work, we observe that many of these algorithms are based on {\em local} rounding procedures that only use a small part of the SDP solution (of size $n^{O(1)} 2^{O(r)}$ instead of $n^{\Omega(r)}$). We give an algorithm to find the requisite portion in time polynomial in its size. The challenge in achieving this is that the required portion of the solution is not fixed a priori but depends on other parts of the solution, sometimes in a complicated iterative manner. Our solver leads to $n^{O(1)} 2^{O(r)}$ time algorithms to obtain the same guarantees in many cases as the earlier $n^{O(r)}$ time algorithms based on $r$ rounds of the Lasserre hierarchy. In particular, guarantees based on $O(\log n)$ rounds can be realized in polynomial time. We develop and describe our algorithm in a fairly general abstract framework. The main technical tool in our work, which might be of independent interest in convex optimization, is an efficient ellipsoid algorithm based separation oracle for convex programs that can output a {\em certificate of infeasibility with restricted support}. This is used in a recursive manner to find a sequence of consistent points in nested convex bodies that "fools" local rounding algorithms.Comment: 30 pages, 8 figure

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

On Quadratic Programming with a Ratio Objective

Quadratic Programming (QP) is the well-studied problem of maximizing over {-1,1} values the quadratic form \sum_{i \ne j} a_{ij} x_i x_j. QP captures many known combinatorial optimization problems, and assuming the unique games conjecture, semidefinite programming techniques give optimal approximation algorithms. We extend this body of work by initiating the study of Quadratic Programming problems where the variables take values in the domain {-1,0,1}. The specific problems we study are QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2}, and Normalized QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum d_i x_i^2}, where d_i = \sum_j |a_{ij}| We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an $\tilde{O}(n^{1/3})$ algorithm for QP-ratio. We also obtain an $\tilde{O}(n^{1/4})$ approximation for bipartite graphs, and better algorithms for special cases. As with other problems with ratio objectives (e.g. uniform sparsest cut), it seems difficult to obtain inapproximability results based on P!=NP. We give two results that indicate that QP-Ratio is hard to approximate to within any constant factor. We also give a natural distribution on instances of QP-Ratio for which an n^\epsilon approximation (for \epsilon roughly 1/10) seems out of reach of current techniques

The power of sum-of-squares for detecting hidden structures

We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: the twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables. We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of roughly degree d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program. Using related ideas on SoS algorithms and low-degree matrix polynomials (and inspired by recent work on SoS and the planted clique problem by Barak et al.), we prove new nearly-tight SoS lower bounds for the tensor and sparse principal component analysis problems. Our lower bounds for sparse principal component analysis are the first to suggest that going beyond existing algorithms for this problem may require sub-exponential time

Strongly Refuting Random CSPs Below the Spectral Threshold

Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with $n$ variables and $m$ clauses, there is a value of $m = \Omega(n)$ beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when $m/n = \omega(1)$). Intuitively, strong refutation should become easier as the clause density $m/n$ grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as $k$-SAT and $k$-XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, $m/n \ge \widetilde O(n^{k/2-1})$, and the clause density at which instances become unsatisfiable with high probability, $m/n = \omega (1)$. In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random $k$-XOR instances with clause density $m/n \ge \widetilde O(n^{(k/2-1)(1-\delta)})$ in time $\exp(\widetilde O(n^{\delta}))$ or in $\widetilde O(n^{\delta})$ rounds of the sum-of-squares hierarchy, for any $\delta \in [0,1)$ and any integer $k \ge 3$. Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at $\delta = 0$, and brute-force refutation at the satisfiability threshold when $\delta = 1$. We also leverage our $k$-XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors