We propose an analytical framework for studying parallel repetition, a basic product operation for one-round twoplayer games. In this framework, we consider a relaxation of the value of projection games. We show that this relaxation is multiplicative with respect to parallel repetition and that it provides a good approximation to the game value. Based on this relaxation, we prove the following improved parallel repetition bound: For every projection game G with value at most ρ, the k-fold parallel repetition G ⊗k has value at most val(G ⊗k )This statement implies a parallel repetition bound for projection games with low value ρ. Previously, it was not known whether parallel repetition decreases the value of such games. This result allows us to show that approximating set cover to within factor (1 − ε) ln n is NP-hard for every ε > 0, strengthening Feige's quasi-NP-hardness and also building on previous work by Moshkovitz and Raz.In this framework, we also show improved bounds for few parallel repetitions of projection games, showing that Raz's counterexample to strong parallel repetition is tight even for a small number of repetitions.Finally, we also give a short proof for the NP-hardness of label cover(1, δ) for all δ > 0, starting from the basic PCP theorem.

The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a d-regular graph, the edge expansion/conductance Φ(S) of a subset S ⊆ V is defined as Φ(S) = |E(S,V \S)| d|S| . Approximating the conductance of small linear sized sets (size δn) is a natural optimization question that is a variant of the well-studied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1 − ε), and close to 0 expansion.In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following:-We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs.This is the first non-trivial "reverse" reduction from a natural optimization problem to Unique Games.-Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UG-hard to approximate small set expansion.-On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].

Subexponential time approximation algorithms are presented for the U nique G ames and S mall -S et E xpansion problems. Specifically, for some absolute constant c , the following two algorithms are presented. (1) An exp( kn ϵ )-time algorithm that, given as input a k -alphabet unique game on n variables that has an assignment satisfying 1-ϵ c fraction of its constraints, outputs an assignment satisfying 1-ϵ fraction of the constraints. (2) An exp( n ϵ /δ)-time algorithm that, given as input an n -vertex regular graph that has a set S of δ n vertices with edge expansion at most ϵ c , outputs a set S' of at most δ n vertices with edge expansion at most ϵ. subexponential algorithm is also presented with improved approximation to M ax C ut , S parsest C ut , and V ertex C over on some interesting subclasses of instances. These instances are graphs with low threshold rank , an interesting new graph parameter highlighted by this work. Khot's Unique Games Conjecture (UGC) states that it is NP -hard to achieve approximation guarantees such as ours for U nique G ames . While the results here stop short of refuting the UGC, they do suggest that U nique G ames are significantly easier than NP -hard problems such as M ax 3-S at , M ax 3- Lin , L abel C over , and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio. Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every ϵ >0 and every regular n -vertex graph G , by changing at most δ fraction of G 's edges, one can break G into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most n ϵ eigenvalues larger than 1-η, where η depends polynomially on ϵ. The subexponential algorithm combines this decomposition with previous algorithms for U nique G ames on graphs with few large eigenvalues [Kolla and Tulsiani 2007; Kolla 2010].

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 ε worse than the relaxation's value (normalized to lie in [0, 1]), as long as *

We study the computational complexity of approximating the 2-toq norm of linear operators (defined as A 2→q = max v 0 Av| q / v 2 ) for q > 2, as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following:1. For any constant even integer q 4, a graph G is a smallset expander if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2 → q norm. As a corollary, a good approximation to the 2 → q norm will refute the Small-Set Expansion Conjecture -a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n 2/q ) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set Expansion.2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2 → 4 norm of the projector to low degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique Games considered by prior works. This improves on the previous upper bound of exp(log O(1) n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require ω(1) rounds.3. We show reductions between computing the 2 → 4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2 → 4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2 → 4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp( √ n poly log(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2 → 4 norm.

We introduce a method for proving lower bounds on the ecacy of semide nite 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 δ , for some constant δ > 0. This result yields the rst super-polynomial lower bounds on the semide nite extension complexity of any explicit family of polytopes.Our results follow from a general technique for proving lower bounds on the positive semide nite 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) sumof-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for 3-.

The Small-Set Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the Unique Games Conjecture (Khot, STOC 2002). In particular, the Small-Set Expansion Hypothesis implies the Unique Games Conjecture (Raghavendra, Steurer, STOC 2010).Our main result is that the Small-Set Expansion Hypothesis is in fact equivalent to a variant of the Unique Games Conjecture. More precisely, the hypothesis is equivalent to the Unique Games Conjecture restricted to instance with a fairly mild condition on the expansion of small sets. Alongside, we obtain the first strong hardness of approximation results for the Balanced Separator and Minimum Linear Arrangement problems. Before, no such hardness was known for these problems even assuming the Unique Games Conjecture.These results not only establish the Small-Set Expansion Hypothesis as a natural unifying hypothesis that implies the Unique Games Conjecture, all its consequences and, in addition, hardness results for other problems like Balanced Separator and Minimum Linear Arrangement, but our results also show that the Small-Set Expansion Hypothesis problem lies at the combinatorial heart of the Unique Games Conjecture.The key technical ingredient is a new way of exploiting the structure of the Unique Games instances obtained from the Small-Set Expansion Hypothesis via (Raghavendra, Steurer, 2010). This additional structure allows us to modify standard reductions in a way that essentially destroys their local-gadget nature. Using this modification, we can argue about the expansion in the graphs produced by the reduction without relying on expansion properties of the underlying Unique Games instance (which would be impossible for a local-gadget reduction).

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